1. PROFIT MAXIMIZATION AND SUPPLY
Chapter 13
PROFIT MAXIMIZATION
AND SUPPLY
MICROECONOMIC THEORY
BASIC PRINCIPLES AND EXTENSIONS
EIGHTH EDITION
WALTER NICHOLSON
Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.

2. The Nature of Firms
A firm is an association of individuals who have organized themselves for the purpose of turning inputs into outputs
Different individuals will provide different types of inputs
the nature of the contractual relationship between the providers of inputs to a firm may be quite complicated

3. Contractual Relationships
Some contracts between providers of inputs may be explicit
may specify hours, work details, or compensation
Other arrangements will be more implicit in nature
decision-making authority or sharing of tasks

4. Modeling Firms’ Behavior
Most economists treat the firm as a single decision-making unit
the decisions are made by a single dictatorial manager who rationally pursues some goal
profit-maximization

5. Profit Maximization
A profit-maximizing firm chooses both its inputs and its outputs with the sole goal of achieving maximum economic profits
seeks to maximize the difference between total revenue and total economic costs

6.  = TR(q) – TC(q) = P(q)q – TC(q)
Output Choice
Total revenue for a firm is given by
TR(q) = P(q)q
In the production of q, certain economic costs are incurred [TC(q)]
Economic profits () are the difference between total revenue and total costs
 = TR(q) – TC(q) = P(q)q – TC(q)

7. Output Choice
The necessary condition for choosing the level of q that maximizes profits can be found by setting the derivative of the  function with respect to q equal to zero

8. Output Choice
To maximize economic profits, the firm should choose the output for which marginal revenue is equal to marginal cost

9. Second-Order Conditions
MR = MC is only a necessary condition for profit maximization
For sufficiency, it is also required that
“marginal” profit must be decreasing at the optimal level of q

10. Profit Maximization Profits are maximized when the slope ofq*
Profits are maximized when the slope of
the revenue function is equal to the slope of the cost function
revenues & costs TC TR
But the second-order
condition prevents us
from mistaking q0 as
a maximum q0
output

11. Marginal Revenue
If a firm can sell all it wishes without having any effect on market price, marginal revenue will be equal to price
If a firm faces a downward-sloping demand curve, more output can only be sold if the firm reduces the good’s price

12. Marginal Revenue
If a firm faces a downward-sloping demand curve, marginal revenue will be a function of output
If price falls as a firm increases output, marginal revenue will be less than price

13. Marginal Revenue Suppose that the demand curve for a sub sandwich is
q = 100 – 10P
Solving for price, we get
P = -q/
This means that total revenue is
TR = Pq = -q2/ q
Marginal revenue will be given by
MR = dTR/dq = -q/5 + 10

14. Profit Maximization
To determine the profit-maximizing output, we must know the firm’s costs
If subs can be produced at a constant average and marginal cost of $4, then
MR = MC
-q/ = 4
q = 30

15. Marginal Revenue and Elasticity
The concept of marginal revenue is directly related to the elasticity of demand facing the firm
The price elasticity of demand is equal to the percentage change in quantity that results from a one percent change in price

16. Marginal Revenue and Elasticity
This means that
if the demand curve slopes downward, eq,P < 0 and MR < P
if the demand is elastic, eq,P < -1 and marginal revenue will be positive
if the demand is infinitely elastic, eq,P = - and marginal revenue will equal price

17. Marginal Revenue and Elasticity
eq,P < -1
MR > 0
eq,P = -1
MR = 0
eq,P > -1
MR < 0

18. Average Revenue Curve
If we assume that the firm must sell all its output at one price, we can think of the demand curve facing the firm as its average revenue curve
shows the revenue per unit yielded by alternative output choices

19. Marginal Revenue Curve
The marginal revenue curve shows the extra revenue provided by the last unit sold
In the case of a downward-sloping demand curve, the marginal revenue curve will lie below the demand curve

20. Marginal Revenue Curve
As output increases from 0 to q1, total
revenue increases so MR > 0
price
As output increases beyond q1, total
revenue decreases so MR < 0 P1
D (average revenue)
output q1 MR

21. Marginal Revenue Curve
When the demand curve shifts, its associated marginal revenue curve shifts as well
a marginal revenue curve cannot be calculated without referring to a specific demand curve

22. Short-Run Supply by a Price-Taking Firm
SMC
SATC
P* = MR q*
Maximum profit
occurs where
P = SMC
SAVC
output

23. Short-Run Supply by a Price-Taking Firm
SMC
SATC
P* = MR
SAVC
Since P > SATC,
profit > 0
output q*

24. Short-Run Supply by a Price-Taking Firm
SMC
If the price rises
to P**, the firm
will produce q**
and  > 0
q**
P**
SATC
P* = MR
SAVC
output q*

25. Short-Run Supply by a Price-Taking Firm
If the price falls to
P***, the firm will
produce q***
q***
P***
price
SMC
SATC
P* = MR
SAVC
profit maximization
requires that P =
SMC and that SMC
is upward-sloping
output q*
 < 0

