Lecture 18: Input and Output Decisions of a Competitive Firm

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In the last lecture we solved for the profit-maximizing output choice of different kinds of firms: some that faced a downward-sloping inverse demand curve, and others — “competitive” or “price taking” firms — that faced a horizontal inverse demand curve. In this chapter and the next we will analyze the comparative statics of a “competitive” or “price taking” firm: that is, we will examine how such a firm’s output and input decisions vary when the prices of output and inputs vary.

In this lecture we will continue to analyze a firm in the short run: that is, the period of time in which some inputs are fixed. Following our convention, we will assume that capital is fixed and labor is flexible in the short run. We will therefore examine how the firm’s (joint) decision of how much to produce and how much labor to hire in the short run depends on the wage rate and the price at which the firm sells its output. In this chapter, we will continue to use our production function $q = f(L,K) = \sqrt{LK}$, with capital fixed at $\overline K = 32$. In the next chapter we will examine the decision made by the firm in the long run, when it can vary both labor and capital; and we’ll use a slightly different Cobb-Douglas production function.

Output as a function of price: the supply curve

When we defined the revenue function for a competitive firm, we said that its inverse demand function was simply $p(q) = p$, where $p$ is the market price that the firm takes as given. Since a competitive firm’s optimal choice depends on the price, a critical question to ask is how the quantity the firm supplies responds to a change in the market price.

If $p(q) = p$, the total revenue functions becomes simply $r(q) = p \times q$ Therefore the firm’s marginal revenue is equal to the market price: $MR(q) = p$ which means the profit-maximizing condition $MR = MC$ is just $p = MC$; that is, the firm will maximize its profit when price equals marginal cost. Since $MC$ is a function of $q$, our strategy will be to invert this function and solve for the optimal quantity $q$ as a function of the price; this gives the firm’s supply function.

To see how to do this, let’s continue with our example of a firm with the production function $f(L,K) = \sqrt{LK}$, which has capital fixed at $\overline K = 32$; this means that the short-run conditional demand for labor is $L^c(q) = {q^2 \over \overline K} = {q^2 \over 32}$ Therefore its profit function in general is $\pi(q) = pq - (wL^c(q) + r \overline K) = pq - \left( {wq^2 \over 32} + 32r\right)$ Let’s start with the most concrete profit-maximization problem, and one by one shift things from constant

Step 1: Fixed $\overline K$, $w$, $r$, and $p$

If the firm has to pay $w = 8$ for each unit of labor and $r = 2$ for each unit of capital, and sell at a market price $p = 12$ is $\pi(q) = 12q - \left(8 \times {q^2 \over 32} + 64\right)$ If we take the derivative this with respect to $q$, set it equal to zero, and solve for $q$ we have $\begin{aligned} \pi^\prime(q) = 12 - 8 \times {2q \over 32} &= 0\\ \underbrace{8 \times {2q \over 32}}_{MC(q)} &= \underbrace{12}_p\\ q^\star &= {32 \times 12 \over 2 \times 8} = \boxed{24} \end{aligned}$

Step 2: Fixed $\overline K$, $w$ and $r$, variable $p$

Now let’s replace $p = 12$ with just the variable $p$. The profit function now becomes $\pi(q\ |\ p) = pq - \left(8 \times {q^2 \over 32} + 2 \times 32\right)$ Now when we take the derivative this with respect to $q$ set it equal to zero and solve for $q$, we end up with a function of $p$ rather than a number: $\begin{aligned} \pi^\prime(q) = p - 8 \times {2q \over 32}\ &= 0\\ 8 \times {2q \over 32} &= p\\ q^\star(p) &= {32 \times p \over 2 \times 8} = \boxed{2p} \equiv S(p) \end{aligned}$ This is our supply function, $S(p)$. The diagram below illustrates this by showing how the firm optimally responds to each price. Try dragging the price up and down; the quantity will adjust automatically to the profit-maximizing quantity.

See interactive graph online here.

Some things to notice:

When $p > 8$, the firm can run a profit because there is a region for which $TR > TC$.
When $p = 8$, the firm has at most zero profit; $TR = TC$ (and therefore $AR = AC$) at $q = 16$.
When $p < 8$, the firm cannot run a profit, because the entire $TR$ line lies below the $TC$ curve (and the $AR$ line lies below the minimum of the $AC$ curve).

One might ask why a firm would ever run a loss: why produce anything at all, if you’re going to have a negative profit? The answer is that the firm must incur its fixed costs, at least in the short run; so for all intents and purposes those are sunk costs in the short run. When deciding whether or not to produce, the firm must therefore ask if they can at least make more revenue than their variable costs. If they can, then they should produce a positive output in the short run; if they can’t, they should shut down.

Shifts in the supply curve

In the above analysis we held the price of inputs constant. But what happens when we let the wage rate be a variable as well?

