Lecture 5: Productive Efficiency

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Last week we talked about exchange economies in which there was already an “endowment” of goods, without talking about where those goods came from. We started by analyzing how an individual would make decisions on their own; then analyzed market behavior; and finally showed how markets could help everyone in an economy do better than they could on their own.

This week we’re going to do much the same thing, but zoom out to include the production decisions that generate the initial “endowment” of goods in the exchange model. We will be examining the simplest possible production economy: one in which agents are endowed not with goods, but with time – i.e. labor – and production technologies which allow them to produce different goods.

In the first part of this analysis we’ll expand our notion of Pareto efficiency to include production. In particular, the kind of Pareto efficiency we looked at in Lecture 2 is called allocative efficiency: a point was efficient if there was no other allocation in the Edgeworth box which could make at least one agent better off without making any agent worse off. The topic for today is productive efficiency, which we’ll define in a similar way, but in a new context: we’ll look at how agents are using their resources to produce goods, and we’ll say that their use of resources exhibits productive efficiency if there is no way those resources could have been used in a different way to produce at least as much of all goods, and strictly more of at least one good.

Next, we’ll see that when agents can buy and sell the goods they produce at market prices, their resource allocation will be productively efficient.

Finally, we will combine this model of production with the model of trade from last class: essentially, facing a set of market prices, each agent will choose both how much they supply of each good, and how much they demand; and we will show that in competitive equilibrium, the outcome we observe is both productively and allocatively efficient. Essentially, this is a much more robust (and elegant) version of the comparative advantage model you should be familiar with from Econ 1 or high school economics.

The math in this, especially with curved PPFs, gets pretty wonky pretty fast; and since this is the last week we’re doing this material, I don’t really want to get bogged down in a ton of algebra. So for the most part we’re going to do a visual and intuitive investigation of what’s going on, rather than solving math problems. As you read, therefore, try to focus on the big-picture ideas rather than the specifics of the numbers.

Finally, please note that is a new treatment of this material for me, so I don’t know exactly how it will split between Tuesday and Thursday. I’ve made the Tuesday reading a little bit long, and the Thursday reading a little bit short. We’ll see where the natural split is in lecture; my guess is we might only get to the stuff with prices in Thursday’s lecture. Thanks for bearing with me as I work out the kinks!

Resource Constraints and Production Possibilities

Time is the ultimate resource constraint: an hour spent doing one thing is one hour less spent doing something else. Much of economic activity boils down to how we spend our time.

In Econ 50, we saw how time — in the form of labor input, and in conjunction with other inputs — may be transformed into a single good via a production function. Today we will examine the issue of allocating a fixed set of resources across goods, and what effect resource constraints have on what combinations of goods we can produce. After all, an hour spent producing one good is an hour less producing another.

The standard model economists use to analyze tradeoffs like this is sometimes called a “Robinson Crusoe” or “desert island” model. Robinson Crusoe is a little problematic (have you actually read it recently? Sheesh!) so we’ll use the more modern version presented in the movie “Cast Away.” In that movie a FedEx manager named Chuck Noland is stranded on an uninhabited island somewhere in the Pacific after a plane crash. All he has are the resources on the island — stones, branches, coconut trees, fish in the water — and a few FedEx packages from the plane. With these resources, and his own time, he figures out a way to survive, mainly on the fish he catches and the coconuts he’s able to break open. In short, we have the world’s simplest production economy: Chuck has a certain amount of time, which we can divide between fishing or cracking coconuts open.

Let’s call fish “good 1” and coconuts “good 2,” so we’ll call the labor devoted to fish $L_1$ and the labor devoted to coconuts $L_2$. Finally, let’s call the total amount of time Chuck has available to him (say, hours per week) $\overline L$. In this case Chuck’s resource constraint is given by the equation $L_1 + L_2 \le \overline L$ We can visualize his tradeoff of how to spend his time by plotting “labor devoted to fish” and “labor devoted to coconuts” on a graph:

See interactive graph online here.

