Profit-Maximization

Economic Profit u Profit maximization provides the rationale for firms to choose the feasible production plan. u Profit is the difference between revenues and costs.  In the following analysis, we will assume that firms operate within a competitive market, which implies that the selling price is exogenous to the firm.

Economic Profit u When computing profits we must include all inputs used by the firm, valued at their market price. In particular, we must be aware of the opportunity cost of using inputs in the production of the firm rather than on alternative uses. Example: labor of the firm’s owner; capital of shareholders.

Economic Profit u Thus, the economic definition of profits requires that we value all inputs and outputs at their opportunity costs, which means that the economic definition will certainly differ from the accountants definition.

Profits and the Stock Market Value u Often the production process that a firm uses goes on for many years. Inputs that are acquired today (such as buildings or machines) last several years and contribute to the future production of output. Therefore, firms have to value a flow of costs and revenues over time. How should they evaluate those flows?

Profits and the Stock Market Value u Even if we abstract from inflation, receiving (or paying) today is different from receiving (or paying) one year from now. To measure future money flows we need to calculate the present value of these future flows, i.e. the value of these flows from a today’s perspective.

Profits and the Stock Market Value u The present value is calculated with the help of the (real) interest rate, which can be thought of giving the price of money in different moments of time.

Profits and the Stock Market Value u Therefore, one can say that the present value of the firm is the present value of all future profits. In the case of corporations where the capital consists of many shares, the dividends correspond to shares of profits. Thus, the stock market value of the firm’s share is, in a world of certainty, equal to the correspondent fraction of the present value of the firm’s future profits.

Economic Profit u A firm uses inputs j = 1…,m to make products i = 1,…n. u Output levels are y 1,…,y n. u Input levels are x 1,…,x m. u Product prices are p 1,…,p n. u Input prices are w 1,…,w m.

The Competitive Firm u The competitive firm takes all output prices p 1,…,p n and all input prices w 1,…,w m as given constants.

Economic Profit u The economic profit generated by the production plan (x 1,…,x m,y 1,…,y n ) is

Economic Profit u Output and input levels are typically flows. u E.g. x 1 might be the number of labor units used per hour. u And y 3 might be the number of cars produced per hour. u Consequently, profit is typically a flow also; e.g. the number of euros of profit earned per hour.

Economic Profit u Fixed Inputs are those the quantity of which the firm cannot vary. As we have seen, only in the short run can we have this category. u Variable Inputs are those the quantity of which can be freely chosen by the firm. In the long run, all inputs are variable.

Economic Profit u Quasi-Fixed inputs are those that do not depend on the quantity of output being produced, but only apply as long as the firm is producing a positive amount of output. There can easily be quasi-fixed factors in the long run.

Economic Profit u Suppose the firm is in a short-run circumstance in which u Its short-run production function is

Economic Profit u Suppose the firm is in a short-run circumstance in which u Its short-run production function is u The firm’s fixed cost is and its profit function is

Short-Run Profit-Maximization u The firm’s problem is to choose the production plan that attains the highest possible profit, given the firm’s constraint on choices of production plans. u Q: What is this constraint?

Short-Run Profit-Maximization u The firm’s problem is to locate the production plan that attains the highest possible iso-profit line, given the firm’s constraint on choices of production plans. u Q: What is this constraint? u A: The production function.

Short-Run Profit-Maximization u The profit maximization problem facing the firm can be written as: The first-order condition for this Max problem is: It turns out that the production plan that maximizes profits has a nice economic interpretation: at the optimal production plan, the value of the marginal product of an input must equal its price.

Short-Run Profit-Maximization is the marginal revenue product of input 1, the rate at which revenue increases with the amount used of input 1. If then profit increases with x 1. If then profit decreases with x 1.

Short-Run Iso-Profit Line A €  iso-profit line contains all the production plans that provide a profit level € . A $  iso-profit line’s equation is I.e.

Short-Run Iso-Profit Lines has a slope of and a vertical intercept of

Short-Run Iso-Profit Lines Increasing profit y x1x1

Short-Run Profit-Maximization x1x1 Technically inefficient plans y The short-run production function and technology set for

Short-Run Profit-Maximization x1x1 Increasing profit y

Short-Run Profit-Maximization x1x1 y Given p, w 1 and the short-run profit-maximizing plan is And the maximum possible profit is

Short-Run Profit-Maximization x1x1 y At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximal iso-profit line are equal.

Short-Run Profit-Maximization; A Cobb-Douglas Example Suppose the short-run production function is The marginal product of the variable input 1 is The profit-maximizing condition is

Short-Run Profit-Maximization; A Cobb-Douglas Example Solvingfor x 1 gives That is, so

Short-Run Profit-Maximization; A Cobb-Douglas Example is the firm’s short-run demand for input 1 when the level of input 2 is fixed at units.

Short-Run Profit-Maximization; A Cobb-Douglas Example is the firm’s short-run demand for input 1 when the level of input 2 is fixed at units. The firm’s short-run output level is thus

Comparative Statics of Short-Run Profit-Maximization u An increase in p, the price of the firm’s output, causes –an increase in the firm’s output level (the firm’s supply curve slopes upward), and –an increase in the level of the firm’s variable input (the firm’s demand curve for its variable input shifts outward).

Comparative Statics of Short-Run Profit-Maximization The equation of a short-run iso-profit line is so an increase in p causes -- a reduction in the slope, and -- a reduction in the vertical intercept.

Comparative Statics of Short-Run Profit-Maximization x1x1 y

u An increase in w 1, the price of the firm’s variable input, causes –a decrease in the firm’s output level (the firm’s supply curve shifts inward), and –a decrease in the level of the firm’s variable input (the firm’s demand curve for its variable input slopes downward).

Comparative Statics of Short-Run Profit-Maximization The equation of a short-run iso-profit line is so an increase in w 1 causes -- an increase in the slope, and -- no change to the vertical intercept.

Comparative Statics of Short-Run Profit-Maximization x1x1 y

Long-Run Profit-Maximization u Now allow the firm to vary both input levels. u Since no input level is fixed, there are no fixed costs.

Long-Run Profit-Maximization u Both x 1 and x 2 are variable. u Think of the firm as choosing the production plan that maximizes profits for a given value of x 2, and then varying x 2 to find the largest possible profit level.

Long-Run Profit-Maximization x1x1 y Larger levels of input 2 increase the productivity of input 1. The marginal product of input 2 is diminishing.

Long-Run Profit-Maximization u Profit will increase as x 2 increases so long as the marginal profit of input 2 u The profit-maximizing level of input 2 therefore satisfies

Long-Run Profit-Maximization u Profit will increase as x 2 increases so long as the marginal profit of input 2 u The profit-maximizing level of input 2 therefore satisfies u And is satisfied in any short-run, so...

Long-Run Profit-Maximization u The input levels of the long-run profit-maximizing plan satisfy u That is, marginal revenue equals marginal cost for all inputs. and