Lecture 20: The Efficiency of Markets

The social planner’s problem

Any economic model of equilibrium, such as the partial equilibrium model we analyzed in the last chapter, makes some assumptions about what motivates agents to engage in some economic behavior; posits an economic environment in which they interact; and finally, posits an equilibrium outcome to which the system converges.

When evaluating such a model, we may ask ourselves whether the outcome it describes represents a “good” outcome. There are plenty of economic models with “bad” outcomes; for example, one famous game theoretic model is called the “Prisoners’ Dilemma,” and illustrates that in certain circumstances, agents maximizing their own individual payoffs actually do worse than if they could coordinate with one another.

To answer the question of whether the outcome of a model is “good,” we often invoke the thought experiment of a disinterested, benevolent “social planner.” By disinterested we mean that this is someone who is not one of the agents in the model, and so has no personal stake in the outcome. (This is different from, say, an auctioneer who is designing a system to maximize their own revenue.) By benevolent we mean that this “social planner” is seeking to maximize the total social welfare in an economy.

How would such a “social planner” answer the fundamental economic questions of how much of a good to produce, how to produce it, and who should consume it? We might posit some criteria:

• We want the quantity of the good to produce, $Q^*$, to maximize the total “welfare” — that is, the total benefit to people minus the total cost of production.
• We want to produce this quantity $Q^*$ in the lowest-cost way possible.
• We want to distribute this quantity $Q^*$ to the people who value it the most.

Let’s approach the first question first, by analyzing a simple model in which there is just one consumer and one firm. We’ll then move on to a model of multiple consumers and multiple firms to address the other two questions.

Let’s start by analyzing the last point: how much to produce to maximize total surplus.

Choosing the quantity that maximizes “total welfare”

To analyze total welfare, let’s posit that there is a single firm producing sandwiches, and a single consumer who buys sandwiches. Let’s look first at the benefit to the consumer; then at the cost to the firm; and then bring together benefit and cost to find the optimal quantity.

Total benefit for the consumer

For reasons that will be made clear later, let’s assume that the consumer’s preferences over sandwiches ($x_1$) and money spent on other goods ($x_2$) may be represented by the quasilinear utility function \(u(x_1,x_2) = 10x_1 - \tfrac{1}{2}x_1^2 + x_2\) Note that this utility function is of the form \(u(x_1,x_2) = v(x_1) + x_2\) Since good 2 is “money spent on other things,” the price of a unit of good 2 must just be $p_2 = 1$; and furthermore, since having one more dollar raises utility by 1, (Note: One way to confirm this is to solve the utility maximization problem using the Lagrange method, and see that the value of the Lagrange multiplier $\lambda$ must be 1.) we’re actually assuming that we’ve normalized this utility function so that utility is actually measured in dollars.

With this interpretation, the function \(v(x_1) = 10x_1 - \tfrac{1}{2}x_1^2\) may be thought of as the total benefit function, giving the total benefit (in dollars) of consuming $x_1$ sandwiches. Visually, we can see this value two ways: as the height of the curve $v(x_1)$ at quantity $x_1$, or as the integral of the marginal benefit curve, $v^\prime(x_1)$, from 0 to $x_1$:

This “total benefit” is sometimes referred to as gross consumer’s surplus.

Total cost to the firm

Again for reasons that will be made clear later, let’s analyze the case of a firm with no fixed costs; specifically, one whose cost of making $q$ sandwiches is given by \(c(q) = q + \tfrac{1}{4}q^2\) As we’ve done many times, we may graph the total and marginal costs. Let’s note, though, that the total (variable) costs of producing $q$ units may be given by the integral of marginal costs from 0 to $q$:

Note that from a social welfare perspective, the area under the MC curve represents the total opportunity cost of using the variable inputs required to produce $q$ units of output, in terms of value given up producing other goods. In other words: suppose this total variable cost represents the cost of the labor required to produce $q$ units of output. This variable cost, $wL(q)$, represents the market value of that labor: that is, the value that same labor could bring elsewhere in the economy if it were used to produce something else. Thus we can think of the total cost as being the benefit lost to consumers from other goods that aren’t produced because the labor to produce them is being used to make sandwiches instead.

Total welfare

So, if this were the only firm and consumer in the economy, what would the optimal quantity of sandwiches, $Q^\star$, be?