Given his resource constraints, Chuck is going to be limited in the number of fish he can catch, and the number of coconuts he can break open. Some “bundles,” or combinations of outputs, will be possible for Chuck to produce; others will not. We will call Chuck’s “feasible set” of output bundles his production possibilities set. The boundary between that set and the set of bundles he cannot produce we’ll call his production possibilities frontier. This is essentially the same concept as a budget set, only with the limiting resource being time rather than money.

If Chuck is not using all of his resources – or is using them inefficiently – he will produce at a point strictly within his production possibilities set. At such a point, he can produce more fish without necessarily producing fewer coconuts, and vice versa. Along the PPF, however, he faces a tradeoff: if he wants to produce more fish, he needs to produce fewer coconuts. Therefore, for a single agent, productive efficiency means producing at a point along one’s own PPF.

Chuck’s PPF is related to his resource constraints via his production functions: depending on how he can transform labor into fish and coconuts, different combinations of output will be available to him. Note that, just like a budget set or an Edgeworth Box, the PPF exists in good 1 - good 2 space!

We’re going to introduce a small notational change, though. We’re still going to say that $X = (x_1,x_2)$ represents a consumption bundle in good 1 - good 2 space; but to distinguish consumption from production, we’re going to say that the bundle Chuck produces is $Y = (y_1,y_2)$.

Example 1: Linear PPF

Let’s start with a simple case of linear technologies: assume that Chuck has 12 hours of labor, and with each hour of labor he can catch 2 fish or collect 1 coconut. This simple setup is really saying that there is an economy with a single resource (labor) which can be used in one of two production functions: the quantity of fish, $y_1$, is given by $y_1 = f_1(L_1) = 2L_1$ and the quantity of coconuts, $y_2$, is given by $y_2 = f_2(L_2) = L_2$ For any division of labor $(L_1,L_2)$, the production functions tell us how many fish and coconuts Chuck will produce. Therefore, the set of feasible production possibilities depends on the set of feasible resource allocations.

We can illustrate this by viewing the resource constraint and the PPF side-by-side, as shown below. Drag the orange dot all the way to the right: we can see that if Chuck devotes all 12 hours of labor to producing fish, he can produce $12 \times 2 = 24$ fish. Likewise, if you move the orange dot all the way up and to the left, you can see that if Chuck devotes all 12 hours to coconuts, he can produce $12 \times 1 = 12$ coconuts.

See interactive graph online here.

Again, this is just like a budget line: instead of money, Chuck has $\overline L = 12$ hours. Each fish “costs” half an hour to produce, and each coconut “costs” one hour to produce. Hence, we can think of this as being like a budget line where $p_1 = {1 \over 2}$, $p_2 = 1$ and $m = 12$.

Opportunity Cost and the Marginal Rate of Transformation

Anywhere along the PPF, Chuck cannot make more of both goods: if he wants to produce more fish, he needs to produce fewer coconuts, and vice versa. The slope of the PPF measures the rate at which his available technology allows him to trade off between two goods. In particular, it represents the opportunity cost of producing an additional unit of good 1, in terms of units of good 2 given up. We call this slope the marginal rate of transformation, or MRT.

As with the other slopes we’re familiar with in good 1 - good 2 space (i.e., the price ratio and the MRS), we will treat the MRT as a positive number, since we know it represents a tradeoff and will (nearly) always be negative.

The curvature of the MRT is clearly related to the nature of the production functions for the two goods. To see how, think of what happens as we move to the right along the PPF. When there’s only one input — labor — this means shifting a single hour of labor from producing good 2 to producing good 1. Since the $MP_L$’s of the production functions measures the amount of output produced by the last hour of labor, it follows that spending one less hour producing good 2 means we give up approximately $MP_{L2}$ units of good 2; and likewise, when we spend one more hour producing good 1, we gain $MP_{L1}$ more units of good 1. Therefore, $MRT = \frac{\Delta x_2}{\Delta x_1} = \frac{MP_{L2}}{MP_{L1}}$ For example, in the derivation of the PPF above we had $f_1(L_1) = 2L_1$ $f_2(L_2) = L_2$ therefore $MP_{L1} = df_1(L_1)/dL_1 = 2$ $MP_{L2} = df_2(L_2)/dl_2 = 1$ Hence $MRT = \frac{MP_{L2}}{MP_{L1}} = \frac{1}{2}$ Indeed, if you look at the graph of the PPF above, you can see that it has a constant slope of $-\frac{1}{2}$. In other words, when the marginal products are constant, the opportunity cost of producing your first unit of good 1 is the same as the opportunity cost of producing the last (or any other); so the MRT is constant, and the PPF is a straight line.

