Dynamic Efficiency and Sustainable Development
Methods to define sustainability
Content (Day 1)
Slides 14 - Renewable Resources: Fish
Introduction to Fisheries and Renewable Resources
This lecture introduces the concept of bioeconomic models, focusing specifically on fisheries as our first official bioeconomic model. When discussing earth economy modeling, we often take for granted that models linking environmental and economic factors have existed for a very long time, well before the recent explosion of research on Earth economy modeling. The bioeconomic model presented here couples information about the growth of a species, in this case fish, with an economic model to determine the profit-maximizing yet sustainable catch rate.
After working through the mathematics of fisheries, the lecture concludes with a simulation game designed to illustrate the basic concepts learned, while also applying them to previously discussed topics such as open access and commons dilemmas. This lecture is somewhat more math-heavy than others, but the equations presented are particularly useful for understanding the explicit linkage between the environment and the economy.
Bioeconomics: Linking Biology and Economics
Bioeconomics, as the name suggests, connects biological concepts to economic analysis. In this case, we build a relationship between the stock levels of fish and the subsequent growth of fish populations.
The Stock Dynamics Equation
The fundamental equation for stock dynamics states that the stock in time period T+1 is equal to the stock in period T, plus some growth function that itself is a function of the size of the stock. Mathematically, this is expressed as:
S_{T+1} = S_T + G(S)
Before examining the specific functional form of the growth function, consider what happens when a lake is completely full of fish. When there are very high numbers of individuals of any species, the population does not grow as fast because individuals are competing for limited resources. Conversely, when there is only one individual, the population also does not grow very fast. This creates a relationship described by an upside-down U-shaped curve, where at very low population levels, fish do not grow very fast simply because there are not enough breeding pairs, and at very high populations, they also do not grow as fast because something in their environment limits further growth beyond the carrying capacity.
The Squirrel and Acorn Example
The squirrel and oak tree relationship provides an excellent illustration of these population dynamics. Oak trees have evolved a characteristic where they occasionally produce a bumper crop of acorns. They do this because it means there will be so many acorns that there will not be enough squirrels to eat them all, allowing some acorns to get planted. The timing of these bumper crops is more or less random.
What follows is a predictable pattern: after a huge crop of acorns, the next year sees a huge increase in the squirrel population. However, the year after that brings massive suffering for squirrels as the population crashes due to insufficient food resources. This pattern is observable in nature, with population booms always followed by population busts.
The Logistic Growth Model
The specific functional form of the growth function G(S) is derived from biological studies of animal populations and represents the best equation for describing their growth dynamics.
The Growth Function Equation
The growth function is expressed as:
G(S) = R \cdot S \cdot \left(1 - \frac{S}{K}\right)
In this equation, R represents the intrinsic growth rate and K represents the carrying capacity.
Understanding the Intrinsic Growth Rate
The intrinsic growth rate represents the maximum growth rate under the very best conditions when nothing is slowing down population growth. If we ignore the right-hand side of the equation, it simply states that if there are S number of individuals, growth will occur at this rate. For example, if there are 100 fish and R is 0.1, then 10 new fish would be produced if there were no effects from overpopulation. This represents the fastest possible growth rate, which would lead to unlimited growth if there were no constraints.
The Carrying Capacity Effect
However, stocks cannot grow forever because they eventually hit some sort of limit. This is modeled by the term (1 - S/K), which represents how close the population is to the carrying capacity.
Consider what happens when S exactly equals K. If K is 1,000 and S is 1,000, then S/K equals 1, and 1 minus 1 equals 0. This zero gets multiplied by the intrinsic growth rate, meaning that regardless of what R or S values are, the growth will be zero. This represents the situation where all the squirrels have eaten all the acorns and there is nothing left to eat, preventing any population growth.
To the extent that S is less than K, the growth term remains positive. For example, if S were 900 and K were 1,000, the fraction would be 0.9, and 1 minus 0.9 equals 0.1. While this is still a small number, it produces positive growth because it is not multiplied by zero.
Graphical Representation of Growth Dynamics
Growth as a Function of Stock
The first key plot combines information from the growth equation to show growth as a function of stock. This can be verified by plugging the equation into a graphing calculator.
Starting with the easy case: if S is zero, there is nobody to reproduce, so the growth rate is zero. No new fish can appear where there are no fish.
Conversely, when S equals K (the carrying capacity), there is also no growth. At this point, labeled S_K, the stock is exactly at the carrying capacity. It is important to note that this graph shows growth, not total population. At the carrying capacity, there would still be fish, but there would be no extra growth since the population change rate is zero.
When the growth function is plotted for all values between zero and the carrying capacity, it reveals the logistic growth curve, a famous function that describes the growth dynamics of many physical populations. Different species may have different R and K values, but all exhibit this general shape with the two zero points discussed and positive growth for everything in between.
Maximum Sustainable Yield (MSY)
One particularly important point on this curve is where growth reaches its maximum, called the Maximum Sustainable Yield (MSY). At this stock level, growth is maximized because the population is far enough from the carrying capacity that resources are not limiting, but not so low that there are insufficient breeding pairs. The stock level at MSY is denoted S_{MSY}.
This framework sets up an important debate between two different worldviews. Environmentalists tend to favor operating a fishery at the maximum sustainable yield because it appears to be the best option with the greatest growth rate. However, economists often advocate for maximum sustainable profit instead. This represents the classic tension between environmental and economic perspectives.
Stock Progression Over Time
A companion plot shows the progression of stock over time, with stock on the vertical axis. This plot represents essentially the derivative of the stock dynamics. When stock is close to zero, the slope is zero, but as stock increases, growth begins and accelerates. The stock curve starts by curving upward.
At some point, the growth rate begins to flatten as the population crosses the maximum sustainable yield and the carrying capacity overcrowding effect takes hold. There is an inflection point where the curve transitions from being convex in one direction to convex in the other, which provides another way to identify the maximum sustainable yield.
Stable and Unstable Equilibria
The growth rate is zero at two different places on the curve, but these points differ in their stability properties.
At the upper zero point (carrying capacity), if the population is bumped down due to fishing, the next period will see it grow back toward the equilibrium because growth is positive. Perturbations away from this stock level tend to diminish over time as the population grows back. This is called a stable equilibrium.
The lower zero point (zero population) represents an unstable equilibrium. If there are zero fish, the population will stay at exactly zero. However, if fish are dropped into a lake (such as from stocking programs), bringing the population slightly above zero, it will not return to zero. Instead, the fish will start breeding and the population will increase toward the carrying capacity. A perturbation away from this point does not return, making it unstable.
In this case, instability is beneficial. However, stable and unstable points can also be arranged where an unstable equilibrium represents a bad outcome, such as the transition away from Holocene climate conditions due to climate change.
As a side note, fish are typically seeded into lakes by dropping them from slow-flying planes or helicopters from about 200 feet. The minnows survive the fall remarkably well, as fish are apparently good at diving from that height. This bumping away from the unstable zero equilibrium initiates population growth toward the carrying capacity.
Incorporating Harvest into the Model
The growth model provides important biological information, but for economic analysis, we need to incorporate harvesting activities.
The Harvest Equation
The stock dynamics equation is modified to include harvest H(T):
S_{T+1} = S_T + G(S_T) - H_T
This is straightforward: if nothing is harvested, the entire stock passes into the next period, but if some is harvested, less will be available.
Connection to Macroeconomic Models
This equation closely resembles the GDP and capital accumulation models from macroeconomics. In the GDP model, capital stock grows depending on how much is saved, while here fish stock grows depending on how much is saved from catching. There are significant similarities between these seemingly dissimilar concepts of fisheries sustainability and macroeconomics.
Defining Sustainability
Sustainability occurs when S_{T+1} = S_T, meaning the stock does not change over time. This could represent a bad sustainable outcome (like zero fish) or a good one (like the carrying capacity). Given enough time, species will move toward their carrying capacity.
On the growth curve, any stock between zero and the carrying capacity has positive growth, meaning there will be more stock the next period. As long as growth is above zero, the stock will constantly and asymptotically move toward the carrying capacity equilibrium, which represents the sustainable yield.
For harvested fisheries, sustainability occurs when the growth rate equals the harvest rate. If harvest equals the amount added to the population, the system remains stable. However, if harvest exceeds the maximum sustainable yield, the stock will eventually be depleted because more is being taken than the growth rate each period.
The Economics of Fishing: Profit Maximization
The bioeconomics component involves fisher people who are profit maximizers responding to fish prices and marginal costs of production.
The Profit Function
The fisher person’s profit (π) is defined as:
\pi = P \cdot H(S) - MC(S) \cdot H(S) = (P - MC(S)) \cdot H(S)
The profit equals the price multiplied by the harvest, minus the marginal cost (which is a function of stock) multiplied by the harvest. This simplifies to the difference between price and marginal cost, multiplied by the actual catch.
Stock-Dependent Marginal Cost
A key element making this a bioeconomic model is that marginal cost depends on the stock level. This reflects the reality that in actual fisheries, as stock levels decrease, the cost of catching the marginal unit of harvest increases. If there are very few fish in a lake, it takes more time to catch each fish.
This stock-dependent marginal cost is decreasing in stock, meaning that when stock is high, there is an additional incentive for the profit-maximizing firm to let the stock grow large. When stock is high, catching fish is easy and requires less time, fuel, and effort.
Numerical Example: Comparing MSY and Maximum Profit
Setting Up the Example
Consider a specific growth curve:
G(S) = 0.3S \cdot \left(1 - \frac{S}{30}\right)
In this example, R equals 0.3 and the carrying capacity K equals 30 (with all values in thousands). In practice, regulatory agencies like the DNR estimate these values through field research, with different values for different lakes and fish species.
For any stock level, harvest can be calculated. At a stock level of 25:
H(25) = 0.3 \times 25 \times \left(1 - \frac{25}{30}\right) = 1.25 \text{ (thousands)}
With a price of $10 per unit, profit would be $10 times the harvest.
