14.4 Market Supply

In Part III, we analyzed the problem of a profit-maximizing firm that bought resources (labor and capital) and used those resources to produce goods, which it then sold. We defined a perfectly competitive firm as one which was a price taker in both input and output markets: specifically, we assumed that they could buy any quantity of labor and captial at prices $w$ and $r$ respectively, and they could sell any amount of output at price $p$.

Just as we wrote the quantity of a good demanded by an individual consumer as $d(p)$ and the market demand as $D(p)$, let’s write the amount of a good supplied by a firm as $s(p)$ and the market supply as $S(p)$. And just as we aggregated the individual demands of all consumers in a market to get $D(p)$, we need to aggregate the quantity supplied by all firms to determine the market supply curve $S(p)$. Thus, if we have $N_F$ firms, and $s^j(p)$ is the supply function for firm $j$, we can write the overall market supply $S(p)$ as \(S(p) = s^1(p) + s^2(p) + s^3( p) + \cdots + s^{N_F}(p)\) or, more succinctly, \(S(p) = \sum_ {j=1}^{N_F}s^j(p)\) which we can read as “the total quantity of a good supplied at price $p$ is the sum, for each $j$ from 1 to $N_F$, of the quantity supplied by each firm $j$ at that price.”

Just as we did with demand, let’s look at an example in which there are $N_F$ identical firms, just to keep the math simple. In this case, if each firm has the individual supply function $s(p)$, the total market supply is just the number of firms times the amount supplied by each firm: \(S(p) = \sum_{i = j}^{N_F}s^j(p) = N_Fs(p)\) For example, let’s suppose that in some market, there are $N_F$ firms who each have access to the Cobb-Douglas production function \(f(L,K) = \sqrt{LK}\) We showed in Chapter 13 that, if capital is fixed at $\overline K$, the short-run supply function for such a firm is \(s(p) = {\overline Kp \over 2w}\) Therefore the total amount supplied by the market is \(S(p) = N_Fs(p) = {N_F \overline Kp \over 2w}\) The following diagrams show this situation. The diagram on the left shows the supply curve for each firm; the diagram on the right shows the market supply.

Again, you can play with the sliders to see how the individual supply curve and market supply curves are affected.