# 15.4 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 Chapter 14 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:
\(\mathcal{L}(A, E, S, T) = \overbrace{8 \ln A}^{TB^A} + \overbrace{8 \ln E}^{TB^E} - (\overbrace{S^2 + 8}^{TC^S} + \overbrace{2T^2 + 4}^{TC^T}) + \lambda[(S + T) - (A + E)]\)
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. :)