26. Short-Run Supply by a Price-Taking Firm
The positively-sloped portion of the short-run marginal cost curve is the short-run supply curve for a price-taking firm
it shows how much the firm will produce at every possible market price
firms will only operate in the short run as long as total revenue covers variable cost
the firm will produce no output if P < SAVC

27. Short-Run Supply by a Price-Taking Firm
Thus, the price-taking firm’s short-run supply curve is the positively-sloped portion of the firm’s short-run marginal cost curve above the point of minimum average variable cost
for prices below this level, the firm’s profit-maximizing decision is to shut down and produce no output

28. Short-Run Supply by a Price-Taking Firm
SMC
The firm’s short-run supply curve is that portion of the SMC curve that is above minimum SAVC
SATC
SAVC
output

29. Short-Run Supply Suppose that the firm’s short-run total cost curve is
STC = 4v + wq2/400
If w = v = $4, then the cost curve becomes
STC = 16 + q2/100
This implies that short-run marginal cost is
STC/q = 2q/100 = q/50

30. Short-Run Supply
Profit maximization requires that price is equal to marginal cost
P = SMC = q/50
This means that the supply curve (with q as a function of P) is
q = 50P

31. Short-Run Supply
To find the firm’s shut-down price, we need to solve for SAVC
SVC = q2/100
SAVC = SVC/q = q/100
SAVC is minimized when q = 0
the firm will only shut down when the price falls to 0

32. Short-Run Supply Short-run average costs are given by
SATC = STC/q = 16/q + q/100
SATC is minimized when
SATC/q = -16/q2 + 1/100 = 0
q = 40
SATC = SMC = $0.80
For any price below $0.80, the firm will incur a loss

33. Profit Maximization and Input Demand
A firm’s output is determined by the amount of inputs it chooses to employ
the relationship between inputs and outputs is summarized by the production function
A firm’s economic profit can also be expressed as a function of inputs
(K,L) = Pq – TC(q) = Pf(K,L) – (vK + wL)

34. Profit Maximization and Input Demand
The first-order conditions for a maximum are
/K = P[f/K] – v = 0
/L = P[f/L] – w = 0
A profit-maximizing firm should hire any input up to the point at which its marginal contribution to revenues is equal to the marginal cost of hiring the input

35. Profit Maximization and Input Demand
These first-order conditions for profit maximization also imply cost minimization
they imply that RTS = w/v

36. Profit Maximization and Input Demand
To ensure a true maximum, second-order conditions require that
KK < 0
LL < 0
KK LL - KL2 > 0
Capital and labor must exhibit sufficiently diminishing marginal productivities so that marginal costs rise as output expands

37. Profit Maximization and Input Demand
The first-order conditions can generally be solved for the optimal input combination
K* = K*(P,v,w)
L* = L*(P,v,w)
These input choices can be substituted into the production function to get q*
q* = f(K,L) = f [K*(P,v,w),L*(P,v,w)] = q*(P,v,w)

38. quantity supplied = q*(P,v,w)
Supply Function
The supply function for a profit-maximizing firm that takes both output price (P) and input prices (v,w) as fixed is written as
quantity supplied = q*(P,v,w)
this indicates the dependence of output choices on these prices

39. Supply Function
The supply function provides a convenient reminder of two key points
the firm’s output decision is fundamentally a decision about hiring inputs
changes in input costs will alter the hiring of inputs and hence affect output choices as well

40. Producer Surplus in the Short Run
A profit-maximizing firm that decides to produce a positive output in the short run must find that decision to be more favorable than a decision to produce nothing
This improvement in welfare is termed (short-run) producer surplus
what the firm gains by being able to participate in market transactions

41. Producer Surplus in the Short Run
If the firm was prevented from making such transactions, output would be zero and profits would equal -SFC
Production of the profit-maximizing output would yield profits of *
The firm gains *+ SFC
this is producer surplus

42. Producer Surplus in the Short Run
SMC
price P* q*
If the market price
is P*, the firm will produce q*
Producer surplus is the
shaded area below P*
and above SMC
output

43. Producer Surplus in the Short Run
In mathematical terms, producer surplus is given by

44. Producer Surplus in the Short Run
Because SFC is constant, changes in producer surplus as a result of changes in market price are reflected as changes in short-run profits
these can be measured by the changes in the area below market price above the short-run supply curve

45. Producer Surplus in the Long Run
By definition, long-run producer surplus is zero
fixed costs do not exist in the long run
equilibrium profits under perfect competition with free entry are zero
In long-run analysis, attention is focused on the prices of the firm’s inputs and how they relate to what they would be in the absence of market transactions

46. Revenue Maximization
When firms are uncertain about the demand curve they face or when they have no reliable notion of the marginal costs of their output, the decision to maximize revenues may be a reasonable rule of thumb for ensuring their long-term survival

47. Revenue Maximization
A revenue-maximizing firm would choose to produce that level of output for which marginal revenue is zero
Because we know that MR = P[1+(1/eq,P)], MR=0 implies that eq,P = -1
demand will be unit elastic at q*