Example 2: A Curved PPF

In the previous example, Chuck’s MRT was constant (i.e. the PPF was a straight line) because the marginal products of labor were constant. However, if the marginal products of labor change, the MRT will also change.

For example, suppose Alison is also stranded on a desert island, but her production functions exhibit diminishing marginal products of labor: $y_1 = f_1(L_1) = \sqrt{12L_1}$ $y_2 = f_2(L_2) = \sqrt L_2$ Her marginal products of labor aren’t constant: $MP_{L1} = {df_1(L_1) \over dL_1} = \sqrt{\frac{3}{L_1}}$ $MP_{L2} = {df_2(L_2) \over dL_2} = \sqrt{\frac{1}{4L_2}}$ and the MRT changes as you move along the PPF: $MRT = \frac{MP_{L2}}{MP_{L1}} = \sqrt{L_1 \over 12L_2}$ This is increasing as we move along the PPF to the right — that is, as we devote more labor to good 1, $L_1$ increases and $L_2$ decreases, and the slope of the PPF gets steeper.

Let’s think about why the MRT increases as you move to the right along the PPF for these production functions. Because they exhibit diminishing marginal products of labor, as Alison spends more time fishing ($\uparrow L_1$) and less time on coconuts ($\downarrow L_2$), her $MP_{L1}$ is decreasing and $MP_{L2}$ is increasing. Intuitively, each hour of additional fish production produces less and less additional fish, while each hour she subtracts from coconuts production would have produced an increasing number of coconuts. Therefore, her opportunity cost of producing an additional fish is increasing as she produces more fish.

Visually, you can see this in the following set of diagrams. The large square diagram on the left shows the PPF if Alison has $\overline L = 16$ hours of labor, and these two production functions; the two smaller diagrams show the production functions for fish and coconuts. Try shifting labor by moving the points in the smaller diagrams left or right; and see what happens to the marginal products of labor and the $MRT$:

See interactive graph online here.

Productive efficiency with two producers

So far we’ve derived the PPF and analyzed the MRT for a situation with a single producer. Let’s now introduce a second producer in each of the scenarios, and see how this affects our idea of productive efficiency.

Linear PPFs and Comparative Advantage: Chuck and Wilson

Let’s start with Chuck, and assume he meets another marooned agent named Wilson. (Yes, I know Wilson is a volleyball, and can’t produce anything. Work with me here, we need another agent.) Wilson also has $\overline L = 12$ hours, but is more productive than Chuck in producing both goods: he can produce 3 fish or 3 coconuts per hour! Therefore, he can produce at most 36 fish, or 36 coconuts. You can see his PPF, alongside Chuck’s for comparison, in the diagram below:

See interactive graph online here.

For example, suppose both Chuck and Wilson were evenly distributing their labor between fish and coconuts. This would result in Chuck producing 12 fish and 6 coconuts, and Wilson producing 18 fish and 18 coconuts; therefore, between them, they would produce 30 fish and 24 coconuts. We can see this choice as an allocation in an Edgeworth Box with those dimensions, as shown below:

See interactive graph online here.

Is this production choice productively efficient? Again, we say that a joint production decision is efficient if we couldn’t reallocate resources to produce more of at least one good without making less of any goods. From this standpoint, it’s easy to see that this is not, in fact, a productively efficient choice.