We can posit that the total welfare in the society from $Q$ sandwiches, $W(Q)$, would be the total benefit to the consumer minus the total cost to the firm: \(\begin{aligned}\textcolor{#2ca02c}{W(Q)} &= \textcolor{#1f77b4}{TB(Q)} - \textcolor{#d62728}{TC(Q)}\\ &= \textcolor{#1f77b4}{10Q - \tfrac{1}{2}Q^2} - \textcolor{#d62728}{\left[Q + \tfrac{1}{4}Q^2\right]}\end{aligned}\) To maximize this difference, we can take the derivative and set it equal to zero. Unsurprisingly, this chooses the point where the the marginal benefit of the last unit consumed by the consumer is just equal to the marginal cost to the firm of producing that good: \(\begin{aligned}\textcolor{#2ca02c}{W^\prime(Q)} = \textcolor{#1f77b4}{MB(Q)} - \textcolor{#ff7f0e}{MC(Q)} &= 0 \\ \textcolor{#1f77b4}{MB(Q)} &= \textcolor{#ff7f0e}{MC(Q)} \\\textcolor{#1f77b4}{10 - Q} &= \textcolor{#ff7f0e}{1 + \tfrac{1}{2}Q}\\Q^\star &= 6\end{aligned}\)

Visually, we can see total welfare as either the vertical distance between $TB(Q)$ and $TC(Q)$, or the area above the $MC(Q)$ curve and below the $MB(Q)$ curve. In the diagram below, drag the quantity to the right and left and see what happens to total welfare. Confirm for yourself that welfare is maximized at $Q^\star = 6$:

Why is producing any quantity other than 6 suboptimal?

• If $Q < 6$, then $MB > MC$; so producing additional units would benefit the consumer more than they would cost the firm.
• If $Q > 6$, then $MB < MC$; so the last few units produced cost the firm more than they benefited the consumer.

Now that we know what the optimal quantity to produce is, let’s look at why competitive equilibrium results in this optimal quantity.

Consumer and producer surplus

Up to now there has been no mention of prices; all we were doing was solving for the optimal quantity. So, how does the equilibrium price in competitive markets result in the optimal quantity? Fundamentally, our model of consumer choice tells us that consumers maximize their utility by setting their marginal benefit equal to the price, while firms set price equal to marginal cost. Therefore, as long as both consumers and firms are price takers and “take” the same price, markets will set consumers’ marginal benefit equal to firms’ marginal cost.

Let’s break that down a bit more precisely, by analyzing how total welfare may be thought of as the sum of consumer’s surplus and producer’s surplus.

Consumer’s Surplus

Recall that our consumer in this example had the quasilinear utility function \(u(x_1,x_2) = v(x_1) + x_2\) where $x_2$ was “dollars spent on other goods.” If this consumer has income $m$ and can buy as much good 1 at price $p_1$, then the amount of money they’ll have to spend on other goods if they buy $x_1$ units of good 1 is just $x_2 = m - p_1x_1$. Therefore we can write their utility just in terms of $x_1$ as \(u(x_1) = v(x_1) + m - p_1x_1\) Note that this is their income $m$, plus their total benefit from good 1 $v(x_1)$ minus their total expenditure on good 1, $p_1x_1$. We can therefore define their (net) consumer’s surplus as their total benefit from good 1, minus the total expenditure: \(CS(x_1) = \overbrace{v(x_1)}^{TB} - \overbrace{p_1x_1}^{TE}\) Visually, this is the vertical distance between the total benefit curve and a line with slope $p_1$. It’s also the integral, from 0 to $x_1$, of the marginal benefit minus the price — that is, the area of total benefit, minus the green rectangle with width $x_1$ and height $p_1$ representing their total expenditure, $p_1x_1$:

Intuitively, consumer’s surplus is therefore the sum of the value the consumer gets from each unit, above and beyond the price they have to pay for it.

If you set any price in the diagram above, you can see that CS is maximized when the consumer buys the quantity that sets their $MB = p$. Intuitively, if they buy less than that amount, they are giving up their potential surplus from the additional units which would bring them a benefit above and beyond the price; and if they buy more than that amount, they’re paying more for the last units than they are receiving benefit from those units. (This reduction in surplus is shown as the negative number in the shaded area below the price line and above the MB curve.)