Calculating Profit at Maximum Sustainable Yield
If the MSY corresponds to a stock of 15, the total revenue at this level can be calculated:
TR = 10 \times 0.3 \times 15 \times \left(1 - \frac{15}{30}\right) = 10 \times 0.3 \times 15 \times 0.5 = 22.5 \text{ (thousands)}
This gives total revenue of $22,500 assuming MSY.
However, profit maximization requires considering costs as well as revenue. Total cost is marginal cost multiplied by harvest. Using a simple linear marginal cost function:
MC(S) = 30 - S
This function satisfies the requirement that total cost is low when stock is high (easy fishing) and high when stock is low (difficult fishing).
At MSY (S = 15):
TC = 30 - 15 = 15 \text{ (thousands)}
Therefore, profit at MSY equals:
\pi_{MSY} = TR - TC = 22,500 - 15,000 = 7,500
This generates a profit of $7,500, which seems reasonable. However, this calculation did not involve actual profit maximization; it simply assumed the MSY level without justifying why that level was chosen.
Finding the Maximum Sustainable Profit
A profit-maximizing individual would set marginal revenue equal to marginal cost.
First, derive the total revenue function:
TR = 10 \times 0.3S \times \left(1 - \frac{S}{30}\right) = 3S - \frac{S^2}{10}
Taking the derivative to find marginal revenue:
MR = \frac{d(TR)}{dS} = 3 - \frac{S}{5}
The marginal cost derivative from the total cost function (30 - S) gives:
MC = -1
Setting marginal revenue equal to marginal cost:
3 - \frac{S}{5} = -1
Solving for S:
3 + 1 = \frac{S}{5} 4 \times 5 = S S^* = 20
The maximum sustainable profit occurs at a stock level of 20.
Comparing the Two Approaches
At S* = 20:
TR = 10 \times 0.3 \times 20 \times \left(1 - \frac{20}{30}\right) = 20,000
TC = 30 - 20 = 10,000
\pi_{MSP} = 20,000 - 10,000 = 10,000
The maximum sustainable profit equals $10,000.
Even though the catch is less at the profit-maximizing level (total revenue of $20,000 compared to $22,500 at MSY), the profit is higher ($10,000 compared to $7,500). This occurs because costs are falling, and the fishery benefits from the fact that it is easy to catch the first few fish when stock remains high.
Intuition Behind the Result
Why would a profit-maximizing fisher person not want the maximum number caught? The answer lies in costs. Because marginal costs depend on the stock level, there is an incentive to make fishing easy by maintaining higher stock levels. Less time, less fuel, and less effort are required when fish are plentiful. With a flat marginal cost, the fisher person would want to maximize catch. But because of this bioeconomic component linking costs to stock levels, higher profits are achieved by not fishing down to the maximum sustainable yield.
Graphically, maximization involves finding the point where total revenue and total cost curves are farthest apart, which occurs where the slopes of the two lines are equal.
Introduction to Open Access and the Tragedy of the Commons
The logic presented assumes the fisher person has perfect control and ownership of the lake. However, the situation changes dramatically when considering open access commons resources.
The Open Access Problem
If other people can enter the fishery, they will fish. The economic logic of keeping fish in the lake to lower catching costs only holds when there is perfect ownership. When the fishery is a commons, the tragedy of the commons emerges, which was discussed in previous lectures.
The Fishery Simulation Game
A simulation game has been developed to illustrate these concepts intuitively. The game incorporates the full bioeconomic model, including growth dynamics as a function of stock, as well as supply and demand effects where increased catch simultaneously drives down supply.
Different scenarios can be explored: - Solo scenarios where entry can be restricted, allowing investigation of profit-maximizing behavior - Free entry scenarios where other boats enter the fishery
Under free entry conditions, when fishing is profitable, more boats spawn. Eventually, participants overcatch the commons, and the stock may collapse depending on behavior, causing reduced overall welfare.
Additional game variations include scenarios with pirates (where boats can ram other ships) and sharks (which chase the most profitable boats). While these additions diverge from ecological foundations, they add challenge and engagement to the simulation.
The exercise involves trying different scenarios with approximately one and a half minutes each to achieve maximum profit, building intuition about fisheries economics applied to open access cases before formally discussing the mathematical framework in subsequent lectures.
Content (Day 2)
Introduction and Review
Today we continue our exploration of fisheries economics, building on the foundational concepts from the previous lecture. Fisheries represent one of the oldest and most successful models where economists have been able to link standard economic theory with biological understanding. This integration of disciplines makes fisheries a particularly valuable case study in environmental economics.
The primary focus of this lecture is to use fisheries economics to illustrate the concept of open access. Open access is one of our three major market failures and represents a classic commons problem. Whenever you have a commons, you have a commons dilemma, and the fundamental reason for this is that open access means more and more people are going to utilize the resource. Fisheries serve as a perfect example of this dynamic.
We will also examine some of the data on the status of the world’s fisheries and what we can learn by applying our analytical tools to understand current conditions. Finally, we will review for the upcoming midterm examination.
Review of Key Concepts from Previous Lecture
In the previous lecture, we spent considerable time discussing two key concepts that do not need to be written down again but are worth reviewing. We talked about the growth function and how it leads to total revenue, and we talked about profit maximization as a function of stock. We used our analytical tools to determine how a profit-maximizing firm would conduct their profit maximization differently when there are different levels of stock.
This led to some key conclusions, most notably that although many fisheries managers aim for managing at the maximum sustainable yield, when we instead examine the situation from a profit perspective rather than yield, we find that the maximum sustainable yield is actually worse than if you aimed for maximum sustainable profit. The key takeaway is that MSY (Maximum Sustainable Yield) is not equal to MSP (Maximum Sustainable Profit). The difference between total revenue and total cost is greatest at a point that does not correspond to the maximum sustainable yield. When you have the ability for a single person to strategically choose how much to fish, they have an incentive to not maximize yields but to maximize profit by leaving extra fish in the water so that they are easier to catch.
However, this strategy obviously requires being able to prevent other people from coming in and catching those fish.
Open Access in Fisheries
Now we take our same tools and apply them to a more realistic case: open access. While some fisheries are private, generally speaking, lakes are big, oceans are huge, and it is very often the case that even the laws do not try to restrict the number of people. This is true on the open oceans. Additionally, enforcement is difficult, and there is significant illegal fishing that goes on.
There is an excellent visualization from the New York Times that shows illegal fishing boats moving around the map over time. You can plot where boats are using their transponders and combine this with data on unregistered fishing vessels to see where illegal fishing is happening, which is predominantly around Africa.
The Fundamental Problem of Open Access
The fundamental difference we are building toward is that we no longer have the perfect case where we can aim for MSP. In open access, firms will enter. The logic of keeping fish around so they are easier to catch breaks down because somebody else is going to get them if you do not.
Firms will enter so long as there is any profit at all to be extracted from the fishery. They will enter so long as the total revenue is greater than the total cost. At both the MSY and the MSP, the total revenue is indeed greater than the total cost, but there are other areas to the left where you would have much lower stocks. An individual fisher person would not want to operate there because it lowers their profit, but if you are comparing it to a profit of zero (not fishing at all because you have not yet entered the fishery), that little positive slice of profit, while less than the optimal, can be captured by a new entrant. Of course, this hurts existing fishers and lowers their profits too, but from the individual’s perspective, they have an incentive to capture that extra bit of profit. This continues all the way down until the point where total revenue is no longer greater than total cost.
In other words, we will have firms entering and driving down the stock until total revenue equals total cost.
Assessing Open Access Outcomes
How can we assess this? How can we figure out how to use our profit maximization approach to demonstrate that profit in open access is going to be driven down to zero? This is that profit curve from before, and that is why we preferred being at the maximum profit point. But if there is any profit, there will be incentive for firms to enter.
We will build on our intuition from the previous graphical analysis, but we will also introduce an alternate way of looking at this.
Method 1: The Profit-Stock Approach
To review, we have so far talked about the biological growth curve, which essentially says that harvest depends on stock. We then coupled it with economic detail on marginal benefit and marginal cost, where we said that profit depends on stock. These approaches are correct and useful, but there is a problem: fisher folk do not choose stock.
If you are a DNR officer, maybe you are choosing the stock in a sense when you are choosing how many permits to issue or how many fish to literally stock into the lake. But if you are an individual fisher person, you do not actually get to choose that amount. Instead, you just get to choose what you do, which we will describe as your fishing effort.
Method 2: The Effort-Based Approach
Instead of Method 1, the profit-stock approach, we will have a second method based on effort. This makes sense because people choose their effort; they do not choose their stock. If you have your own lake, maybe you could say that, but not if you are talking about the ocean.
Defining the Model Parameters
Our growth function is going to stay exactly the same. Growth is based on the intrinsic rate, R, of growth of that stock, S, times 1 minus S over the carrying capacity: G = R × S × (1 - S/K). This is exactly what we had before.
We are going to add one more key component called effort, in this case, fishing effort. We will now say that harvest is parameterized by this function: H = Q × E × S, where Q is the catchability coefficient (a data point on the catchability of the fish), E is the effort (which could be thought of as hours of fishing or similar measures), and S is the same as before—the stock.
Now we have these two equations. They both have stock in them, but what we will see is that we can use this to reframe our equations to express the optimal effort, rather than optimizing profit based on the stock level.
Linking Effort and Growth Through Stock
Looking at these equations, we will use them to solve for the optimal effort and see what we can learn about sustainability in that case. Looking ahead, the fact that they both share stock (S) means we can relate effort and growth through the stock.
Another trick we will use is to define the sustainable harvest function. There is nothing that says harvest has to be sustainable—we could do unsustainable harvest by just catching them all. But it will be useful to relate these two by assuming sustainability, which means setting H equal to G. If H equals G, the harvest equals the growth, and that is obviously sustainable because we are taking out as much as is coming in the next period.