To see why, note that Chuck’s MRT (i.e. the slope of his PPF, and his opportunity cost of producing fish) is lower than Wilson’s. In particular, Chuck can produce a fish giving up only half a coconut; Wilson, on the other hand, must give up an entire coconut for every fish he produces. Because Chuck can produce fish at a lower opportunity cost (in terms of coconuts) than Wilson, we say that Chuck has a comparative advantage in the production of fish. Likewise, Wilson’s opportunity cost of a coconut is one fish, while Chuck’s is two fish; hence Wilson has a comparative advantage in the production of coconuts.

Suppose, then, that Chuck produces 6 more fish, and Wilson produces 6 fewer. If you drag the dots in the diagram above, you can see that while they’re producing the same amount of fish, they’re now producing 3 more coconuts…so the Edgeworth Box has grown in size! In other words, since we were able to increase production of one good without decreasing production of the other, our first production choice (in which both Chuck and Wilson split their time evenly) wasn’t efficient.

Visually, we can see whether a point is efficient or not by plotting the PPFs within the Edgeworth box implied by the production choice. Move the production choice back to the original allocation, and check the box labeled “Show PPFs in the Edgeworth Box;” this will show the PPFs in the Edgeworth Box diagram, with Chuck’s origin being the bottom-left corner and Wilson’s the top-right. You can see that there is an overlap in the area below and to the right of the allocation in the Edgeworth box. This is a little like the overlap we saw in their indifference curves, but it has a slightly different interpretation: allocations within that area would have the same total amount of fish and coconuts, but both Chuck and Wilson would be producing at a point inside their own individual PPFs. Put another way, the same total amount of fish and coconuts could be produced with less total labor.

So what would be an efficient point? To answer this, suppose we say that Chuck and Wilson together would like to produce 30 fish. What is the maximum number of coconuts they could produce? If we repeat the exercise above, we can see that they can get another 3 coconuts if Chuck produces 6 additional fish, and Wilson produces 6 fewer. But at this point Chuck is producing his maximum number of fish (i.e., he is completely specializing in fish), while Wilson is producing 6 fish and 30 coconuts. Therefore, if they want 30 fish, they can collectively have at most 30 coconuts; and they do so by Chuck specializing in fish, and Wilson dividing his time between fish and coconuts. At his point, we can see that there is no overlap of their PPFs within the Edgeworth box.

More generally, for any quantity of fish the two of them can produce, there is some maximum amount of coconuts that they would also produce, and vice versa. These points constitute the joint PPF for Chuck and Wilson: that is, the boundary between combinations of fish and coconuts that they can jointly produce, and those they cannot – i.e., the set of productively efficient output combinations. To see the joint PPF in this case, select “Show joint PPF” in the diagram above.

And in fact, what you can see is that to get to the joint PPF, at least one of the two must be specializing in the good for which they have comparative advantage:

If they both specialize (Chuck produces only fish, and Wilson only coconuts), the produce at the point $Y = (24,36)$ – i.e., the kink.
If Chuck specializes and Wilson mixes, they produce along the orange portion of the joint PPF to the right of the kink. Along this segment, the MRT is 1, which is Wilson’s MRT, because producing another fish requires Wilson to fish a bit more and collect coconuts a bit less.
If Wilson specializes and Chuck mixes, they produce along the blue portion of the joint PPF to the left of the kink. Along this segment, the MRT is ${1 \over 2}$, which is Chuck’s MRT, because producing another fish requires Chuck to fish a bit more and collect coconuts a bit less.

This makes sense: if we start at the vertical intercept where both of them are producing coconuts, and think to ourselves “Who would we want to start producing fish?” – the answer has to be Chuck, because he can produce those first fish at a lower opportunity cost. Hence, the PPF starts out with the same slope as Chuck’s PPF, until Chuck is fully specializing in fish. Beyond that, Wilson would also need to produce some fish; so the PPF has the same slope as Wilson’s to the right of that point. At the kink itself, therefore, Chuck is specializing in fish, and Wilson is specializing in coconuts.

Let’s take this intuition a bit further and find the joint PPF for two agents with curved PPFs.