Step 3: fixed $\overline K$ and $r$, variable $p$ and $w$

For our next step, let’s keep capital fixed at $\overline K = 32$ and $r = 2$ as the price of capital, but let both $w$ and $p$ be variables. Our profit function now becomes $\pi(q\ |\ w,p) = pq - \left(w \times {q^2 \over 32} + 2 \times 32\right)$ Now when we take the derivative this with respect to $q$ set it equal to zero and solve for $q$, we end up with a function of $p$ and $w$: $\begin{aligned} \pi^\prime(q) = p - w \times {2q \over 32} &= 0\\ w \times {2q \over 32} &= p\\ q^\star(p,w) &= {32 \times p \over 2 \times w} = \boxed{16p \over w} \equiv S(p\ |\ w) \end{aligned}$ Again, we can see that when $w = 8$, this is the same as our supply function above: $q^\star(p) = 2p$. But this gives us an additional insight into how the supply curve shifts when there’s a change in wage. Intuitively, if the wage rate rises, the marginal cost of producing any quantity increases, so at any given price, the firm will produce less output.

Step 4: everything is variable!

Let’s take this all the way to its conclusion! If we let $r$ and $\overline K$ be variable as well, so our profit function is the general $\pi(q) = pq - (wL(q) + r \overline K) = pq - \left( {wq^2 \over \overline K} + r\overline K \right)$ we can derive our supply function as a function of the prices $w$, $r$, and $p$, as well as the fixed level of capital $\overline K$: $\begin{aligned} \pi^\prime(q) = p - w \times {2q \over \overline K} &= 0\\ w \times {2q \over \overline K} &= p\\ q^\star(w,r,p,\overline K) &= \boxed{\overline Kp \over 2w} \equiv S(p\ |\ w,r, \overline K) \end{aligned}$

Note that a change in $r$ doesn’t affect the supply function, because $r$ only affects fixed costs and the supply decision is based on marginal costs. However, a change in $\overline K$ does affect the optimal output, because if impacts the marginal cost. You can see this by changing the wage rate in the diagram below and seeing how it affects the supply curve:

See interactive graph online here.

As you can see:

an increase in the market price results in an upward movement along the supply curve.
an increase in the market wage rate results in an inward shift of the supply curve. That’s because an increase in the wage rate raises the marginal cost of producing each unit of output, so at any given market price the firm will produce fewer units.
an increase in capital results in an outward shift of the supply curve. That’s because an increase in the amount of capital (with this production function, at any rate) lowers the marginal cost of producing each unit of output, so at any given market price the firm will produce more units.

The profit-maximizing demand for labor

We can use the same process as above to derive the profit-maximizing demand for labor, $L^\star(w,r,p)$. In fact, there are three ways we can do so!

First approach: plug $q^\star(w,r,p)$ into $L^c(q)$

The first is to simply solve for the optimal $q^\star(w,r,p)$, and plug that into the conditional demand for labor. Let’s go back to the case where $r = 2$ and $\overline K = 32$ to keep things simple; this would imply that $\begin{aligned} L^\star(w,p) &= L^c(q^\star(w,p))\\ &= {\left[q^\star(w,p)\right]^2 \over 32} \end{aligned}$ Plugging the supply function $q^\star(w,p) = {16p \over w}$ into this gives us $L^\star(w,p) = {\left[{16p \over w}\right]^2 \over 32} = {8p^2 \over w^2}$ For example, if $p = 12$, we can plot the demand for labor as a function of its price, $w$:

See interactive graph online here.

As we might expect, like the supply function itself, this is increasing in $p$ (the firm will hire more workers when the value of what they produce goes up) and decreasing in $w$ (the firm will hire fewer workers if it has to pay them more.) Therefore, just as a change in $p$ caused a movement along the supply curve, while a change in $w$ caused a shift of the supply curve, here the inverse is true: a change in $p$ causes a shift of the labor demand curve, while a change in $w$ causes a movement along the labor demand curve:

See interactive graph online here.

Second approach: write profit as a function of $L$, not $q$

The second approach is equivalent, and yields some interesting insights: we can write the profit of the firm not in terms of quantity, but in terms of labor. After all, a choice of quantity is a choice of labor, and vice versa.