This allows us to link our two effort-based equations. If G equals H, we can say Q × E × S (the H side) equals R × S × (1 - S/K) (the G side).
Deriving the Sustainable Harvest Function
This is the first step toward defining a sustainable harvest function. We will eventually use this to show what happens in open access, but we need to manipulate it to make it easier to use.
Step one is to replace H with QES, as we just did. Then we perform several algebraic steps to solve this.
First, we have an S on the outside of both sides, so we can divide both sides by S to get: Q × E = R × (1 - S/K). Next, we can divide both sides by R to get: (Q × E)/R = 1 - S/K. Now we want to isolate S by itself, so we rearrange to get: 1 - (Q × E)/R = S/K. Finally, we multiply both sides by K and write it in a more convenient form: S = K × (1 - QE/R).
This is a more convenient way of expressing the sustainable harvest function. So long as our effort, E, is defined according to these parameters and set equal to S, we will have a sustainable harvest. Obviously, if we change our effort, it might not be sustainable.
This is useful because now we can use it to figure out what individual agents, if they are profit maximizing, are going to do, but here choosing their effort level instead of the stock level.
Eliminating Stock from the Equation
The second step is that from our effort equation H = QES, we can rearrange to express S: S = H/(Q × E).
This allows us to simplify further. We can actually eliminate the stock. Plugging in this definition from our harvest rule into the sustainable harvest equation: H/(Q × E) = K × (1 - QE/R).
This involves many steps and substantial algebra, which is why this is a 3,000 level course. We still do not have too many derivatives, but we will get to one shortly.
We want to rearrange this to express the sustainable harvest with H on the outside. We take the equation and move QE over: H = Q × E × K × (1 - QE/R).
Why is this useful? We have now derived a sustainable harvest rule that does not have the stock in it; it just has the effort in it. This is useful because we can then use it to figure out the sustainable maximum solution or the profit-maximizing solution, but where we are headed is using it to derive what will happen in open access.
Origin of Model Parameters
The catchability coefficient and intrinsic growth rate come from biology. DNR officers or biology PhDs actually go out to lakes and test these values. Catchability can be obtained from surveys, looking at how many fisher folk were out there, how many hours they spent, and how many of each species they caught. In reality, the data would be different Q values for each fish species.
For example, muskies are very hard to catch, so if you looked at survey data from fisher folk, they would have a low Q for muskies, but something like a largemouth bass is pretty easy to catch and would have a higher Q value.
The intrinsic growth rate is even easier because you can actually simulate it in a lab—literally have a big pond and see what happens, then determine which coefficient for R best matches the observed data on the growth function. We are not going to get into the data collection here; we are users of that data. There is a whole set of disciplines behind collecting that data, including field biology and surveying fisher folk.
This is what makes it interesting: this is not just arbitrary theory. It is theory in the sense that we are simplifying to say the logistic growth equation is useful, but models are wrong, yet sometimes they are useful. The model is useful not because fish literally follow that equation, but rather because when we do good biology and get these Q and R values, it is pretty predictive. It connects data to reality with an underlying theory of why it works.
Numerical Examples of Effort-Based Analysis
Let us do an example with real numbers to make things more specific and simplified.
We are going to do the same thing where we have a function, but now it is going to be harvest in terms of effort. Let us start by defining our sustainable yield function as this simplified case: H = E × (1 - E/80). This looks similar to our logistic growth with the same dynamics, but we are now looking at it in terms of E, the effort coefficient.
The arc of what we will do involves three examples of using this approach. First, we will look at maximum sustainable yield, but expressed as effort. Then we will do net benefit maximizing (calling it this because we are now thinking about all of society, not just the profit of a firm). Finally, we will use that to show what happens if we allow open access.
This is probably the most mathematically challenging day of the course. Mathematics can be enjoyable, and this integration of economics and biology is particularly engaging.
Example 1: Maximum Sustainable Yield in Terms of Effort
Our first example uses the sustainable yield function to get the MSY calculations. What is the effort, not the stock, associated with MSY?
When we had our growth curve with stock on the x-axis, if we wanted to find the maximum, we took the derivative and found where it was zero. This is a fundamental finding in optimization theory: you can always find an optimum of a curve by looking at all the points where the derivative equals zero. Now we are re-expressing this in terms of effort instead of stock, and the equation has the same curved shape, so we will do the same thing.
We take the derivative of our harvest function and set it equal to zero to figure out what effort level corresponds to the maximum. To do this, we first distribute the E: H = E - E²/80. Taking the derivative with respect to E: dH/dE = 1 - 2E/80, which simplifies to 1 - E/40.
To solve this, we set the derivative equal to zero. Looking at the expression 1 - E/40 = 0, for something to equal zero, it needs to be 1 minus 1. What level of E would make E/40 equal to 1? The answer is 40, because 40/40 = 1. This implies that E for MSY equals 40.
So that is maximization theory applied to fisheries. We can go further to determine the actual harvest associated with that optimal effort. The harvest at maximum sustainable yield is 20 (which can be verified by substituting E = 40 into the original harvest function).
Example 3: Open Access
Why have we spent so much time on this? Because we can now move to the really interesting case of what happens if, instead of maximizing society’s net benefits, other fisher folk are allowed to enter freely. We will use the same net benefit function.
The net benefit of E is: NB(E) = 10E × (1 - E/80) - 5E.
Instead of doing the derivative and finding the optimum, we will use what we call the open access trick. This is actually a shortcut because we know that if there is any benefit to society (expressed as total revenue being greater than total cost), new firms will enter. We know this will happen as long as it is rational, which means it will continue until total revenue and total cost are equal.
This gives us useful information: in open access, the net benefit will be equal to zero. This is different—we are not taking the derivative and setting it equal to zero; we are taking the actual function and setting it equal to zero.
Let us simplify. The net benefit of E can be written as: NB(E) = E × (5 - E/8). Setting this equal to zero for our open access trick, we find it has two solutions. The first is obviously E = 0, but that is not what happens—we are not saying that open access causes everyone to quit fishing. Instead, people enter and take away all the benefit, converting it into their own profit.
So we solve the other condition: 0 = 5 - E/8. This gives us 5 = E/8, and solving for E gives us E = 40.
This is the level of effort associated with open access—much higher than the socially optimal effort of 20. We have used the same mathematics, looked at open access using the open access trick (knowing that people keep entering until net benefit goes to zero), and demonstrated that what we expected to happen did indeed occur: too much fishing effort results from open access.
Why This Matters: The Bioeconomic Model
This analysis is useful because it represents a bioeconomic model that combines economic theory with biology and biological theory. What does it accomplish? It lets us make predictions. There are numerous studies that use the coefficients on catchability and effort calculations to model what will happen in different oceans.
Research by Palomares et al. examines different oceans, and the basic conclusion is that conditions are worsening. This is concerning because much of the world’s population depends on fish for protein.
You can also plot how well we are doing compared to maximum sustainable yield, and in many cases, performance falls well below zero, indicating suboptimal management. This ties back to what we can observe happening in the oceans and also suggests potential policy changes. Taxes or other policies could change this calculus and make it no longer optimal for open access to result in over-exploitation.
The Key Takeaway
The key takeaway is that open access equals over-exploitation. This is a restatement of what we observed in our market failures analysis, but with considerably more detail. This is exactly the tragedy of the commons. The tragedy of the commons means that open access leads to more and more fisher people entering, and this inevitably leads to tragedy because it drives the stock down to the point where nobody is profitable—which is bad for both humans and fish.
Midterm Review
Shifting gears to prepare for the midterm examination, students should examine the practice midterm and the detailed walkthrough provided. Some content is straightforward, such as supply and demand analysis, which most students handled well.
One caveat is that the exam will ask students to figure out tax revenue, which requires re-solving supply and demand with a tax and using those steps to calculate the actual tax revenue.
Key Problem Type: Uniform Standards vs. Firm-Specific Approaches
A particularly important question type involves two different marginal abatement cost curves, MAC1 and MAC2, with two different policy cases. The first case is where everybody has a uniform reduction in their pollution. When plotted, this involves a vertical line at whatever uniform emissions level is set. From there, you can determine total costs by seeing where this line intersects each marginal cost curve and summing them up to assess societal efficiency.
The critical insight that distinguishes successful answers is as follows: if instead you are asked to consider a firm-specific approach rather than a uniform standard, the key element is to start by setting MAC1 equal to MAC2. In a question where specific curves are given, this means setting those expressions equal to each other, then solving for E1 and E2.
Graphically, the MAC curves remain unchanged, but we are saying we can have different levels of emissions that are not uniform. This is illustrated by a horizontal line instead of a vertical dotted line, where E2 differs from E1. The key insight is that one firm can abate emissions much more cheaply than the other, and the society-optimizing, cost-minimizing approach is to let that firm do more abatement. Allowing the more effective firm to use their ability to abate pollution more cost-effectively means we can achieve the same standard as the uniform approach but with much greater benefits to society.
Transcript (Day 1)
All right, let’s get started, everybody. Welcome to Monday. We’re going to talk about fisheries, renewable resources specifically, and I just wanted to make one note. I realize I did not update the name. We’ve already done development, but if you’re looking for the slides, that’s where it is, it’s on the development page. The slides are named correctly, though. You can also get it from the main page, just wanted to make sure that wasn’t confusing.
So yes, we’ll be talking about fisheries. A few quick reminders, though. We have our midterm coming up on Friday. I wasn’t able to get it out over the weekend, but I will have the practice questions out to you by 3 o’clock today. So you’ll have plenty of time to look over those, and they’ll have detailed solutions. It will review some of the key math that we’ve done in the past, and give you much more specific solutions. I spent a long time creating the very detailed solution guides that I hope will help.