Curved PPFs: Alison and Bob

We analyzed Alison’s curved PPF above; let’s now introduce her to a new friend, Bob. (This is kind of like the “origin story” for our Edgeworth Box model.) Like Alison, Bob has $\overline L = 16$ hours of labor and can produce fish according to the production function $f_1(L_1) = \sqrt{12L_1}$ However, he is better at cracking open coconuts: his production function for good 2 is $f_2(L_2) = \sqrt{6L_2}$ This means his MRT is $MRT = {MP_{L2} \over MP_{L1}} = \sqrt{L_1 \over 2L_2}$ Their two PPFs may be seen in the diagram below, along with the Edgeworth box diagram showing their (joint) production decision:

See interactive graph online here.

To determine productive efficiency, let’s use some intuition. We saw before that an allocation for Chuck and Wilson wasn’t efficient if there was an overlap of their PPFs in the Edgeworth box. In order for their to be no such overlap for Alison and Bob, since their PPFs are curves, it means their PPFs must be tangent to one another: that is, their MRT’s must be the same. This makes sense: as we saw above, if their MRT’s were different, they could produce more of both goods by shifting fish production to the person with the lower opportunity cost of producing fish, and shifting coconut production to the person with the lower opportunity cost of producing coconuts.

For example: at the initial production choice shown above, at which Alison and Bob are each producing 10 fish, Alison’s MRT is about 0.3, while Bob’s is about 0.74. Since these aren’t equal, this isn’t an efficient production choice: in particular, since Alison’s opportunity cost of producing fish is lower than Bob’s, they could produce the same 20 fish and have more coconuts if Alison produced a few more fish and Bob produced a few less. (You can see that this isn’t efficient if you show the PPFs in the Edgeworth box.) Now drag Alison’s production up from 10 to 12, and Bob’s down from 10 to 8. They’re still producing 20 fish in total, but now their coconut production has increased from 9.55 to 10, and we can see that their MRT’s are now both equal to 0.5. This is a productively efficient choice, and the total quantities of fish and coconuts lie on their joint PPF, as you can see if you check the box to show it.

One way to find the joint PPF is to choose some output combination for Alison; see what her MRT is; and then adjust Bob’s production until his MRT is the same. You’ll end up with an Edgeworth box in which their PPFs are tangent, and the total production will lie somewhere along the joint PPF. (It’s possible, of course, to solve for the equation of the joint PPF, but it’s cumbersome and well beyond the scope of this course.)

It’s not obvious that the world in which we live is productively efficient. However, in the happy world of perfect competition, there’s one way that we can achieve efficiency, which will (hopefully) come as zero surprise to you at this point: the market.

Selling fish and coconuts: prices and productive efficiency

Let’s wrap up today’s lecture by thinking about what our four castaways would do if there was a market for fish and coconuts: that is, the price of fish is $p_1$, and the price of coconuts is $p_2$, and everyone can sell their fish and coconuts at these prices. (We’ll get to the demand side on Thursday; for now, we’re just thinking about prices and production.) How can each of the agents maximize the market value of their production choice $Y = (y_1,y_2)$ – that is, $p_1y_1 + p_2y_2$?

We can return to a gravitational-pull style argument to analyze each producer’s optimal choice. If one of them spends an additional hour fishing, they will produce $MP_{L1}$ fish, each of which they can sell for $p_1$; so their increase in revenue from spending another hour fishing is approximately $\frac{\text{dollars}}{\text{hour fishing}} \approx MP_{L1} \frac{\text{fish}}{\text{hour fishing}} \times p_1 \frac{\text{dollars}}{\text{fish}}$ Because that hour spent fishing is one less hour collecting coconuts, in that hour they forgo $MP_{L2}$ coconuts, each of which they could have sold for $p_2$; so his decrease in revenue from spending one less hour collecting coconuts is $\frac{\text{dollars}}{\text{hour collecting coconuts}}\approx MP_{L2} \frac{\text{coconuts}}{\text{hour collecting coconuts}} \times p_2 \frac{\text{dollars}}{\text{coconut}}$ They should therefore devote more time to fishing, and less to collecting coconuts, when $MP_{L1} \times p_1 > MP_{L2} \times p_2$ If we cross multiply, we can see that this is the same as the condition $\frac{p_1}{p_2} > \frac{MP_{L2}}{MP_{L1}}$ or ${p_1 \over p_2} > MRT$ Conversely, they should devote less time to fishing, and more to collecting coconuts, when $p_1/p_2 < MRT$.