To be more precise: the general form for a firm’s profit in the short run is $\pi = pq - wL - r\overline K$ Before, we wrote $L$ in terms of $q$, which allowed us to write the profit as a function of the choice variable $q$: $\pi(q) = pq - wL^c(q) - r \overline K$ In order to do this, we inverted the production function to get $L^c(q)$. However, we could just as easily write the entire profit expression in terms of $L$, using the fact that $q = f(L)$: $\pi(L) = pf(L) - wL - r \overline K$ Now if we take the derivative of this with respect to $L$ we get the condition $\pi^\prime(L) = p \times MP_L - w$ The second term of this expression is the wage rate: hiring another unit of labor costs you $w$. But what does it benefit you? The first term is called the marginal revenue product of labor: it describes how much additional revenue the firm receives from an additional unit of labor. As always, our units help us: $MRP_L = p {\text{dollars} \over \cancel{\text{unit of output} } } \times MP_L {\cancel{ \text{units of output} } \over \text{hour of labor} } = p \times MP_L {\text{dollars} \over \text{hour of labor} }$ Intuitively, if the $MP_L$ tells you how many additional units of output another hour of labor produces, the $MRP_L$ tells you the monetary value of those units (i.e., the number of units times the price at which each is sold).

Let’s compare this to our previous derivative: before, we had $\pi(q) = pq - wL(q) - r \overline K$ so $\pi^\prime(q) = p - w \times {1 \over MP_L}$ The first term here was the marginal revenue from another unit of output (i.e. its price); the second was the marginal cost of that unit of output (the wage rate times the amount of labor required to produce it). Now we have $\pi^\prime(L) = p \times MP_L - w$ So, now the first term is the marginal revenue from another unit of labor, as described above; and the second term is the marginal cost of that unit of labor (i.e. its wage).

Third approach: isoprofit lines and the production function

You might have noticed that both of those derivatives are fundamentally the same equation: $p = w \times {1 \over MP_L} \iff p \times MP_L = w$ The third way we could write that equation is $MP_L = {w \over p}$ The units of both sides of this are units of output per unit of labor. This makes sense: if we plot the production function in a graph with labor on the horizontal axis and output on the vertical axis, the $MP_L$ is the slope of that function, so its units have to be units of output per unit of labor.

Now, we can actually treat this production function as a “constraint” for a firm that wants to maximize profit. The “indifference curves” of the firm will increase as you move up and to the left: up because you’re selling more output, which makes you money; and to the left because you’re hiring fewer workers, which saves you wages.

What are these “indifference curves”? They are isoprofit lines: lines that represent the same amount of profit. In particular, if $\pi = pq - wL - r\overline K$ then we can write $q = {\pi + wL + r\overline K \over p}$ If we write this as $q = {\pi + r\overline K \over p} + {w \over p} \times L$ this is a line in $L-q$ space with an intercept of $(\pi + r\overline K)/p$ and a slope of $w/p$.

Now, every possible combination of $(L, f(L))$ is going to have an associated profit; the object of the firm is to make the most profit possible. Where does this happen? When at a point where the isoprofit line is tangent to the production function: that is, where $w/r = MP_L$.

This is perhaps better seen visually. The diagram below shows our production function. Try dragging the point along the production function until you find the highest possible profit:

See interactive graph online here.

Our “gravitational pull” argument applies here: to the left of the optimum at $L = 18$, the $MP_L > w/p$; this means that workers are more productive than the real wage, so the firm should hire more labor. On the other hand, to the right of the optimal point, $MP_L < w/p$, the real wage is higher than worker’s productivity, so the firm should reduce its labor (and its output). At the point of tangency, the firm is on the highest possible isoprofit line, so it’s maximizing its profit.

Theory of the firm: a summary

In this brief unit, we have essentially taken what you learned in Econ 1 about the way firms make supply decisions, and done it with calculus. Hopefully I haven’t given you many new insights about the economics beyond what you already knew, but with any luck you can see how calculus can help us analyze a firm’s marginal decisions.

Fundamentally, any firm in this model is making a single choice: how much output to produce. Downstream of that, it makes a choice of how to produce that output (i.e. how much labor and capital to use). In the short run, this is a trivial decision: every quantity $q$ is associated with exactly one value of $L$. In the long run, the firm needs to additionally solve a cost-minimization problem to find $L^c(q)$ and $K^c(q)$.

Finally, firms’ optimal decisions are impacted by the environment in which they’re making the decision. If the firm is a price taker in input markets, its output decision will depend upon the prices of its variable inputs. Likewise, if the firm is a price taker in output markets, its output decision will depend on the price at which it can sell its product. If a firm has market power, it doesn’t take the price as given; but it does take the demand curve as given, and must choose the optimal amount given the demand it faces.

There are a million obvious extensions to this model. One of the practice exam questions you’ll be doing imagines a firm with market power that can invest in advertising to boost the demand for its product. Firms can also have more than two inputs, or more than one product. They could invest in their production technology to be able to do more with less.

But the main point I want you to have taken away from these last few weeks is that each decision a firm makes balances its marginal revenue and its marginal cost; and that in particular, a competitive firm balances price and marginal cost. That key insight will drive the fundamental result of our next unit: that markets, under certain assumptions, will be efficient.

It’s time to bring supply and demand together!

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