One thing I’ll mention is there will be a question on the optimal abatement between firms when the firms have dissimilar abatement cost curves. That was one question that tricked up a number of people, and so I’ve given a really detailed explanation of the math in this practice sheet that we’ll send around.
Today, we are diving into fisheries, and this is our first official bioeconomic model. When we’re talking about earth economy modeling, we’re kind of taking it for granted that there have been models that link something environmental and something economic. That’s been done for a very long time, long before the sort of recent explosion of research on Earth economy modeling. We’re going to take a look at one of the most classic ones, and that’s going to be a bioeconomic model that couples information about the growth of a species, in this case fish, and how that is dynamically computed with an economic model to figure out what would be the profit-maximizing, but yet still sustainable catch rate.
After we get through the mathematics of fisheries, then we’re going to end with what will be, I hope, a fun little simulation, a game that I spent way too much time on, that will illustrate the basic concept we’ll learn today, as well as applying it back to something we’ve learned about before with open access and commons dilemmas.
This one’s going to be a little bit more math-heavy than some of our other ones, but it’s a really useful set of equations for understanding the explicit linkage of the environment and the economy. Bioeconomics, then, you might guess from the name, is something coming from biology connected to economics. In this particular case, we’re going to be building a relationship between the stock levels of fish and the subsequent growth of fish.
First off, let’s get a few equations on the board. We’re going to be talking today about the stock, S(T+1). That’s a stock in time period one into the future is going to be equal to the stock in period T, plus some growth function. That is also going to be a function of the size of the stock.
What’s this saying? Let’s, before even getting into the specific functional form, think about what happens when you have a lake that is completely full of fish. There are so many fish. What happens? This could be true of any species. Squirrels and acorns, fish and the things that fish eat. Do they grow as fast? Does the population grow as fast when there are tons of individuals of that species?
No, because they’re competing. But what if there’s only one? Does it grow very fast? No. So we’re going to have this relationship, it’s going to be an upside-down U-shaped curve, and we’ll draw it in a minute, where at very low levels of population, fish don’t grow very fast, and this is true of basically any animal, simply because there’s not enough breeding pairs.
However, when we get to really high populations, which we’ll characterize in a minute with something called the carrying capacity, it also is the case that they don’t grow as fast, and that’s because there’s something in their environment that limits further growth beyond that carrying capacity. Actually, the squirrels example is maybe the best one.
One thing that’s kind of funny about squirrels and the oak trees that they rely on is every once in a while, oak trees have evolved this characteristic where they’ll have a bumper crop of acorns. They do this because that means there’ll be so many acorns that there won’t be enough squirrels to eat them all, so some of them get planted. It’s random, more or less, when the tree does this.
So some of those get planted. The problem is, what happens the next period, the next year, to the squirrel population? A ton more. You can actually observe this. You can see one year there’ll be a huge crop of acorns, the next year there’ll be a huge amount of squirrels, and then what happens the year after that? To the squirrel population, massive squirrel suffering. It’s the brutal reality. They had a really great year, and it’s always followed by a really, really bad year, and so you can see this. We’re not going to be talking about squirrels, though, just because that’s too close to home. We’re going to talk about fish. And we’re going to draw out that specific relationship.
This is the basic equation, but I haven’t said what that function looks like. This just comes from literally biology, studying animals and looking at what is the best relationship or best equation for describing their growth dynamics.
I’m just going to write it down, and then I’m going to define these in a minute. So the G is the function that takes the argument S, the stock, is going to be R, which we’ll call the intrinsic growth rate, times that stock. And then we’re going to do this part here, 1 minus S over K. R is the intrinsic growth rate. K is the carrying capacity.
Let’s break this down. First off, what does intrinsic growth rate mean? It means, in the very best case, when there’s nothing slowing it down, essentially, what’s its maximum growth rate? If, for a second, we ignore the right-hand side, it’s just saying if there are S number of individuals, there will be this rate of growth. You can think about if there’s 100 fish, then we’re going to say 100 times this, maybe it’s 0.1 or something, is the number of new fish that would happen if there was nothing related to overpopulation. So it’s the fastest possible growth rate. That would just be something that would grow forever. The stock gets bigger, it grows more. The stock gets bigger, it grows more. But we’ve already said that this is going to be something that can’t go on forever, because eventually we hit some sort of limit. That’s what’s going to be modeled over here.
Basically the maximum growth rate here is going to be multiplied by something that is a little bit less than 1. It’s going to be 1 minus S divided by K. Essentially, this component here, S divided by K, is how close to the carrying capacity are we.
Here’s a question. What would happen if S exactly equaled K? Let’s walk it through. If, doesn’t matter what it is, let’s say K is 1,000. If S is 1,000, we’d have 1,000 divided by 1,000 is 1, and so we’d have 1 minus 1 is 0. The zero would be multiplied by the intrinsic growth rate, and so it doesn’t matter what R is, or what S is, if we have this term here going to be zero, it’s going to result in no growth at all. That’s just an example of all the squirrels have eaten all the acorns, and there’s nothing to eat, and they don’t survive.
But to any extent that you have an S that is less than K, you can see this might be just, like, if it was 900 over 1,000, this would be just 0.9, and so it would be 1 minus 0.9. Still a small number, 0.1, but now this would have a positive value, because it’s not multiplied by the zero over there on the right.
The first key plot that we’re going to have is going to combine this information into a plot that shows growth as a function of stock. If you wanted to, you could always just plug this equation here into a graphing calculator. But let’s just talk it through.
We’ll start with the easy one. If S is zero, there’s nobody to reproduce. I’m pretty comfortable just jumping to the conclusion that if S stock here is zero, the growth rate is zero. No new fish where there aren’t any fish.
Then conversely, we just talked about this case where when the S equals the K, we also have no growth. We’re going to label that one S_K, that would be where the stock is exactly equal to the carrying capacity.
One thing to note, this is the growth. This isn’t the total number here, so this isn’t saying there aren’t any fish. In this case, it would just be that the fish are exactly at the carrying capacity, so there’d be no extra growth. This is kind of like a change map, depending on the stock.
But then if we plot it out, this function, it’s actually a very famous function. It’s called the logistic growth curve, and all sorts of physical populations exhibit some version of this growth function. You might have different Rs, you’ll have different Ks, but it’s going to have these two points that we already talked about, but then for anything in between, it’s positive.
We’ll call out one particular point where it hits its maximum, and this is going to be what we call the MSY, or Maximum Sustainable Yield. What is that? We’re saying that if somehow we can lower our stock to this point, that’ll maximize the growth, and that’s because we’ve gotten us far enough away from where the carrying capacity means we’re limiting our resources available, but we’re still not getting too low where there aren’t enough breeding pairs.
This is just taking these two bits of information, the stock one with the growth one plugged into it, to express it in a nice, straightforward way. And maybe one more label I’ll have is the stock at the MSY is the one right there.
We’re setting this up because we want to talk about what is really a big debate, change in worldviews between two different groups of people. Those that would like to see a fishery operated where it was at the maximum sustainable yield. I’ll give you a foreshadowing. Environmentalists tend to love that. It seems like the best, the most growth rate, that must be the best. But we’re going to contrast that in this lecture with another way of looking at it, which is not maximum sustainable yield, but maximum sustainable profit. You can sort of see, again, it’s going to be that classic trope of environmentalists versus economists.
One last plot that I’ll also have is a companion to this plot. In some ways, it’s easier to think about, because instead of a dynamics plot where you have stock and growth, let’s just play this out. We’re going to derive from that, this plot over here, which is going to show the progression over time, and now we’re putting stock on the vertical axis.
What we’re going to have here is this is essentially the derivative of the stock. The growth is the change in the stock. When it’s close to zero, it’s going to have a slope of zero, but then as the stock increases, if you have anything above that, it’ll start to grow, and the more the stock increases over time, the growth rate will actually go up. This is going to be curving upwards.
But at some point, it’s going to be the case that the growth rate starts to flatten off, and that’s just because at some point, we cross our maximum sustainable yield and start to have this carrying capacity overcrowding effect coming in.
There’s an inflection point. This is where it’s no longer going up at an increasing rate. Where it hits that inflection point from being convex this way to convex that way. It’s just another way of identifying the maximum sustainable yield. This one’s a little easier to interpret because you can see that if you’re down here, it’s going to grow, grow, grow, but if you’re past that point, it’s going to start growing at a decreasing rate.
Just a couple of points on this graph that I want to highlight is that the growth rate is zero in these two different places. You might be tempted to say that they would be the same. When we have a growth here, or a growth here, it’s going to be a zero value in growth. But the difference is in stability, this is a technical point, but it’s an important one.
If we were at this upper zero point and we bumped it down, something happened, like we caught a bunch of fish, what’s going to happen? Well, the next period, it’s going to grow back towards this place, because we have positive growth. Perturbations away from this stock level here will tend to go away over time, just because it’ll grow back into that case.
This one is what’s called an unstable equilibrium. If you had zero fish, it’ll stay at exactly zero fish. But what happens if you drop a bunch of fish in a lake, and suddenly you’re a little bit above zero? Is it going to return to zero? Not according to this one. Those are going to start breeding and go up, up, up, like this. A perturbation away from this point is considered unstable because it’s not going to return there.
It’s a good thing in this case. You also can have unstable and stable points that are flipped around, where it’s an unstable, bad one. Like, we’re in a good condition, and once we fall away from it, it’s no longer good, like the Holocene and climate change, for instance.
Side note, does anybody know how they seed fish into a lake if they’re trying to stock it? Yep, exactly. I saw that one time, and it’s the funniest thing. Well, the one I saw was with a slow-flying plane. Like, 200 feet above, they’re just shooting little minnows off the back of it. Turns out that’s about the right way to do it. The fish are just fine. Apparently, they’re good at diving from that height. I thought that was funny. But yeah, the point is, if you did that, if you dropped a bunch of fish from a helicopter, it’ll bump it away from this stable equilibrium and shoot up to here, and they’ll start spawning, hopefully.