Maximizing revenue with curved PPFs

Let’s apply this logic first to Alison. The following graph shows one possible production choice along her PPF. The market prices are shown by the sliders. Initially, the prices are $p_1 = 2$, $p_2 = 4$, $y_1 = 6$, and $y_2 = 3.61$. Therefore, the market value of this production choice is $m(Y) = p_1y_1 + p_2y_2 = 2 \times 6 + 4 \times 3.61 = 26.41$ However, this isn’t the best she can do. Her MRT at this point is about 0.14, which is less than the price ratio of $p_1/p_2 = 2/4 = 0.5$. Therefore, she can increase the value of her output if she moves her production to the right along her PPF, producing more fish and fewer coconuts. If you drag her production point to the right, you can see that the value of her output is maximized at the point $(12,2)$:

See interactive graph online here.

Notice that as long as Bob faces the same prices as Alison, he too will choose a point along the PPF where his MRT is equal to the price ratio. This means that both Alison and Bob will set their MRT equal to a common value, meaning that their MRT’s are equal – and that their joint decision is productively efficient! Furthermore, because both of their MRT’s are equal to the price ratio, it follows that the MRT at the point along the PPF is also equal to the price ratio – so Alison and Bob together, even though they’re not coordinating, not only choose a point along the joint PPF, but they choose the point with the highest market value!

See interactive graph online here.

Maximizing revenue with linear PPFs

OK, this has been a lot. Let’s finish by looking at something you probably learned about in high school: complete specialization with linear PPFs.

In the special case of a linear PPF, the above argument means that unless $p_1/p_2 = MRT$, Chuck will completely specialize in either fish or coconuts. To see why, let’s return to our concrete example from above.

In the derivation of our first PPF, we said that in one hour, Chuck could produce 2 fish or 1 coconut. If the price of fish is $p_1 = 3$ and the price of coconuts is $p_2 = 4$, this means that he could either make $MP_{L1} \times p_1 = 2 \text{ fish} \times €3/\text{fish} = €6$ producing and selling fish, or $MP_{L2} \times p_2 = 1 \text{ coconut} \times €4/\text{coconut} = €4$ producing and selling coconuts. Since this is true of every hour he has, he should only produce fish.

For more general prices, he should only produce fish if $2p_1 > p_2$, or $p_1/p_2 > 1/2$; that is, if he should produce only good 1 if $p_1/p_2 > MRT$!

The following graph illustrates this point. Bundle $Y$ is Chuck’s production choice; the green line passing through it is all the other bundles which the same monetary value. Try moving $Y$ along the PPF to see how Chuck can maximize the value of what he produces. Then, change the prices, and see how his optimal production choice changes!

See interactive graph online here.

Of course, the same is true for Wilson – and just like the situation with Alison and Bob, we can see that when both Chuck and Wilson respond to the market prices, they will always choose a productively efficient point – that is, a point along their joint PPF:

See interactive graph online here.

Again, note that even though Chuck and Wilson are not coordinating, they are choosing a point along the joint PPF that maximizes the total combined value of their production: that is, it’s “as if” a social planner were maximizing the joint value of production. But no social planner is needed: the “invisible hand” of the market ensures that the total (market) value of the goods is maximized.

Summary and next steps

Today we looked at what made a production choice efficient; we analyzed efficiency with two producers in the case of linear PPFs and curved PPFs; and we saw that if all producers take a common set of market prices as given, and choose the output combination that maximizes the value of what they produce, they will choose a productively efficient output combination (i.e., one that lies along their joint PPF) – and what’s more, they will choose the most valuable such combination.

In our next and final lecture on this topic, we’ll bring demand back in, and see how the two agents’ preferences determine which productively efficient point is chosen.

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