Profit Maximization PDF

11 Profit Maximization So far in our analysis of the firm’s problem, we have been able to solve for the most efficient way to produce a quantity of output given costs of inputs. But how does a firm decide what quantity to produce? In these notes we will explore this deci...

11 Profit Maximization So far in our analysis of the firm’s problem, we have been able to solve for the most efficient way to produce a quantity of output given costs of inputs. But how does a firm decide what quantity to produce? In these notes we will explore this decision. Concepts Covered Profit Maximization in Perfect Competition The Firm’s Short Run and Long Run Decision Connecting Cost Minimization and Profit Maximization 11.1 Profit Maximization in Perfect Competition Profit Function We assume that a firm’s primary goal is to maximize its profit. We define the firm’s profit as its total revenue minus its total cost. We define revenue as the price the firm receives for its output times the amount it can sell R(Q) = p(Q) · Q In the last notes, we showed how to derive a firm’s cost as a function of Q. For now we will just leave that as a general function C(Q). Then we can write profit π as π(Q) = R(Q) − C(Q) Profit Maximization The profit function above depends only on one variable so we can find the value of Q that maximizes profit by taking a simple first order condition. dπ(Q) dR(Q) dC(Q) dR(Q) dC(Q) = − = 0 =⇒ = dQ dQ dQ dQ dQ In the last notes, we defined the derivative of the cost function with respect to Q as the marginal cost (MC) and we can define the derivative of revenue with respect to Q as marginal revenue (MR) so the condition above tells us that the firm will maximize profits when MR=MC. It is worth thinking about what this condition means intuitively. Remember that marginal cost can be interpreted as the cost of producing one more unit of output. Marginal revenue is the additional revenue from producing one more unit. If the revenue you receive from the next unit is more than the cost (M R > M C), it’s worth it to produce that unit. If the additional revenue 1 were less than the cost (M R < M C), then the firm would be losing money by producing that additional unit, so it would be better off not producing it. By this logic, the firm will want to keep increasing production until it hits the point where goods start becoming more costly than revenue (assuming marginal cost is increasing as Q increases) In general, it is possible for the price of a good to depend on the quantity produced. If the firm has some monopoly power, then the firm’s quantity decision will also affect the price. If the firm wants to sell a lot, it would need to decrease the price to increase the quantity demanded for its product. In this case, we have dR dp R(Q) = p(Q) · Q =⇒ = p(Q) + Q dQ dQ The first term captures the increase in revenue the firm receives if prices stay constant. The second term captures the fact that if the firm wants to sell more, it has to decrease its price, so the increase in revenue will be less than the current price. As an example, let’s imagine if a firm sets a price of 10, it will sell 5 units of its product. To increase the quantity it sells to 6, let’s assume it has to decrease its price to 9. What is the marginal revenue from the 6th unit? Well if the price had stayed constant at 10, it would have added 10 to the firm’s revenue (this is the first term in the equation above). But to get this increase, it had to decrease prices for all six of its sales by 1. So we need to subtract 6 (which corresponds to the 2nd term) and get a marginal revenue of 4. In all, revenue was (10)(5) = 50 before the price change and (9)(6) = 54 after the price change. Note that the decrease in price needed to increase quantity depends on how sensitive demand is to price changes. If demand is not very sensitive to price changes (demand is inelastic), the firm will need to decrease price by so much that it ends up losing revenue by producing an extra unit. On the other hand, if demand is elastic, the firm doesn’t have to change the price much to get large increases in quantity, so it can do better by producing a larger quantity. Profit Maximization with Perfect Competition For now, we are going to constrain our discussion to a case where the above concerns go away. In particular, we will assume that the firm is operating in a perfectly competitive market. One way to think about this assumption is to think that the firm is so small relative to the overall size of the market that it can sell as much as it wants without affecting the market price. We call such a firm a “price taker” because they take price as given. Therefore, for a perfectly competitive firm, R = pQ =⇒ M R = p The firm always receives an additional p units of revenue for every unit it produces. Using our profit maximizing condition from above, this result implies that a firm maximizes profit when MC = MR = p The firm maximizes its profit by choosing a quantity so that the price is equal to marginal cost. If marginal cost were higher than price, the firm would be better off reducing its output and if it were less than price, the firm could earn more profit by increasing output. 2 11.2 The Firm’s Short Run and Long Run Decision The Short Run Supply Curve Thinking back to introductory microeconomics, we plotted a supply curve as a quantity produced for every possible price. We can do the same exercise now with a deeper understanding of why the firm would want to produce a certain quantity at a certain price. Remember that a firm in a perfectly competitive market will always set p = M C(Q) To find the firm’s supply curve we simply solve this function for Q. Assuming marginal cost is increasing in Q, the supply curve will be an upward sloping curve. As the price increases, the firm produces a larger quantity. The Firm’s Short Run Decision First, let’s define a few concepts. Looking at the firm’s cost function, we will define the variable cost (V C) as the part of the cost function that depends on quantity and the fixed cost (F C) as the portion that does not. In the short run, we will assume that the firm’s fixed cost cannot be changed. We can then divide by Q to get the average variable cost and average fixed cost VC FC AV C = AF C = Q Q And average cost is the sum of these (AC = AV C + AF C). When marginal cost is greater than average variable cost at the point where MC=P, the firm will want to produce a positive (non-zero) quantity in the short run. To see why, remember that under perfect competition, marginal revenue is always equal to price. Therefore, average revenue is also equal to price. Since we have assumed M C > AV C, then this also means that average revenue (price), is greater than average variable cost. Ignoring the fixed cost, which cannot be changed, the firm will increase its profits by producing at the point where P = M C. Now imagine we had a situation where marginal cost was less than average variable cost M C < AV C Since we set price equal to marginal cost, this means p < AV C Multiply both by Q pQ < V C In other words, the total revenue from producing Q units is less than the variable cost of producing those Q units. Note that the firm can always do better than this. If they set Q = 0, then both revenue and variable costs go to zero. The takeaway from this discussion is that using the marginal cost curve as a supply curve only makes sense when M C = p > AV C. So our short run supply curve is the portion of the marginal cost curve that is above the average variable cost curve and 0 everywhere else. If the price is too low for the firm to produce a positive quantity, we say that the firm shuts down in the short run. 3 Long Run Entry and Exit Above, we looked at the firm’s variable cost, but ignored the fixed portion. We do that under the assumption that in the short run the firm cannot change its fixed costs. Even if it decided the price were too low to actually produce output, it would still have to pay the fixed cost. To connect this idea to reality, we could think of this being the case where the firm already has a factory set up. Even if they don’t produce anything, they still have to pay rent and upkeep for the factory. However, in the long run, the firm would be able to sell its factory and exit the market entirely. It will want to exit whenever its total profits are negative, or, equivalently, when it’s average revenue (which in a competitive market is equal to price and marginal cost) is less than average cost (including the fixed cost) AR = p = M C < AC Example Let’s assume that a firm has cost function given by C(Q) = Q2 + 100. The firm maximizes profits by setting M C = 2Q = p, so our supply curve is given by 1 Q= p 2 In this case, AV C = Q is always less than M C = 2Q so the firm will always produce in the short run. The average cost is C 100 AC = =Q+ Q Q If we set this expression equal to marginal cost, we can find the cutoff point where the firm wants to leave the market 100 AC = M C =⇒ Q + = 2Q =⇒ Q = 10 Q And plug this Q back into marginal cost to find the price p = M C = 2Q = 20 So in this case the firm will exit the market in the long run whenever p < 20 Summary The above analysis gave us two important cutoff points. First, in the short run, a firm will shut down (produce 0) if M C < AV C In this time period where the firm’s fixed cost cannot be changed, it might still produce temporarily at a negative profit. However, in the long run, it will exit the market whenever M C < AC Remember that these conditions relied on our assumption that the firm is operating in a perfectly competitive market. Future economics classes will explore what happens when we allow for market power. 4 11.3 Connecting Cost Minimization and Profit Maximization In the last notes we showed how to derive a cost function from the firm’s cost minimization problem. In particular, we first solved for the contingent demands for capital and labor as a function of Q, w, and v K ∗ = K(Q, w, v) L∗ = L(Q, w, v) And then plugged these into the firm’s cost C(Q, w, v) = wL∗ + vK ∗ Since w and v are assumed to be determined in a market, the firm cannot set these values. However, they can choose their quantity to maximize profit. Therefore, after solving for the cost function, we can plug this equation into the firm’s profit and solve the equation as we did above. If we assume that labor is a variable input (i.e. can be changed in the short run), while capital is a fixed input (can only be changed in the long run), then we can also separately solve for the firm’s costs in the short run. For example, let’s assume we have a Cobb-Douglas production function Q = F (K, L) = K 1/2 L1/2 But that in the short run, capital is fixed at K = 100. Then we could rewrite the production function in the short run as Q = 10L1/2 This makes it really easy to solve for the contingent demand for labor. All we have to do is invert the function Q2 L∗ = 100 And our cost function is ∗ Q2 C(Q, w, v) = wL + vK = w + v(100) 100 5 Appendix: Another Approach In the previous section, we described a 2 step approach to figuring out the firm’s profit maximizing decision given wages, rental rates, and prices. First, we calculated the optimal quantity of capital and labor to produce a given quantity Q, which gives us the cost function. Then we used this cost function to solve for the optimal profit. However, another way to approach the problem would be to write everything in terms of capital and labor. In particular, we could write profit as π = pF (K, L) − wL − vK And then set up the profit maximization problem as max π K,L Again this is an unconstrained maximization problem in two variables, K and L. Taking first order conditions ∂π ∂F ∂F =p − w = 0 =⇒ p =w ∂L ∂L ∂L ∂π ∂F ∂F =p − v = 0 =⇒ p =v ∂K ∂K ∂K The first condition says that the wage should be equal to the marginal product of labor times the price. We can think of this as the contribution of the marginal worker to the firm’s revenue (MPL is how much they can produce and p is the price the firm can sell the product). Similarly, the rental rate of capital is set equal to the contribution of capital to a firm’s revenue. Solving these two equations for K and L would give us the quantities of capital and labor the firm chooses as a function of wages, the rental rate of capital and the price of its output. Note that this result is not the contingent demand for capital and labor because we have implicitly found the profit maximizing quantity so Q is already set. To find the profit maximizing quantity, we could plug these optimal values for K and L back into the production function. 6