That’s just the growth, but this is an economics class, we want to do something with that. We don’t just want to have fish grow for the sake of growing, we want to do something like harvest.
We’re going to add in a new term. Keep those on the boards and keep pointing to them. I could have modified the equations we have, because I’m only going to add a little bit, but now we’re going to have a new stock dynamics that’s going to take into account the fact that we also might want to fish, or have a harvest, which will be H, H of T.
S in T plus 1 is now going to be equal to what we had before, which is the stock in period T, plus some growth that depends on the size of that stock, that’s that curve. But now, we’re going to subtract out something, which is going to be H of T. Pretty straightforward. If we did nothing, the whole stock would pass into the next period, but if we harvest some, less of that will be available.
Let me see if anybody gets this point. It’s a tough question, but it’s an intuitive one. Have we seen something like this before, this equation? Obviously not fish, but anything else in this class.
Yes, it’s exactly like GDP. This is savings in this case, and this is consumption. Exactly, so we are… it’s kind of funny, but that’s actually why I put macroeconomics at the beginning, is because macroeconomics is a really good way of thinking about any sort of dynamic growing thing. With GDP, we had capital stock growing, depending on how much you saved. Well, here we have a fish stock growing, depending on how much you saved, in this case, saved from catching.
There’s tons and tons of similarity between these really dissimilar-sounding concepts of fisheries and sustainability of renewable resources and macroeconomics.
Now that we have this equation in place, we can define a few words a little bit more detailed. What is sustainability? It’s any case where S(T+1) is equal to S(T). That’s a boring equation. But it’s basically saying that something is sustainable if the stock doesn’t change over time. We could have a bad sustainable, like zero, that’s not very good, but we could also have this sustainable point at the carrying capacity. Given enough time, species will move towards their carrying capacity.
We can actually see that on this graph, because if we had a stock anywhere in between, like here, we would say, well, what’s the growth for this time period? It’s here. It’s something positive, which means that there’ll be more stock the next period, and as long as this is above the zero axis, which it is everywhere in this case, it’ll constantly asymptotically, at least, be moving towards that equilibrium, and there is where we would finally have this sustainable yield.
A converse that we can get from this is for that to be the case when we have fisheries that might be harvested. An equivalent one is to say it’s going to happen where the growth rate as a function of the stock is equal to the harvest rate. That sort of makes sense. If you harvest off as much as was added to the population, that’s going to be stable.
All these things conspire together, though, where whenever the stock is, or the growth over time is positive, meaning we haven’t harvested it down to zero, it’s going to continue to grow.
But what happens if we go too high? What happens if the harvest is way up here? If we’re harvesting above the maximum sustainable yield? Remember, this G function here is just the growth, but if we have a harvest level that is higher than this, we subtract this out, and we would eventually deplete our stock, because we would be taking more than the growth rate each period.
That’s harvest. But we haven’t said anything about what’s driving this harvest. So now we get into the bioeconomics, economics-y part of this. We’re going to say that our fisher folk, or fisher people, are going to be your basic profit maximizers. Just like before. But they’re going to respond to a price of fish, P, and a marginal cost of production.
Such that the fisher person’s profit, we’ll call pi, as is the case, is going to be the price multiplied by the harvest. We’re going to add a letter in here. We’re going to let them choose their level of harvest based on the stock.
So basically, how much do they catch multiplied by the price. You sell your fish at the price, but they’re profit maximizers, and we always know that they take into account both the revenue, price times production, and what’s called marginal cost, which itself is going to be a function of the stock. They incur that cost for each unit of harvest, S, that they produce.
We can simplify this around. It’s really the difference between the price and the marginal cost. Whatever that margin there is, multiplied by how much they actually catch, that’s going to be their profit, and that’s what they want to maximize. One of the key elements that makes this a bioeconomic model, though, is in the marginal cost here. I made it a function of S, and this is going to be reflecting the fact that in a real-life fishery, as you have a lower and lower stock level, what do you think happens to the cost of catching the marginal unit of harvest?
If there are very few fish in the lake, how much time does it take to catch a fish? More. This is a really new thing. We’re going to now have the fact that the marginal cost is going to depend on the stock. It’s going to be decreasing in stock, meaning that whenever we have a really high stock, there’s another incentive, in this case, to the profit-maximizing firm to let that stock get big, because simply now it’s really easy to catch fish. You can catch a whole lot of fish in a small amount of time if there’s a ton of fish in the lake.
Now I want to put this all together and give a few examples of how we can build towards profit maximization in this case. Before, we just had the sort of general logistic growth curve. We’re now going to fill it in with a specific growth curve.
We’re going to have 0.3, that’s going to be the R, the intrinsic growth rate, multiplied by S. And then we’re also going to have a carrying capacity effect, where it’s going to be 1 minus S over 30.
Just to note, this means that R, in this case, is 0.3, the carrying capacity was 30. In real life, the DNR or whoever will go around and estimate these values, and so we have different values for different lakes and different fish species and stuff, so this just comes from literally from the data.
But as soon as we have this, we can now say, for any stock level, what’s our harvest going to be? For a stock level of 25, just to try one out, I’m just putting the 25 in here, in all the places where it shows up. In this particular example, I have everything in thousands of dollars, but this would give us a harvest at that stock level of 25 equal to $1,250.
But what we really want is not H, but profit, is just P times H. 10 times that. This is what the profit-maximizing fisher person is going to do.
That’s just for an arbitrary S, but a sort of logical one to try out is, what if we have a stock set equal to the maximum sustainable yield? Now, why might we start there? Well, like I said, we’re going to have this divergence of two different opinions on whether we should be an environmentalist and maximize the sustainable yield versus the maximum sustainable profit. But let’s start off with that environmental one, the MSY.
In this particular case, I’ve given you a little bit of data here. Before, that was just in general terms, but now we’re going to say the MSY, which corresponds with this line, is going to be 15.
So if the MSY is 15, and we want to ask what’s the total revenue at that level, let’s calculate it out. 0.3 times 15, 1 minus 15 divided by 30. We can see that’s 5, that’s half, and so we can easily, from there, get 2.25, in thousands. You simplify this down and put it in your calculator, but we’re going to, again, multiply it by price.
Including that in gets us total revenue, assuming MSY, of $22,500.
We don’t maximize total revenue, right? We learned in Econ 101 that we actually maximize profit, which depends both on the total revenue and on the costs. That’s the second bit of information we need to know. So let’s add that in.
The total costs, we’ve already got some of the ingredients for it, we already identified marginal cost, but it’s going to be marginal cost, which we’ve already argued depends on the stock, because it’s very fast to catch fish when there’s a ton of stock. That marginal cost is then just multiplied by how much the fisher person actually chooses. We’re going to make it easy on ourselves. Let’s have a real easy equation, the 30 minus S line here, which I’ve now plotted there.
This fits all of the things we need, as we can see that the total cost when the stock is really high is low. That means you can just stand on the dock and catch a bunch of fish, but eventually it gets really, really high, as it becomes harder and harder to catch fish.
Importantly, we can now also figure out what the profit at MSY would be. TC at MSY, this one’s easy, 30 minus the stock at MSY equals 15. Everything’s in thousands, so 15,000.
So now we can finally get profit equals TR minus TC equals 22,500 minus 15,000 equals 7,500. Just to be clear, that’s profit at MSY.
Pretty good, we’re making profit, that’s nice. But now we’re going to flip gears and contrast this with how a non-environmentalist but a profit maximizer would choose to do this. The catch here is, we just didn’t do any profit maximization here. We simply asked, okay, let’s do the MSY level of fishing, and see how much profit we had. But we didn’t say why we chose MSY. Maybe it looks good, because it’s got the greatest growth, right? So maybe that’s why we did it. But it’s not actually profit maximizing.
How would a profit-maximizing individual approach this same set of parameters? Hopefully you’ve had enough econ where your spidey sense should be tingling right now. Whenever you think of profit maximizer, you’re probably going to set one marginal equal to another marginal. Back in Econ 101, it was always MR equals MC. We’ve got TR and TC, so you could sort of make a guess on where we’re going with this.
In this case, the total revenue is going to be exactly the same as before, but we’re not just going to jump to the assumption that it’s 15. Instead, we’re going to take our total revenue curve and figure out how to get the marginal revenue curve from this. I was wrong the other day when I said there were no more derivatives. There are a couple of derivatives, but they’re really easy, and I’ll walk you through them. I’m not going to test you on being able to take a challenging derivative.
We’ll walk through this one hand-in-hand. Our TR curve, just to repeat it, is 0.3S times 1 minus S over 30. So how would you take the derivative of that? Let’s simplify it a little bit first.
We’re also going to just skip a step by plugging the price right on in. So the price is $10. 10 times 0.3S times 1 minus S divided by 30.
The easiest way to simplify this is we’re going to distribute this 0.3S. We’ll quickly call it 3S because of that price. We’re going to distribute it to the 1, and then also over here to the S minus 30, and so this is going to give us something that’s easier to take the derivative of.
3S, let’s multiply by 1, so it’s just 3S. But now we’re going to take 3S, multiply it by that, and so here, you know for sure we’re going to have an S squared. But then it’s going to be a 3 divided by 30, and so that would be 1 over 10, and we’ll just jump right to the answer. It gets us 3S minus S squared over 10, and now the derivative.
We’re going to take the derivative of that, it’s much easier. What’s the derivative of 3S? Just gets rid of the S. It becomes a 3. And what’s the derivative of S squared divided by 10? The 2 comes down here, so 2 divided by 10 makes this 1/5th instead, and so it just becomes S divided by 5, and so that will give us what we want, our marginal revenue as a function of the stock is 3 minus S over 5.