Producer’s Surplus

For firms, producer’s surplus is defined as their total revenues minus their total variable costs. (This is slightly different than profits, because profits are defined as total revenues minus all total costs, including fixed costs.) To avoid dealing with this disparity, we’ve been looking at a firm with only variable costs in this example; hence producer’s surplus and profits are the same for this example.

Intuitively, producer’s surplus is therefore the sum of the revenue the producer gets from selling each unit, above and beyond the cost they have to pay to produce it.

If you set any price in the diagram above, you can see that PS is maximized when the firm produces the quantity that sets their $MC = p$. Intuitively, if they produce less than that amount, they are giving up their potential surplus from the additional units which would bring them a benefit above and beyond the cost; and if they produce more than that amount, they’re paying more to produce the last units than they are receiving revenue from selling those units. (This reduction in surplus is shown as the negative number in the shaded area above the price line and under the MC curve.)

Total surplus

Let’s bring the consumer and the producer together into a market. Let’s have $Q$ be the quantity bought and sold in the market, and $P$ be the market price. Using the analysis above, we can see that consumers try to maximize $CS = TB(Q) - P \times Q$, while firms try to maximize $PS = P \times Q - TC(Q)$. Note that if we add these together, we get \(\begin{aligned} CS + PS &= TB(Q) - P \times Q + P \times Q - TC(Q)\\ &= TB(Q) - TC(Q)\\ \end{aligned}\) which was our definition of total welfare, or $W(Q)$. Note that this implies that a “social planner” cares only about the benefit to the consumer and the cost to the firm, and doesn’t care at all about the transfer of money between them.

Remember that the social planner wanted to choose the quantity that set $MB(Q) = MC(Q)$. Well, since the consumer sets $P = MB(Q)$ to maximize their surplus, and the firm sets $P = MC(Q)$ to maximize its surplus, having a single price in the market coordinates their activities so that $MB(Q) = MC(Q)$. Therefore, at the competitive market equilibrium, total welfare is maximized:

Allocative efficiency of markets

In the above analysis, we modeled the economy “as if” it were a single representative consumer and a single firm, and found that competitive markets resulted in the overall efficient quantity of a good. What, though, if consumers and firms are different? What can we say about the allocative efficiency of markets: that is, for a given overall quantity of a good, do markets distribute that good in the most efficient way possible?

Let’s return to our example from last time of Adam and Eve, and Subway and Togo’s. Let’s use the following notation to keep all our quantities straight: \(\begin{aligned} A &= \text{ number of sandwiches Adam consumes}\\ E &= \text{ number of sandwiches Eve consumes}\\ S &= \text{ number of sandwiches Subway produces}\\ T &= \text{ number of sandwiches Togo's produces}\\ \end{aligned}\) If you’ll recall, Adam and Eve’s total benefit from sandwiches, measured in dollars, were given by \(v^A(A) = 8 \ln A\) \(v^E(E) = 4 \ln E\) and Subway and Togo’s total cost functions were \(c^S(S) = S^2 + 8\) \(c^T(T) = 2T^2 + 4\) Let’s now frame the social planner’s problem as a constrained optimization problem: what values of $A, E, S$, and $T$ would maximize total welfare (total benefit to consumers minus total costs to firms) \(W(A, E, S, T) = 8 \ln A + 4 \ln E - (S^2 + 2T^2)\) subject to the constraint that the total number of sandwiches produced equals the total amount consumed: \(A + E = S + T\) As usual, we can set up a Lagrangian: \(\mathcal{L}(A, E, S, T) = \overbrace{8 \ln A}^{TB^A} + \overbrace{4 \ln E}^{TB^E} - (\overbrace{S^2 + 8}^{TC^S} + \overbrace{2T^2 + 4}^{TC^T}) + \lambda[(S + T) - (A + E)]\) The first order conditions of the optimization problem are \(\begin{aligned} {\partial W \over \partial A} &= \overbrace{8 \over A}^{MB^A} - \lambda = 0 & \Rightarrow \lambda = {8 \over A} \equiv MB^A\\ {\partial W \over \partial E} &= \overbrace{4 \over E}^{MB^E} - \lambda = 0 & \Rightarrow \lambda = {4 \over E} \equiv MB^E\\ {\partial W \over \partial S} &= -\overbrace{2S}^{MC^S} + \lambda = 0 & \Rightarrow \lambda = 2S \equiv MC^S\\ {\partial W \over \partial T} &= -\overbrace{4T}^{MC^T} + \lambda = 0 & \Rightarrow \lambda = 4T \equiv MC^T\\ {\partial W \over \partial \lambda} &= (S + T) - (A + E) = 0 & \Rightarrow A + E = S + T\\ \end{aligned}\) What do these tell us? The first two tell us that the social planner will allocate sandwiches between Adam and Eve such that their marginal benefit from the last sandwich is the same; that is, \({8 \over A} = {4 \over E} \Rightarrow A = 2E\) Because Adam likes sandwiches twice as much as Eve does, this means that Adam should consume twice as many sandwiches as Eve.

