top-rated free essay

Profit Maximization

By hammerbang Feb 26, 2013 2684 Words
Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Econ 401 Price Theory

Chapter 19: Profit Maximization Problem
Instructor: Hiroki Watanabe

Summer 2009

1 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

1

2

3 4

5 6 7

Introduction Overview Short-Run Profit Maximization Problem Definitions Short-Run Profit Maximization Problem Solution to Short-Run Profit Maximization Problem Example Interpretation Comparative Statics Long-Run Profit Maximization Problem Solution to Long-Run Profit Maximization Problem Tangency Condition & Technical Rate of Substitution Factor Demand Returns to Scale and Profit Mazimization Problem Summary 2 / 49

Intro Overview

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Corresponds to Ch5 utility maximization problem.

( )
∗ 1

= ϕ1 (p, m)

p

ϕ(p, m)
∗ 2

= ϕ2 (p, m)

m

3 / 49

Intro Overview

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Q: How many chefs do we need to maximize the profit?
1 2

You’ll have more revenue as your sales increases. Hiring too many chefs will reduce the productivity.

4 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

1

2

3 4

5 6 7

Introduction Overview Short-Run Profit Maximization Problem Definitions Short-Run Profit Maximization Problem Solution to Short-Run Profit Maximization Problem Example Interpretation Comparative Statics Long-Run Profit Maximization Problem Solution to Long-Run Profit Maximization Problem Tangency Condition & Technical Rate of Substitution Factor Demand Returns to Scale and Profit Mazimization Problem Summary 5 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Definitions

w = (wC , wK ) denotes the factor price (unit price of inputs). The total cost associated with the input bundle (xC , xK ) is TC(xC , xK ) = wC xC + wK xK . The total revenue from y is TR(y) = py or TR(xC , xK ) = pf (xC , xK ).

The economic profit generated by the production plan (xC , xK , y) is π(xC , xK ) = pf (xC , xK ) − wC xC − wK xK .

6 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Definitions

The competitive firm takes output price p and all input prices (w1 , w2 ) as given constants (price taker assumption). Output and input levels are typically flows. (To compute flows, you need to specify a duration of period on which flows are measured. Stock doesn’t require that.) xC = the number of labor units used per hour. y = the number of cheesecakes produced per hour.

Accordingly, profit is usually a flow. Other examples: income (f), GDP (f), capital stock (s), bank balance (s).

7 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Definitions

Fixed Cost Fixed cost is a cost that a firm has to pay for the fixed input. Kayak’s has to pay the rent (wK ) even when y = 0. ¯ Suppose the size of kitchen if predetermined at xK . ¯ FC = wK xK . Fixed cost may or may not be a sunk cost (cost not recouped, regardless of future actions) depending on the timing: 1 2

It is sunk after Kayak’s paid the rent. Not if Kayak’s has not paid the rent.

8 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Short-Run Profit Maximization Problem

In the short run, the firm solves the short-run profit maximization problem (SPMP): Short-Run Profit Maximization Problem (SPMP) Kayak’s maximizes its short run profit given p, (wC , xK ): ¯ maxxC π(xC , xK ) = ¯ pf (xC , xK ) ¯ −wC xC − wK xK

total revenue total cost ¯ = pf (xC , xK ) − wC xC − FC.

9 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Solution to Short-Run Profit Maximization Problem

Iso-Profit Line ¯ An Iso-profit line at π contains all the production plans ¯ (xC , xK , y) that yield a profit level of π. We do not care if the production plan is actually feasible. The iso-profit line simply represents the collection of plans that yields the same π. Let’s say xK = 1, wK = 1 and FC = 1 · 1 = 1. π = py − wC xC − FC wC FC + π ⇒ y= xC + . p p Higher π means higher y-intercept. The slope of iso-profit is wC . p ¯ E.g., p = 1, (wC , wK ) = (1, 1) and xK = 1. 10 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Solution to Short-Run Profit Maximization Problem

Isoprofit 10 9 8 Cheesecakes (y) 7 6
4 6 2 0 −2 −4 8 4 6 2

π=py−wCxC−FC
0

−2

5 4 3 2 1 0 0
0 −2 −4 −6 −8 0 −1 2 0 −2 −4 −6 −8

1

2

3

4 5 6 Chefs (xC)

7

8

9

10
11 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Solution to Short-Run Profit Maximization Problem

Some of the production plans (xC , xK , y) cannot be chosen (not feasible) because of the technological constraint: y ≤ f (xC , xK ). (1)

Which production plan yields the highest profit level while satisfying (1)? ¯ E.g., y = f (xC , xK ) = 8xC .