The other thing we need, though, is going to be quite a bit easier to take the derivative of. We’re going to also get the marginal cost from the total cost. Total cost is going to be that 30 minus S, and the derivative gives us a marginal cost of negative 1 is the derivative of the cost. That’s essentially the slope. Obviously, it’s a linear line, the derivative is the slope, and we made it real simple on ourselves there.
Now, this is where your spidey sense should really be tingling. We’ve got a marginal revenue, we’ve got a marginal cost, let’s set them equal to each other. That is just going to be marginal revenue. 3 minus S divided by 5 equals negative 1, so bring the 1 over there to make a 4. We’ll just make the S over 5 over there.
Now we want S by itself, so we multiply by 5, so 4 times 5 is 20. And so, yes, there’s the math on the screen as well. This is the maximum sustainable profit.
From there, we can get a little bit more information by plugging it back into what we have. We could figure out what would be the catch at the maximum sustainable profit, but we still need to calculate the actual profit, which would be simply plugging in what we found of S star equals 20, back into our total revenue.
That gets us a total revenue of $20,000. That’s a little bit less than maximum sustainable yield. Why is this good? Well, the answer is because now, when we subtract out the cost. The cost at that level is going to be, coming from our total cost curve, so 30 minus 20, our total cost is just $10,000, and so the maximum sustainable profit is going to be equal to 10,000.
Instead of maximum sustainable yield, the maximum sustainable profit equals 10,000.
What do we see? Even though the catch is less, we’re going to be somewhere over here. We get less total caught, 20,000 compared to 22,500. Because the costs are falling, we’re benefiting from the fact that it’s easy to catch the first few fish, and we have a higher profit.
What’s nice, we’ve done similar stuff like this before. Whenever you have total curves, essentially maximization is trying to find the part where the two lines, so this is total revenue and total cost, are the farthest apart. You can sort of eyeball it. But it basically is where the slope of those two lines is going to be the same.
This profit at MSY is the $7,500. The two have converged a little bit. The maximum sustainable yield profit is $7,500, but the maximum sustainable profit is going to be that $10,000. That is achieved by not going all the way to the maximum sustainable yield.
Does that feel intuitive? I think so. What’s going on? Why would a fisher person not want to have the maximum number caught? Costs. Here, the fact that their costs are dependent on the stock would give them this incentive to make it easy for themselves, essentially. Overstock the fishery so that they don’t have to spend much time, less fuel, etc. If we had a different total cost curve, like if we had a flat marginal cost, then they would want to. But because it’s got this bioeconomic component, it is something where they are going to want to get this greater profit.
Here, this one’s getting a bit complex, so you don’t need to remember this one. But here, we’re going to plot another curve, which is just the subtraction of the total cost from the total revenue. Here, now, we can see that it actually is the optimum of the profit.
I wish I’d saved more time. Because this is the pivot then, and maybe we’ll pick it up on the next lecture. But basically, the logic that we’ve learned here is that it was valuable for the fisher person to keep the fish in the lake because it lowers their cost of catching. But what would happen if other people could enter?
They would fish. This logic assumes that they have perfect control, perfect ownership of the lake. Maybe they own the whole thing. But what if it was a commons? That’s something we’ve talked about. We talked about before how there’s always a tragedy of the commons, and we’re going to see next lecture how we can actually use this to get to the tragedy of the commons.
But I want to play it intuitively first. So do load up this game. There’s the QR code. I should admit, I spent way too much time on this one, and unlike my other games, this one’s actually, I think, a little bit more fun.
It’s way down here. You might have already clicked on it when we were doing the public goods and commons resources. We didn’t talk about it because we didn’t yet have the mathematics of open access.
I basically made the app to illustrate the point, and I’m like, this could be made more fun, and so what did I do? I actually made it into a game, and for this one, it gives you the basic controls. This one I do think you need a laptop for. I tried to get the touchscreen to work, but that was surprisingly hard. Your middle mouse button will change the zoom, and if you click middle mouse button, you can look around. Pressing W drives your boat.
Under the hood, we’ll talk more about the mathematics next time, but the little profit that pops up, that is how much profit they made based on the growth dynamics, as well as another element, which we’ll incorporate next class, which is that the more you catch, it’s simultaneously going to drive down the supply.
If you want to get an intuitive sense for this, you can play out different scenarios. One thing I’d note is if you want, you can click on the advanced one. This actually has a full-fledged bioeconomic model underneath it. Here’s the growth curve that we just talked about as a function of the stock, and it’s also going to have supply and demand.
I’ll write up more about this, and it’ll obviously fit into one of the questions, but we’ll then do it with a few other circumstances, like what’s the profit maximizing if you can restrict people from entering in the solo scenario? But how does that change when you have free entry?
As soon as you have free entry, look, there’s other boats. They also make their profit, and if they’re profitable, more boats spawn, but eventually, they overcatch the commons, and the stock may or may not, depending on their behavior, collapse, causing less overall welfare. What I’m going to have you do is, there’s a bunch of different scenarios. You give a minute and a half in each scenario to try to get the maximum profit that you can.
Just because I was feeling punchy, I added another one where there’s pirates, because I thought that would be fun, and totally unrelated to anything in this course, but now you can ram the other ships just to get rid of them. And then there’s also one with pirates and sharks.
In this one, we’re going to diverge a whole lot from ecological foundations, where here, the sharks eat the boats, because I thought that would be more fun. Now you have to do your profit maximization while dodging the sharks, and so let’s just watch. I’m going to zoom out. Now there’s a shark somewhere in the water. Very easy. He chases whoever is the most profitable.
This is totally not realistic economics or ecology, but I just thought it was fun. Whenever you have a game, you have to have something that makes it be a challenge. Anyways, I’ll be assigning that as the exercise. We’ll do this in advance of talking about the intuition of fisheries, but applied to the open access case.
Try out those, and more information will come on what to answer for the weekly questions number 6. With that, expect to see the practice midterm coming here by 3 this afternoon. We’ll have a chance to talk about any questions on it, also on Wednesday, but then the midterm itself will be on Friday.
Thank you all.
Transcript (Day 2)
Alright, let’s get started.
Today we’re going to pick up and return to what we were talking about last lecture, which is fisheries. I love fisheries because it was one of the oldest models where economists were really able to link up standard econ theory with something that biologists understood really well.
And so what we’re going to do today is, first, we’re going to continue on with the economics of fisheries, specifically using it to illustrate the concept of open access. We’ve already seen this before. It’s one of our three big market failures. It’s a commons. Whenever you have a commons, you have a common dilemma, and the reason for that is open access means more and more people are going to go and utilize this, and so fisheries are a perfect example of that.
We’ll then also just take a quick look at some of the data. What’s the status of the world’s fisheries, and what can we learn using the tools we’re going to apply to understand what’s going on there? And then finally, we’ll review for the midterm, if there’s time, but that might be of interest to people, so I’ll try to reserve some time for that.
Okay, so first off, let’s get a little bit of review going.
So, we spent a lot of time, you don’t need to write this one down again, but we talked about two key things on the last lecture. We talked about the growth function, how that leads to the total revenue, and we talked about profit maximization as a function of stock.
We used our tools to say, how would a profit-maximizing firm do their profit maximization differently when there are different levels of stock?
Now, that’s an interesting question, and we saw some key conclusions like this one, which is that although a lot of fisheries managers aim for managing so that it’s the maximum sustainable yield, we saw that when we instead looked at it not from yield, but from a profit perspective, we saw that the maximum sustainable yield was actually worse than if you went for the maximum sustainable profit. And so the key takeaway there is MSY is not equal to MSP. Maximum Sustainable Yield and Maximum Sustainable Profit.
I’m going to go with this notation from now on. And so that’s all we saw, is that this bar here, this difference, is greatest at this point, and that is not the one that corresponds to here. And the key finding is that when you have the ability for a single person to strategically choose how much to fish, they are going to have this incentive to not maximize yields, but maximize profit by leaving extra fish in the water, so that they’re easier to catch.
Okay, so that’s all good, but that obviously requires being able to prevent other people from coming in and catching those fish.
And so now we’re going to take our same tools and apply it to actually a more realistic case, which is open access.
There are some fisheries that are private, but generally speaking, lakes are big, oceans are huge, and so it’s very often the case that even the laws don’t try to restrict the number of people. This is true on the open oceans. But also, it’s pretty hard to enforce. There’s all sorts of illegal fishing that goes on.
I’ll send this link around later if I remember, but there is a really great illustration from the New York Times. They showed a visualization of illegal fishing boats and them moving around the map over time, because you can plot where boats are using their transponders, but you can then also combine this with data on which are unregistered fishing vessels to see where is this illegal fishing happening, and it’s mostly around Africa.
But what we’re going to build up to is the big difference that we don’t have this perfect case where we can aim for MSP, but that in open access, firms will enter. And so suddenly, this logic of, yeah, let’s keep those fish around so it’s easy to catch, breaks down because somebody else is going to get them. So firms will enter.
And they’ll do this so long as there’s any profit at all to be able to eke out of the fishery. And so they’re going to enter so long as the total revenue is greater than the total cost.
And so, just going back a slide for a second, we have, at both the MSY and the MSP, that the total revenue is indeed greater than the total cost, but there’s all these other areas to the left, where you would have much lower stocks. And an individual fisher person wouldn’t want to do that, because it lowers their profit, but if you’re comparing it to a profit of zero, like not fishing at all because you haven’t yet entered this fishery, well, that little positive slice, yes, it’s less than that, but I can get that, I can take that for myself. And of course it’s going to hurt this person, it’s going to lower theirs too, but from the individual’s perspective, they have an incentive to gobble up that extra bit of profit all the way down until this point here, where the total revenue is no longer greater than the total cost.
And so in other words, we’ll have firms entering, driving down the stock, until that is no longer true.
And what we’re going to do for much of the rest of today is, how can we assess this? How can we figure out how to use our profit maximization approach to show that profit in open access is going to be driven down to zero here? This is that profit curve before. That’s why we liked being here, but if there’s any profit, there’s going to be incentive for firms to enter.