Likewise, the next two tell us that the social planner will allocate the production of sandwiches such that the last sandwich produced by Subway has the same marginal cost — that is, uses the same amount of resources — as the last sandwich produced by Togo: \(2S = 4T \Rightarrow S = 2T\) So, since Subway has a lower cost of producing sandwiches, it should produce twice as many sandwiches as Togo.

OK, so let’s go ahead and solve this system of equations. We can use the first four FOC’s to write $A$, $E$, $S$, and $T$ in terms of $\lambda$: \(\begin{aligned} A &= {8 \over \lambda}\\ E &= {4 \over \lambda}\\ S &= {\lambda \over 2}\\ T &= {\lambda \over 4} \end{aligned}\) Plugging this into the final FOC gives us \(\begin{aligned} {8 \over \lambda} + {4 \over \lambda} &= {\lambda \over 2} + {\lambda \over 4}\\ {12 \over \lambda} &= {3\lambda \over 4}\\ \lambda^2 &= 16\\ \lambda &= 4 \end{aligned}\) and therefore $A = 2$, $E = 1$, $S = 2$, and $T = 1$.

Recall what we found in Chapter 14:

• Adam set $MU^A = p$, which resulted in his optimal condition $A = 8/p$
• Eve set $MU^E = p$, which resulted in $E = 4/p$
• Subway set $MC^S = p$, which resulted in $S = p/2$
• Togo’s set $MC^T = p$, which resulted in $T = p/4$

These are the exact same conditions as the first four FOC’s of the social planner’s problem, with the price $p$ serving the same mathematical purpose as the Lagrange multiplier $\lambda$. Then, to solve for the market equilibrium price, we set demand equal to supply: \(\begin{aligned} D(p) &= S(p)\\ {8 \over p} + {4 \over p} &= {p \over 2} + {p \over 4}\\ {12 \over p} &= {3p \over 4}\\ p^2 &= 16\\ p &= 4 \end{aligned}\) This is the exact same thing as plugging the first four FOC’s into the last one (i.e. the constraint).

OK, but what does that mean?

We’ve established that the market price serves the same purpose (and is mathematically equivalent to) the $\lambda$ for a social planner’s constrained optimization problem. But what does that mean intuitively?

Think about the problem of how a given quantity of a good should be produced. If it’s being produced in the most efficient way possible, you couldn’t reallocate some of the production from one firm to another and have a lower cost of production. This means the marginal cost of producing the last good must be equal across firms. In other words, $MC^S = MC^T$. The $\lambda$ in the social planner’s problem is set equal to each of the firms’ marginal costs; in a competitive market, this is achieved by all firms setting $MC = p$.

Likewise, think about the problem of how a given quantity of a good should be distributed. If Adam and Eve got a different marginal benefit from the last sandwich consumed, in terms of their marginal rate of substitution of dollars for sandwiches, then the one who valued their last sandwich less would be willing to sell that sandwich to the other person, and they would both be made better off. Therefore, if an allocation is efficient, it must be the case that Adam and Eve have the same MRS at their optimal quantity. The $\lambda$ in the social planner’s problem is set equal to each of the consumers’ MRS; in a competitive market, this is achieved by all consumers setting $MRS = p$.

In other words: even with billions of people and millions of firms, as long as everyone sets their own personal marginal benefit of doing something to their own personal marginal cost of doing it, the end result is the same as if a social planner were maximizing overall welfare.

Caveats

There are many. We’ll talk about them in class. :)

Reading Quiz

That's it for today! Click here to take the quiz on this reading.