12 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Solution to Short-Run Profit Maximization Problem

Isoprofit 10 9 8 Cheesecakes (y) 7 6 5 4 3 2 1 0 0 1
1 0 −1 −2 −3 −5 −4 3 2 5 6 3 4 2 1 0 −1 −2 −3 1 0 −1 −2 −3 −5 −4 −6 −7 −8 −5 −4 8 7 5 6 3 4 2 1 0 −1 −2 −3

π = py − wC xC − F C √ −7 −6 f(xC , xK ) = 8xC 0 y= ¯ 1 9 −8 − −

2

3

4 5 6 Chefs (xC)

7

8

9

10
13 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Solution to Short-Run Profit Maximization Problem

Recall:
1

2

¯ The slope of production function when xK = xK denotes the marginal product of xC . The slope of iso-profit is wC . p

¯ Kayak’s profit is maximized at (xC , xK , y) where the production function is tangent to the iso-profit curve. Tangency Condition ¯ At the optimal production plan (xC , xK , y), wC p ¯ = MPC (xC , xK ).

14 / 49

Intro Example

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Example ¯ Suppose p = 1, w = (1, 1), xK = 1 and ¯ y = f (xC , xK ) = 8xC . 1 2

What is the fixed cost? Which production plan maximizes the short-run ¯ profit? (MPC (xC , xK ) = 2 ). 2xC

Tangency condition:

2 2xC

= 1.

15 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Interpretation

What does the tangency condition mean? ¯ = MPC (xC , xK ) ¯ ⇒ wC = pMPC (xC , xK ) ∆y ⇒ ∆TC = p ∆x ∆xC C ⇒ additional cost of hiring a chef = additional revenue. ¯ What if wC > pMPC (xC , xK )? ¯ What if wC < pMPC (xC , xK )? wC p

16 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Interpretation

17 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

1

2

3 4

5 6 7

Introduction Overview Short-Run Profit Maximization Problem Definitions Short-Run Profit Maximization Problem Solution to Short-Run Profit Maximization Problem Example Interpretation Comparative Statics Long-Run Profit Maximization Problem Solution to Long-Run Profit Maximization Problem Tangency Condition & Technical Rate of Substitution Factor Demand Returns to Scale and Profit Mazimization Problem Summary 18 / 49

Figure:

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Q: How does Kayak’s respond to wage increase or price reduction?

19 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

20 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

21 / 49

Intro

SPMP

Comparative Statics

LPMP Figure:

Factor Demand

Returns to Scale

Σ

+ ↑ wC reduces xC and y+ . ↓ p reduces x+ and y+ .
C

Discussion
1

Does the increase in wK affect the optimal production ¯ plan (xC , xK , y)? ¯ Does the increase in xK affect the optimal production ¯ plan (xC , xK , y)? 1 2

2

No effect (profit gets smaller though). Short-run technology changes. The same amount of xC produces more y. xC ↓.

22 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Example ¯ Suppose p = 1, w = (2, 1), xK = 1 and ¯ ¯ y = f (xC , xK ) = 8xC . MPC (xC , xK ) = ¯ optimal production plan (x+ , x , y+ )? C K 2 2xC

). What is the

23 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Isoprofit 10 9 8 Cheesecakes (y) 7 6 5 4 3 2 1 0 0 1
1 0 −1 −2 −3 −5 −4 3 2 5 6 3 4 2 1 0 −1 −2 −3 1 0 −1 −2 −3 −5 −4 −6 −7 −8 −5 −4 8 7 5 6 3 4 2 1 0 −1 −2 −3

π = py − wC xC − F C √ −7 −6 y = f(xC , xK ) = 9 8xC 10 ¯ −8 − −

2

3

4 5 6 Chefs (xC)

7

8

9

10
24 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Isoprofit 10
5 0

Cheesecakes (y)