And we’re going to build on our intuition that we did in that graph before, but I’m also going to introduce an alternate way of looking at this.
And so, just to rehash, we so far have talked about the biological growth curve, which is saying, essentially, harvest depends on stock. But then we coupled it with economic detail on marginal benefit and marginal cost, where we said, profit depends on stock.
Those are all right and useful, but what’s the problem? Fisher folk don’t choose stock. If you’re a DNR officer, maybe you are choosing the stock, in a sense, when you’re choosing how many permits to put out there, or maybe how many fish to literally stock into the lake. But if you’re an individual fisher person, you don’t actually get to choose that amount. Instead, you just get to choose what you do, which we’re going to describe as their fishing effort.
And so, instead of Method 1, the profit stock approach, we’re going to have a second method based on effort. This makes sense, because people choose their effort, they don’t choose their stock. If you have your own lake, maybe you could say that, but not if you’re talking about the ocean.
Alright, so let’s put some details out there. Our growth function is going to stay exactly the same. And that’s just going to be that our growth is based on the intrinsic rate, R, of growth of that stock, S, times 1 minus S over the carrying capacity. So that’s exactly what we had before. But we’re going to add in one more key component called effort, in this case, fishing effort.
And we’re going to now say that harvest is going to be parameterized by this pretty simple function: Q times E times S, where Q is some data point on the catchability of the fish. So we call it the catchability coefficient. E is the effort, and so this could be thought of as the hours of fishing, or something like that. And then the last one, S, is the same as before, it’s just the stock.
And so now we’re going to have these two equations. They both have stock in them, but what we’re going to see is we can use this to reframe our equations to express the optimal effort, rather than what we were doing over here, which was how to optimize your profit based on the stock level.
Okay, so what can we do? Looking at this, we’re going to use those two equations, essentially, to solve for the optimal effort, and see what we can learn about sustainability in that case. But sort of looking ahead, we can see the fact that they both share that S thing means we can relate the effort and the growth through the stock.
The other thing that we’re going to do as a trick is we’re going to define the sustainable harvest function. There’s nothing that says that harvest has to be sustainable. We could do unsustainable harvest, that’s easy. Just catch them all. But it’s going to be, I’ll argue, really useful to relate these two to say, well, let’s assume that it is sustainable, and that is going to be setting H equals G. We saw this last lecture, that if H is equal to G, the harvest is equal to the growth, that’s going to be pretty obviously sustainable, because we’re taking out as much as is coming in the next period.
But what it lets us do here is it’s going to let us link our two effort-based equations. If G equals H, we can then say Q times E times S, that side, H equals G, is equal to R times S times (1 minus S over K).
So that’s the first step towards defining a sustainable harvest function. We’re eventually going to use this to show what happens in open access, but we’re going to manipulate it a little bit so it’s easier to use.
And so step one, like I just did, is to replace the H with the QES, but then we’re going to do a number of steps to solve this, and let’s go ahead and do that.
So what do we see here? Well, first, we got an S on the outside of both, and so we can just divide both sides by S. So QE equals R(1 minus S over K). We just got rid of that. But we can also then get rid of our R equation on the right-hand side. We’re going to want it over here, so I divide that by R, which is going to give us QE over R equals 1 minus S over K.
And now where we want to go next with this is to try to isolate S by itself, so we can move our 1 over there and get us 1 minus QE over R. And to do that, we’re also going to flip the sign on this. So make this a positive, make that a negative, so 1 minus QE over R equals S divided by K, and we’re getting there. Now we can finally see how to get S by itself, simply by pulling that K over, and now I’ll write it in a more convenient way: S equals K(1 minus QE over R).
Okay, so why have we done this? This is a more convenient way of expressing the sustainable harvest function. And so long as our effort, E, is defined according to these parameters and set equal to S, then we’re going to have a sustainable harvest. Obviously, if we change our effort, it might not be.
This is useful because now we can use it to figure out a little bit more about what individual agents, if they’re profit maximizing or whatever, are going to do, but here, choosing their effort level instead of the stock level.
And so how can we do that? The second step we have is, from our effort equation H equals QES, we can rearrange that to say something about the S. So S equals H divided by QE.
But that’s going to let us simplify a little further. And we can see that we can actually now get rid of the stock. And so, plugging in this definition from our harvest rule over there into this one, I’m going to rearrange it so I match the board: H over QE equals K(1 minus QE over R).
Now, this is a lot of steps, a lot of algebra, this is why it’s a 3,000 level course. Still not too much derivatives, but we will have one that we’ll get to in a moment.
But what we’re going to want to do is rearrange this to express the sustainable harvest with now H on the outside. And so, how do we do that? Pretty simply, we can just take our equation over there, move this QE over to here: QEK(1 minus QE over R).
And why this is useful is we’ve now derived from all that information a sustainable harvest rule that doesn’t have the stock in it. It just has the effort in it. And this is useful because we can then use it to figure out what would be the sustainable maximum solution, or the profit-maximizing solution, but where we’re going is we use it to derive what will happen in open access.
Let me pause there for a second. I know that was a bunch of algebra real quick. Any questions on that? Any clarifications?
We’re going to go through one more example, where we use real numbers instead of algebra, so hold on to your horses, that might be more fun.
Question: Where does the catchability coefficient and intrinsic growth rate come from?
That’s basically biology. And so DNR officers or biology PhDs actually go out to lakes and test. And so catchability, you could get from surveys, looking at how many fisher folk were out there, how many hours did they spend, and how many of each species did they get. And so in reality, the data would be like a different Q for each fish.
Like, my dad likes fishing for muskies, and they’re very hard to catch. So if you looked at survey data from fisher folk, they would have a low Q for muskies, but something like a largemouth bass is pretty easy to catch, and so that would have a higher one.
The intrinsic growth rate is even easier, because that you can actually just simulate in a lab, like literally have a big pond and see what happens, which coefficient for R best matches the observed data on the growth function. And so we’re not going to obviously get into those here, so we’re kind of just users of that data, but there is a whole set of disciplines behind collecting that data, and so field biology or surveying fisher folk and stuff like that.
That’s what’s kind of cool, though, is this isn’t just arbitrary theory. It is theory in the sense that we’re simplifying it down to say that this logistic growth equation is useful, but a phrase that I always go with is: Models are wrong, but sometimes they’re useful.
It’s useful not because fish literally follow that equation, but rather, when we do good biology and get these Q’s and R’s, it’s pretty darn predictive. And so that’s kind of fun. It connects data to reality, with an underlying theory of why it does it.
Any other questions?
Okay, so let’s do an example where we use some real numbers. And I think it’ll make it a little bit more specific. We’ll simplify it just a little bit.
I’ll leave that out for reference. But we’re going to do the same thing, where we have a function, but now it’s going to be harvest in terms of effort.
And so, let’s start off by just defining our sustainable yield function as this simplified case of H equals effort times (1 minus effort divided by 80). So this looks pretty similar to our logistic growth. It’s the same dynamics, but we’re now looking at it in terms of this E, the effort coefficient.
Okay, so the arc of what we’re going to do, we’re going to do three examples of playing with this. First, we’re going to look at what we did when we started out on fisheries, which was what’s the maximum sustainable yield, but in this case, expressed as effort. Then we’ll do the profit maximizing, but I’m actually going to call it net benefit maximizing, because we’re now thinking about all of society, not just the profit of a firm. And then we’ll use that to show how that’s great and all, but the third case will be like, what if we allow open access? What’s going to happen then?
Side note, this is probably the most mathematically challenging of the course days, so hang with me, we’ll make it through together. I love math, and so this is the funnest day. But not everybody likes math.
I started out that way. I was really bad at it, and I got into my PhD, and I had to literally Wikipedia what was a derivative. That’s not a great place in life to be, starting a PhD that assumes you know real analysis, all sorts of other advanced mathematics, and be like, yeah, I forgot it all. But I survived, and I’m here.
Anyways, so what we’re going to do is first use this to get the MSY calculations. What is the effort, not the stock, but the effort associated with MSY?
And so you might recall back on the last one when we had our curve like this, where we had stock. If we wanted to get the maximum, we went over this briefly, but you take the derivative of this and find where it’s zero. This is a fundamental finding in optimization theory, that you can always find an optimum of a curve by looking at all the points where it’s equal to zero. Now we’re re-expressing it, and instead of stock, it’s going to be effort. But you might notice that equation is very similar. It’s also got that curve. And so now we’re going to do the same thing.
So I’ll erase that, just because that was just to make the point briefly. You don’t need to have that one written down.
But we are going to take the derivative of this, set it equal to zero, and figure out what effort level that corresponds to, and that will be the maximum effort.
Okay, so to do this, this is the canonical way of writing these equations. I actually find it less easy intuitively than just distributing that E over. And so you don’t have to do this if you’re smarter than me, but I’m going to say E times 1 is E minus E times E is E squared.
I find this form a lot easier to take derivatives in, because now we’re going to take dH with respect to E. But now it’s a lot easier, I think. So E, let’s lower it a power down, so this is E to the 1, so it goes away, and it’s just a 1. And for this one, E squared is going to go down to E, and the 2 comes over, so it would be 2E divided by 80, which we can simplify as E over 40. So we have our derivative is just 1 minus E over 40.
So now, how do we solve for this? We want to set that equal to zero. And the way I would do it, you can solve it yourself, but I just kind of look at this, and I’m like, well, for something to be zero, it needs to be 1 minus 1. And so, what level of E would correspond with this part here equaling 1? Well, 40. Because 40 divided by 40 is 1. So this implies that E for MSY is equal to 40, because if we plug that 40 in here, 1 minus 1 is 0.
