Section 5.3 Firm supply and market supply
The solution to the perfectly competitive firm’s profit maximization problem plays an important role in economic theory. Let’s go back to the numerical example above to see what this solution reveals.
Consider a perfectly competitive firm with marginal cost \(MC = 4q\text{.}\) First, suppose the firm is able to sell at a price of \(P = 80\text{.}\) According to profit maximization, the firm should choose to produce where \(P = MC\text{:}\) \(q^* = 20\text{.}\) Now let’s ask a comparative statics question: what if the price changes from 80 to 100? How should the firm optimally respond to the price increase? Intuition may suggest the firm would be incentivized to produce more output, and this intuition is confirmed in the model. Setting \(P = MC\) yields \(100 = 4q\) and the new optimal choice is \(q^{**} = 25\text{.}\)
But what does this mean? On the graph above, we plot the marginal cost curve \(MC = 4q\text{,}\) and show how the optimal choice increases when the price shifts upward. When the price increases, the number of units the firm wants to supply also increases! This fact might sound familiar, since we commonly call it the Law of Supply:
- The Law of Supply states that as the price increases, the quantity supplied increases; conversely, if the price decreases, the quantity supplied decreases.
The Law of Supply confirms the intuitive positive relationship between price and quantity for a seller. If a seller receives a higher price for the product it sells, it will be willing to sell more units of it!
We can see even more than this. Let us plot the two price-quantity combinations - one at \(P = 80\) and one at \(P = 100\) - side-by-side with the marginal cost graph. See any similarity? The points correspond exactly to the firm’s marginal cost curve! So, the marginal cost curve matches a collection of price-quantity combinations. We have another common name for a “collection of price-quantity combinations”: a supply curve. Yes, the firm’s supply curve is its marginal cost curve.
This finding demonstrates the origins of a firm’s supply curve. It is derived from the firm’s profit maximization problem, a result of optimal decision making for any possible price. To see this mathematically, let’s look at the firm’s optimal choice problem when the price is not numerical, but is just a parameter \(P\text{.}\) With marginal cost \(MC = 4q\text{,}\) the firm’s profit maximizing choice is
\begin{equation*}
P = 4q
\end{equation*}
\begin{equation*}
q^* = \frac{P}{4}
\end{equation*}
This expression is the firm’s supply function. A supply function gives the mathematical relationship between price and quantity supplied. This formula reveals, for example, that at a price of 80, the firm will optimally want to supply \(\frac{80}{4} = 20\) units of output, and that at a price of 100, the firm will optimally want to supply \(\frac{100}{4} = 25\) units of output.
We can graph this mathematical function, as seen below. Just as we have seen with a demand function, however, keep in mind that a supply function gives the quantity as the dependent variable, and the price as the independent variable. So, often, to graph a supply function more easily, we sometimes want to express price as a function of quantity. In similar fashion, we call this the inverse supply function. This expression is the same as the supply function, just solved out for the other variable. If the supply function is \(q = \frac{P}{4}\text{,}\) then inverse supply is \(P = 4q\text{.}\) Inverse supply is in \(y = mx + b\) form, which makes slope and y-intercept easy to interpret. Here, the slope of supply is 4, and the y-intercept is 0, meaning at a price of 0, the firm is not willing to supply any units of output.
Market supply considers the quantity supplied in the entire market, which incorporates all of the firms in the market and their willingness to supply units of output. Summing supply curves can also happen once we understand where supply functions come from. Just as we saw in the summing of demand curves, summing supply curves happens horizontally. This is easiest to see with an example. Consider two firms: firm 1 has a supply function of \(q_1^* = \frac{P}{2}\text{;}\) firm 2 has a supply function of \(q_2^* = \frac{P}{4}\text{.}\) So, for example, if the price in the market is 20, then firm 1 will supply 10 units to the market and firm 2 will supply 5 units. Collectively, then, the market has 15 units willing to be sold. Algebraically, then, the market supply curve can be computed by summing according to the horizontal axis variable 1 :
\begin{equation*}
Q_{market} = q_1^* + q_2^* = \frac{P}{4} + \frac{P}{2} = \frac{3P}{4}
\end{equation*}
The graphs below show the individual supply curves and the market supply curve, \(Q = \frac{3P}{4}\text{,}\) equivalently expressed as the inverse supply curve, \(P = \frac{4Q}{3}\text{.}\) Notice that the market supply is flatter and shifted further to the right than the individual supply curves. Additionally, as more firms are added to the market, the supply curve will get flatter and shift further right. These facts are consistent with the idea that as an increase in the number of sellers in a market will shift supply to the right. 2
Key terms in this section:
- Law of Supply
- supply curve
- supply function
- inverse supply function
- market supply curve