5

−5

−1 0

0

4

0 0 0.5

2 Chefs (xC)

−2 0

π = py − wC xC − F C √ y = f(xC , xK ) = 8xC ¯

0

−5

−1

0

−1 5

2

−5

−1 0

−1 5

10
25 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

1

2

3 4

5 6 7

Introduction Overview Short-Run Profit Maximization Problem Definitions Short-Run Profit Maximization Problem Solution to Short-Run Profit Maximization Problem Example Interpretation Comparative Statics Long-Run Profit Maximization Problem Solution to Long-Run Profit Maximization Problem Tangency Condition & Technical Rate of Substitution Factor Demand Returns to Scale and Profit Mazimization Problem Summary 26 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Solution to Long-Run Profit Maximization Problem

Long Run A long-run is the circumstance in which a firm is unrestricted in its choice of input levels. Decision-making process in which you can change the size of the store as well as amount of cheese. ¯ xK now becomes xK .

27 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Solution to Long-Run Profit Maximization Problem

Long-Run Profit Maximization Problem (LPMP) Given w and p, in the long run, Kayak’s solves max π(xC , xK ) = pf (xC , xK ) − wC xC − wK xK . xC ,xK

The same condition applies to xK : wK p + = MPK (xC , xK ).

28 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Solution to Long-Run Profit Maximization Problem

Example Kayak’s production function is given by f (xC , xK ) = xC + xK .

Price of a cheesecake is p = 2 and w = (1, 1). MPC (xC , xK ) = MPK (xC , xK ) = 2 1 . xC 1 . 2 xK

+ + What is the optimal long-run production plan (xC , xK , y)?

29 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Solution to Long-Run Profit Maximization Problem

Isoprofit 5
8 4 6 4 6 4 2

4 Cheesecakes (y)

3
4 2

2 0

2
2 0

0 −2

1
0 −2

−2 π = py − wC xC − wK xK 4 − √ y = xC + 1 −4

0 0

1

2 3 Chefs (xC)

4

5
30 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Solution to Long-Run Profit Maximization Problem

Isoprofit 5
8 6 4

4
6

4

2

Cheesecakes (y)

4

3
4 2

2

0

2
2 0

0 −2

1
0 −2

− py π = 2 − wC xC − wK x−4 K √ y = 1 + xK −4

0 0

1

2 3 Size of Kitchen (xK)

4

5
31 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Tangency Condition & Technical Rate of Substitution

Tangency conditions for long-run profit maximization problem: wC p wK p ⇒ = MPC (xC , xK ) = MPK (xC , xK ).



= wK MPK (xC , xK ) −wC = TRS(xC , xK ). wK

wC

MPC (xC , xK )

32 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Tangency Condition & Technical Rate of Substitution

If Kayak’s fires one chef, they can expand the w kitchen area by wC . K

If Kayak’s fire one chef, they need to expand the kitchen area by TRS(xC , xK ). The factor market’s idea of chef’s worth coincides with Kayak’s idea of chef’s worth. More details in Ch20: cost minimization problem.

33 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

1

2

3 4

5 6 7

Introduction Overview Short-Run Profit Maximization Problem Definitions Short-Run Profit Maximization Problem Solution to Short-Run Profit Maximization Problem Example Interpretation Comparative Statics Long-Run Profit Maximization Problem Solution to Long-Run Profit Maximization Problem Tangency Condition & Technical Rate of Substitution Factor Demand Returns to Scale and Profit Mazimization Problem Summary 34 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Change in wC affects xC as well as π. Tangency condition: + pMPC (xC , xK ) = wC . + At each wC , Kayak’s sets xC at which the additional increase in revenue equates wC (factor demand function). Diminishing marginal product: MP (x , x+ ) goes C C

down as xC increases.

K

35 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

36 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Example + Suppose MPC (xC , xK ) = function is given by
1 . 2 xC

The factor demand 1

wC p

=

.

2

xC

37 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Factor Demand 3 p=1 p=2

Wage (w )

C

1 0.5 0 0

1 Chefs (xC)

3
38 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

1

2

3 4

5 6 7

Introduction Overview Short-Run Profit Maximization Problem Definitions Short-Run Profit Maximization Problem Solution to Short-Run Profit Maximization Problem Example Interpretation Comparative Statics Long-Run Profit Maximization Problem Solution to Long-Run Profit Maximization Problem Tangency Condition & Technical Rate of Substitution Factor Demand Returns to Scale and Profit Mazimization Problem Summary 39 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

If a competitive firm’s technology exhibits decreasing returns to scale then the firm has a single long-run profit-maximizing production plan.