So that’s maximization theory, but applied to fisheries. We could go a little farther to say, that’s the optimal effort, but what is the actual harvest associated with that? And I’ll leave that as an exercise to you, but we’ve got E up here. And we’ve got enough information to also get out that the harvest at the maximum sustainable yield is 20.
Okay. So that was Example 1 on using effort.
But now we’re going to move to the second one, which, instead of doing this contrived maximum sustainable yield, we’re going to look at it as not maximizing yield, but maximizing benefit. So the whole last lecture, we talked about how the profit maximizing was different than the yield maximizing. Your intuition might be going off here, that even though we’re expressing it now differently in terms of effort, well, it’s similar functions, the same kind of thing is going to happen here.
And so, let’s do that. And to do this, we’re going to be defining net benefit in terms of the benefits and costs. Standard econ. Which means we’re going to take the price, just assume it’s equal to 10, and we’re going to be multiplying that by the harvest level, but we’re going to be wanting to get the socially optimal one. So we’re going to get H subscript S for socially optimal. And we’re also going to now factor in the costs. We didn’t need the costs when we were just maximizing the yield. That doesn’t have anything to do with costs, and that’s the problem. So we’re going to add in costs of just saying each unit of effort is equal to a cost of 5. And so you can think of this as the wage of the fisher people going out and doing the fishing.
So what lets us get at for the second one, which will be maximizing net benefit, we’re going to parameterize it with this equation of P times H, and I’m just going to go ahead and be subscripting it S, because we’re trying to find the socially optimal harvest. So price times that, minus the cost. So A, which I’ll plug in in a second, is 5 times E.
So that’s going to be our new function. This is related to that, but now has both price and cost included in it. Let’s go ahead and actually just put in some specific numbers. So I already said that it was a price of 10 and that the cost was 5E.
What we’re going to do first is take our net benefits function and plug in the fact that we actually know what the effort is going to be. And so the net benefit of E, we’re going to get from over here. And we’re going to plug it in in multiple locations to get a net benefit of E of the price times E multiplied by (1 minus E over 80), and then minus 5E, or minus AE. So now it’s in the right order. We just plugged in the E equation in there for our effort, and now let’s actually plug in the price and cost we have. So 10E(1 minus E over 80) minus 5E is going to be our net benefit function.
Okay, why are we doing this? Well, we want to maximize it, and so what’s the rule for maximizing something? You take the derivative and set it equal to zero, that’s just the standard way. So we’re going to take the derivative of net benefit, dNB, and we’re going to take that with respect to E, asking how does that net benefit change as the effort level changes, and for that, we can follow the same approach, distribute it out, and take the derivative. But I’ll just jump to the punchline.
You can fill in in your notebook after class, it’s going to be 10 minus E over 4 minus 5. Obviously, we can combine some terms there. Pull the 5 over there, and it’s 5 minus E over 4, and we can simplify that down. Solve for E, and we’re now going to get E star with subscript S is equal to 20.
And plug it into the net benefit function equals 50.
And so, what do we have here? We are going to have less effort than we had in the one that will maximize the catch, but now that we’re taking into account costs, it will have the same dynamic we saw before of it’s going to be better off, because essentially things are easier to catch.
Okay, now why have I spent so much time doing this? It’s because we now can move to the really interesting case of what’s going to happen if, instead of maximizing our society’s net benefits, other fisher folk are allowed to just go ahead and enter. We’re going to be able to use the same net benefit function, so let’s just write it up here for case 3: Open access.
This is going to be based off of the net benefit function, but not optimizing it. So let’s just get that function up here for clarity. So the net benefit of E is equal to 10E multiplied by (1 minus E over 80).
So we’re just taking from what we did over there, but expressing it again. And now, instead of doing the derivative and finding the optimum, we’re going to do what I call the open access trick. It’s actually a shortcut, because we know that if there’s any benefit to society that, in this case would be expressed as total revenue being greater than total cost, new firms will enter. And we know that will happen as long as it’s rational, which means it will continue until the total revenue and total cost are equal.
That actually gets us a useful bit of information. In open access, the net benefit is going to be equal to zero.
So this is different. We’re not going to take the derivative and set it equal to zero. We’re just going to take the actual function and set it equal to zero. And so that makes it pretty straightforward.
Let’s do some simplification to show our net benefit of E. We’re going to simplify to subtract that out: 5E. Then I’m just doing that trick again, where we’re going to take it to pull the 1/8 out, E squared. That’s just one way you could do it.
But frankly, the easier way is to keep it with that E factored out, and so E times (5 minus E over 8), and here we now have something where we can solve using our open access trick, simply setting it to zero.
And to solve this, it’s actually tricky because it’s got two solutions. The first one is obviously set E equals to zero. That would work. But that’s not going to be what happens. We’re looking for the one where the net benefits is driven to zero with some actual catching. It’s like, we’re not going to say that you give open access and everybody decides to quit fishing. Instead, it’s people enter and take away all that benefit, and make it into their profit. And so it’s not going to be this one, it’s going to be the one related to this one.
So this is kind of the trick, and it’s easy to miss this step. But here, we’re going to be solving for 0 equals 5 minus E over 8. This gives us 5 equals E over 8, and you can see where this is going. Pull the 8 up here, gives us E equals 40.
And that’s going to be the level of effort associated with open access. And so, it’s much higher. That’s the point, is we’ve used the same math, looked at open access with using this open access trick of knowing that people are going to keep on entering until the net benefit goes to zero, and we can actually show that what we expected to happen indeed did happen. That indeed too much fishing effort was what happened.
And so, why is this useful? It’s because it’s a nice bioeconomic model that combines econ theory with biology and the biological theory. But what does it do? It lets us make predictions. And so there’s all sorts of studies that try to do what we were talking about earlier, of use the coefficients on catchability and effort calculations and such to try to model what’s going to happen in different oceans. And so, just taking stock, Palomares et al., look at different oceans, and we see the basic conclusion is that things are getting worse. Not great. A lot of society depends on fish for their protein, and so this is not a good thing.
But what you can also do is plot out things like this, and this is how the experts in the literature will do it, looking at how good are we compared to the maximum sustainable yield, and you can see in a lot of cases we’re falling way below zero, and this means we’re not doing as good as we can.
Anyways, you won’t be tested on the data, I just mean to show you as an illustration that this ties it back to what we can observe going on in the oceans, and also give us a sense of what might we want to change. Well, taxes or these policies that we’re going to be coming to would be something that could change this calculus and make it no longer optimal to have the open access result in over-exploitation.
So that’s kind of the takeaway. Open access equals over-exploitation. This is just a restating of what we saw in our market failures, but with a lot more detail now. This is exactly the tragedy of the commons. And so the tragedy of the commons here is just saying the open access is going to lead to more and more fisher people coming in, and that will inevitably lead to tragedy, because it will drive the stock down to the point where nobody is profitable, which is bad for both humans and for fish.
Alright, any questions on that? I’m trying to save some time for comments on the midterm.
Any questions?
Alright. So, changing gears slightly, we got 8 minutes, we’re doing good.
So I think I’ve mentioned this before, but my computer’s dying. I actually bought a new computer this morning, they just released the M5 Max chip for MacBook Pro, so I’ll be a little bit better in subsequent lectures. It doesn’t come until the 17th, but I’m super psyched about it. Anyways, if you haven’t already done this, go ahead and take a look at the practice midterm. I’m going to hand out the quizzes on the way out for today. It was a short quiz, and that makes a problem because there are few questions. And so if you kind of get down the wrong path, and there’s only one question, you’ll either do really good or really poorly.
Remember that you can drop your lowest quiz, which might be this one for some people.
But the midterm will have some similar questions. And so that’s why I spent so much effort making this, which is the walkthrough. Some of this stuff is pretty easy, everybody did good on the supply and demand.
One caveat is we’ll be asking also to figure out what’s the tax revenue, and how do you do that? Well, you just re-solve supply and demand, but with a tax, and you can use that following the steps here to calculate what’s actually the tax. But I want to highlight for you this question here.
This one’s really similar to the quiz, where some people got it wrong, and I want to talk through, just very briefly, what the key insight is.
If you got that key insight, the rest of the math sort of fell in place, and if you didn’t, it was a hard quiz, because the rest of it doesn’t make sense.
But in this case, a question like this, you’re going to have two different marginal abatement cost curves, MAC1 and MAC2. And there’s going to be two different policy cases. Number one is everybody has a uniform reduction in their pollution.
And that’s one where if we plot it out, I have a chart here, but I’ll draw it here. In the quiz, I gave you specific marginal abatement cost curves. We have MAC1 and MAC2. Option 1, where everybody gets a uniform reduction, that’s going to be the case where you just draw a vertical line at whatever that E uniform is set at. And from there, you can look at what’s the total costs, simply by asking, okay, well, where did this hit the marginal cost curve? And then sum those up to see how efficient society was. Most people got that part right.
But the trick is right here. If instead you’re asked to do something where you could allow it not to be a uniform standard, but this other case where it’s a firm-specific approach, the key element that some people missed is to start with this: set MAC1 equal to MAC2. And because in a question like this, we’ll have literally given you this, it would be setting this part equal to this part, and you can then solve for the E1 and E2.
So I’ve walked through the math down here. This won’t be the exact math on the midterm, but it might be somewhat similar. And so do make sure you take a close look at that. Graphically, what that means is the MAC curves haven’t changed at all. But what we’re saying is we can have different levels of the emissions that are not uniform. But it’s illustrated by, instead of a vertical dotted line, a horizontal one, where we would have E2 is different than E1.
And the key insight is just the way I labeled it here, Firm 2 is the one that can abate emissions a whole lot cheaper, for whatever reason. And so the society optimizing, the cost-minimizing approach to do that is going to be let them. Let them use their more effective ability to abate pollution, or at least do it more cost-effectively, so that we can achieve the same standard as the uniform one, but with a lot more benefits to society.
Alright. Any questions on that?
Cool. Well then, let’s call it a day.