24.1 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.
Maximizing total surplus
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 $x_1$:
[ See interactive graph online at https://www.econgraphs.org/graphs/competition/welfare/total_and_marginal_benefit ]
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 $q$:
[ See interactive graph online at https://www.econgraphs.org/graphs/competition/welfare/total_and_marginal_cost ]
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 $Q^\star = 6$:
[ See interactive graph online at https://www.econgraphs.org/graphs/competition/welfare/total_welfare ]
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.