40 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

41 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Figure:

If a competitive firm’s technology exhibits exhibits increasing returns to scale then the firm does not have a profit-maximizing plan.

42 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

43 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Figure:

An increasing returns-to-scale technology is inconsistent with firms being perfectly competitive. What if the competitive firm’s technology exhibits constant returns-to-scale?

44 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

45 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Figure:

So if any production plan earns a positive profit, the firm can double up all inputs to produce twice the original output and earn twice the original profit. When a firm’s technology exhibits constant returns to scale, earning a positive economic profit is inconsistent with firms being perfectly competitive. A CRS firm is compatible with perfect competition only when firm earns zero profit.

46 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

47 / 49

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

1

2

3 4

5 6 7

Introduction Overview Short-Run Profit Maximization Problem Definitions Short-Run Profit Maximization Problem Solution to Short-Run Profit Maximization Problem Example Interpretation Comparative Statics Long-Run Profit Maximization Problem Solution to Long-Run Profit Maximization Problem Tangency Condition & Technical Rate of Substitution Factor Demand Returns to Scale and Profit Mazimization Problem Summary 48 / 49

Figure:

Intro

SPMP

Comparative Statics

LPMP

Factor Demand

Returns to Scale

Σ

Solving profit maximization problem. Comparative statics. Factor demand. Competitive environment and compatible technology.

49 / 49

Cite This Document

Related Documents

  • Marginal Analysis and Profit Maximization

    ...Marginal Analysis and Profit Maximization Task A At the point of profit maximization within any firm, the aspects of both marginal revenue and marginal cost play a major role. The economically working definition of marginal revenue is termed as: the extra revenue that an additional unit of product will bring. It is the additional income from...

    Read More
  • Do Business Firms Maximize Profits?

    ...6: “What Do Firms Try to Maximize, if Anything?” Introduction Do firms really maximize profit? This question has been under debate since the 1940s and 1950s, when a wide number of mainstream neoclassical economists defended the assumption against a group of institutional economists that questioned the assumption as the norm in the indu...

    Read More
  • Supply and Demand and Total Profit

    ...Question 6 Use Figure 6.5 to determine: a)      How many baskets of fish should be harvested at market prices of                                 i.            $9?  The farmer should harvest 3 baskets in order to gain the maximum profit.                       ...

    Read More
  • Profit maximisation is not the sole objective of business.

    ...Profit maximisation has been one of the main aims of the firms. The generally accepted view is the long run will wish to maximize profit. Marginal Cost and Marginal Revenue can be used to find the profit maximising level of output. Marginal cost is the addition to total cost of one extra unit of output. Marginal revenue is the increase in total ...

    Read More
  • Profit Maximisation

    ...factors that would influence the decision of the senior managers and whether they should join the joint venture or not. Profit maximisation Profit maximisation is the process by which a firm determines the price and output level that returns the greatest profit. There are several approaches to this problem. The total revenue - total cost m...

    Read More
  • The View That Shareholder Wealth Maximization Should Always Be the Preferred Objective of a Firm.

    ...expenses, and assets with the objective of maximizing profits and ensuring sustainability. It is concerned with the procurement and use of funds with an aim to use business funds in such a way that the firm’s value and earnings are maximized. It also provides a frame work for selecting a proper course of action and deciding a viable commercial...

    Read More
  • Why Might a Business Firm Pursue Other Objectives Besides the Objective of Maximum Profits? What Objectives Other Than Profit Maximisation Might a Firm Pursue? Is This Possible in a Competitive World?

    ...Why might a business firm pursue other objectives besides the objective of maximum profits? What objectives other than profit maximisation might a firm pursue? Is this possible in a competitive world? The traditional theory of business behaviour tends to make a general assumption that businesses possess the information, market power and motivat...

    Read More
  • The profit maximization is not an operationally feasible criterion". Do you agree? Illustrate your views.

    ...Q.5The profit maximization is not an operationally feasible criterion". Do you agree? Illustrate your views.. A.5. The profit maximization concept does not specify clearly whether it mean short or long-term profit, or profit before tax or after tax. In addition, in the free economy and perfect competition, b...

    Read More

Discover the Best Free Essays on StudyMode

Conquer writer's block once and for all.

High Quality Essays

Our library contains thousands of carefully selected free research papers and essays.

Popular Topics

No matter the topic you're researching, chances are we have it covered.