profit-maximization. economic profit u profit maximization provides the rationale for firms to...

Profit-Maximization

Economic Profit

Profit maximization provides the rationale for firms to choose the feasible production plan.

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 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 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 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

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

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 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

A firm uses inputs j = 1…,m to make products i = 1,…n.

Output levels are y1,…,yn.

Input levels are x1,…,xm.

Product prices are p1,…,pn.

Input prices are w1,…,wm.

The Competitive Firm

The competitive firm takes all output prices p1,…,pn and all input prices w1,…,wm as given constants.

Economic Profit

The economic profit generated by the production plan (x1,…,xm,y1,…,yn) is

p y p y w x w xn n m m1 1 1 1 .

Economic Profit Output and input levels are typically

flows. E.g. x1 might be the number of labor

units used per hour. And y3 might be the number of cars

produced per hour. Consequently, profit is typically a

flow also; e.g. the number of euros of profit earned per hour.

Economic Profit 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.

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 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

Suppose the firm is in a short-run circumstance in which

Its short-run production function isy f x x ( , ~ ).1 2

x x2 2~ .

Economic Profit

Suppose the firm is in a short-run circumstance in which

Its short-run production function is

The firm’s fixed cost isand its profit function is

y f x x ( , ~ ).1 2

py w x w x1 1 2 2~ .

x x2 2~ .

FC w x 2 2~

Short-Run Profit-Maximization

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.

Q: What is this constraint?


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.

Q: What is this constraint? A: The production function.

Short-Run Profit-Maximization 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.

221121~)~,(

1

xwxwxxpfMaxx

12*

1111

2*

1 )~,(0)~,(

wxxpMPwx

xxfp


MPwp

p MP w11

1 1

p MP 1 is the marginal revenue product ofinput 1, the rate at which revenue increaseswith the amount used of input 1.If then profit increases with x1.If then profit decreases with x1.

p MP w 1 1

p MP w 1 1

Short-Run Iso-Profit Line

py w x w x1 1 2 2~ .

A € iso-profit line contains all the production plans that provide a profit level € .A $ iso-profit line’s equation is

I.e.y

wp

xw xp

11

2 2 ~.

Short-Run Iso-Profit Lines

ywp

xw xp

11

2 2 ~

has a slope of

wp1

and a vertical intercept of

w xp2 2~.

Short-Run Iso-Profit Lines

Increasing

profit

y

x1

Slopeswp

1


x1

Technicallyinefficientplans

y The short-run production function andtechnology set for x x2 2~ .

y f x x ( , ~ )1 2


x1

Increasing

profit

Slopeswp

1

y

y f x x ( , ~ )1 2


x1

y

Slopeswp

1

Given p, w1 and the short-runprofit-maximizing plan is And the maximumpossible profitis

x x2 2~ ,( , ~ , ).* *x x y1 2

.

x1*

y*


x1

y

Slopeswp

1

At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximaliso-profit line areequal.

MPwp

at x x y

11

1 2

( , ~ , )* *

x1*

y*

Short-Run Profit-Maximization; A Cobb-Douglas Example

Suppose the short-run productionfunction is y x x 1

1/321/3~ .

The marginal product of the variableinput 1 is MP

yx

x x11

12 3

21/31

3

/ ~ .

The profit-maximizing condition is

MRP p MPp

x x w1 1 12 3

21/3

13 ( ) ~ .* /


px x w

3 12 3

21/3

1( ) ~* / Solving for x1 gives

( )~

.* /xw

px1

2 3 1

21/3

3

That is,( )

~* /x

pxw1

2 3 21/3

13

so xpx

wpw

x121/3

1

3 2

1

3 2

21/2

3 3*

/ /~~ .


xpw

x11

3 2

21/2

3*

/~

is the firm’s

short-run demandfor input 1 when the level of input 2 is fixed at units. ~x2


xpw

x11

3 2

21/2

3*

/~

is the firm’s

short-run demandfor input 1 when the level of input 2 is fixed at units. ~x2

The firm’s short-run output level is thus

y x xpw

x* *( ) ~ ~ .

1

1/321/3

1

1/2

21/2

3

Comparative Statics of Short-Run Profit-Maximization

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).


ywp

xw xp

11

2 2 ~The equation of a short-run iso-profit lineis

so an increase in p causes -- a reduction in the slope, and -- a reduction in the vertical intercept.


x1

Slopeswp

1

y

y f x x ( , ~ )1 2

x1*

y*


An increase in w1, 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).


ywp

xw xp

11

2 2 ~The equation of a short-run iso-profit lineis

so an increase in w1 causes -- an increase in the slope, and -- no change to the vertical intercept.


x1

Slopeswp

1

y

y f x x ( , ~ )1 2

x1*

y*

Long-Run Profit-Maximization

Now allow the firm to vary both input levels.

Since no input level is fixed, there are no fixed costs.


Both x1 and x2 are variable. Think of the firm as choosing the

production plan that maximizes profits for a given value of x2, and then varying x2 to find the largest possible profit level.


x1

y

y f x x ( , )1 22

y f x x ( , )1 2

y f x x ( , )1 23

Larger levels of input 2 increase theproductivity of input 1.

The marginal productof input 2 isdiminishing.


Profit will increase as x2 increases so long as the marginal profit of input 2

The profit-maximizing level of input 2 therefore satisfies

p MP w 2 2 0.

p MP w 2 2 0.


Profit will increase as x2 increases so long as the marginal profit of input 2

The profit-maximizing level of input 2 therefore satisfies

And is satisfied in any short-run, so ...

p MP w 1 1 0

p MP w 2 2 0.

p MP w 2 2 0.


The input levels of the long-run profit-maximizing plan satisfy

That is, marginal revenue equals marginal cost for all inputs.

p MP w 2 2 0.p MP w 1 1 0 and

profit-maximization. economic profit u profit maximization provides the rationale for firms to...

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