# 15.2 Maximizing 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.