48. Revenue Maximization P* q*
If the firm wishes to maximize revenues, it will produce q*
price d
output MR

49. Revenue Maximization
price
SMC
If the firm wishes to maximize profits, it will produce q**
q** P* d
output q* MR

50. Revenue Maximization
Increasing output beyond q** increases revenue but lowers economic profit
price
SMC P* d
output
q** q* MR

51. Constrained Revenue Maximization
A firm that chooses to maximize revenue is paying no attention to its costs
it is possible that maximizing revenues could result in negative profits for the firm
It may be more realistic to assume that these firms must meet some minimal level of profitability

52. Revenue Maximization
Suppose that a firm faces the following demand curve
q = P
Total revenues (as a function of q) is
TR = Pq = 10q - q2/10
Marginal revenue is
MR = dTR/dq = 10 - q/5

53. Revenue Maximization Total revenues are maximized when MR = 0
this means that q = 50
If output is 50, total revenues are $250
If we assume that AC = MC = $4, total costs are $200 and profits equal $50

54. Constrained Revenue Maximization
Suppose that the firm’s owners require a profit of at least $80
Then the firm might seek to maximize revenue subject to the constraint that
 = TR - TC = 10q - q2/10 - 4q = 80

55. Constrained Revenue Maximization
Rearranging the constraint, we get
q2 - 60q +800 = 0 or
(q - 40)(q - 20)=0
The solution q = 40 yields higher revenues than any other output level between 20 and 40
all of these options yield at least $80 profit

56. The Principal-Agent Problem
In many cases, firm managers do not actually own the firm but instead act as agents for the owners
An agent is a person who makes economic decisions for another party

57. The Principal-Agent Problem
Assume that we can show a graph of the owner’s (or manager’s) preferences in terms of profits and various benefits (such as fancy offices or use of the corporate jet)
The owner’s budget constraint will have a slope of -1
each $1 of benefits reduces profit by $1

58. The Principal-Agent Problem*
If the manager is also the owner
of the firm, he will maximize his
utility at profits of * and benefits
of B*
Profits U1
Owner’s constraint
Benefits

59. The Principal-Agent Problem
The owner-manager maximizes
profit because any other owner-
manager will also want B* in
benefits
Profits
B* represents a true cost
of doing business * U1
Owner’s constraint
Benefits B*

60. The Principal-Agent Problem
Suppose that the manager is not the sole owner of the firm
suppose there are two other owners who play no role in operating the firm
$1 in benefits only costs the manager $0.33 in profits
the other $0.67 is effectively paid by the other owners in terms of reduced profits

61. The Principal-Agent Problem
The new budget constraint continues to include the point B*, *
the manager could still make the same decision that a sole owner could)
For benefits greater than B*, the slope of the budget constraint is only -1/3

62. The Principal-Agent Problem
Given the manager’s budget
constraint, he will maximize utility
at benefits of B**
**
B**
Profits
Agent’s constraint
***
Profits for the firm
will be *** * U2 U1
Owner’s constraint
Benefits B*

63. The Principal-Agent Problem
The firm’s owners are harmed by having to rely on an agency relationship with the firm’s manager
The smaller the fraction of the firm that is owned by the manager, the greater the distortions that will be induced by this relationship

64. The Principal-Agent Problem
The firm’s owners will not be happy about accepting lower profits on their investments
they may refuse to invest in the firm if they know the manager will behave in this manner
The manager could work out some contractual arrangement to induce the would-be owners to invest

65. The Principal-Agent Problem
One possible contract would be for the manager to agree to finance all of the benefits out of his share of the profits
results in lower utility for the manager
would be difficult to enforce
They may instead try to give managers an incentive to economize on benefits and to pursue higher profits

66. Important Points to Note:
In order to maximize profits, the firm should choose to produce that output level for which the marginal revenue is equal to the marginal cost
If a firm is a price taker, its output decisions do not affect the price of its output
marginal revenue is equal to price

67. Important Points to Note:
If the firm faces a downward-sloping demand for its output, it can only sell more at a lower price
marginal revenue will be less than price and may be negative
Marginal revenue and the price elasticity of demand are related by the following

68. Important Points to Note:
The supply curve for a price-taking, profit-maximizing firm is given by the positively sloped portion of its marginal cost curve above the point of minimum average variable cost
if price falls below minimum AVC, the firm’s profit-maximizing choice is to shut down and produce nothing

69. Important Points to Note:
The firm’s profit-maximization problem can also be approached as a problem in optimal input choice
this yields the same results as does an approach based on output choices
In the short run, firms obtain producer surplus in the form of short-run profits and coverage of fixed costs that would not be earned if the firm produced zero output

70. Important Points to Note:
In situations of imperfect knowledge, firms may opt to maximize revenues
this means that the firm expands output until marginal revenue is zero
sometimes these decisions may be constrained by minimum profit requirements
Because managers act as agents for a firm’s owners, they may not always make decisions that are consistent with profit maximization