1. Chapter 11
Profit Maximization

2. Chapter Overview The Profit Maximization Condition
Marginal Revenue and Elasticity
Short Run Equilibrium
Short Run Supply Curve for a Price Taking Firm
Profit Function
Producer Surplus
5. Profit Maximization and Input Demand
Marginal Revenue product
Substitution effect
Output effect

3. 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
If firms are strictly profit maximizers, they will make decisions in a “marginal” way
examine the marginal profit obtainable from producing one more unit of hiring one additional laborer

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

5. 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
To maximize economic profits, the firm should choose the output for which marginal revenue is equal to marginal cost

6. 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

7. Marginal Revenue, Marginal Cost, and Profit Maximization
A firm chooses output q*, so that profit, the difference AB between revenue R and cost C, is maximized.
At that output, marginal revenue (the slope of the revenue curve) is equal to marginal cost (the slope of the cost curve).
Δπ/Δq = ΔR/Δq − ΔC/Δq = 0
MR(q) = MC(q)
The second-order condition prevents us from mistaking q which is less than q0 as a maximum

8. 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
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

9. Marginal Revenue from linear demand function
Suppose that the demand curve is
q = 100 – 10p
Solving for price, we get
p = -q/
This means that total revenue is
R = pq = -q2/ q
Marginal revenue will be given by
MR = dR/dq = -q/5 + 10

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

11. Marginal Revenue and Elasticity
The concept of marginal revenue is directly related to the elasticity of the demand curve 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

12. 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

13. Marginal Revenue and Elasticity
eq,p < -1
MR > 0
eq,p = -1
MR = 0
eq,p > -1
MR < 0

14. The Inverse Elasticity Rule
Because MR = MC when the firm maximizes profit, we can see that
The “markup” will fall as the demand curve facing the firm becomes more elastic
If eq,p > -1, MC < 0
This means that firms will choose to operate only at points on the demand curve where demand is elastic

15. Markup Pricing What does it mean? Where is it used? Why?
Markup on price or cost?
Should it be a constant?
How does markup relate to profit maximization?

16. Markup Pricing What does it mean? Markup on price or cost?
It is a pricing technique whereby a certain percentage of cost of goods sold or of price is added to the cost of goods sold in order to obtain the market price.
Markup on price or cost?
Suppose an item costs $50 to produce and is sold for $100.
Whether it is 50 percent markup or a markup of 100 percent.
In the 1st instance the margin is a markup on price (the proportion of the selling price that represents an amount added to the cost of the goods sold)
In 2nd case the markup is the markup on cost (the proportion of the cost of goods sold that is added on to that figure to arrive at the selling price)

17. Profit Maximization & Markup
Profit is maximum when:
Profit maximizing markup is inversely related to price
elasticity as follows:

18. Profit Maximization & Markup
Note: As long as Ep (which is nearly always negative) has an absolute value larger than 1, m will be positive.
Profit maximizing markup is lower the higher the absolute value of the own price elasticity of demand for the good or the service.
Value of Elasticity
Coefficient
100 x -1 / (Ep + 1) =
Optimal percent markup on Cost -2
100 -3 50 -4
33.3 -5 25 -9
12.5
The formula relating to markup on price to elasticity : Markup = -1/Ep

19. Marginal Revenue Curve
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
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)
Quantity per period 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
Market
price
SMC
SAC
P* = MR q*
Maximum profit occurs where P = SMC
SAVC
Quantity per period

23. Short-Run Supply by a Price-Taking Firm
Market
price
SMC
SAC
P* = MR
SAVC
Since P > SAC,
profit > 0
Quantity per period q*

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

25. Short-Run Supply by a Price-Taking Firm
Market
price
If the price falls to P***, the firm will produce q***
q***
P***
SMC
P* = MR
SAC
SAVC
Profit maximization
requires that P =
SMC and that SMC
is upward-sloping
Quantity per period 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
Market
price
SMC
The firm’s short-run supply curve is the SMC curve that is above SAVC
SAC
SAVC
Quantity per period

29. SC(v,w,q,k) = vk1 + wq1/k1-/
Short-Run Supply
Suppose that the firm’s short-run total cost curve is
SC(v,w,q,k) = vk1 + wq1/k1-/
where k1 is the level of capital held constant in the short run
Short-run marginal cost is

30. SAVC = SVC/q = wq(1-)/k1-/
Short-Run Supply
The price-taking firm will maximize profit where
p = SMC
Therefore, quantity supplied will be
To find the firm’s shut - down price, we need to solve for SAVC
SVC = wq1/k1-/
SAVC = SVC/q = wq(1-)/k1-/

31.  = Pq - C(q) = Pf(k,l) - vk - wl
Profit Functions
A firm’s economic profit can be expressed as a function of inputs
 = Pq - C(q) = Pf(k,l) - vk - wl
Only the variables k and l are under the firm’s control
the firm chooses levels of these inputs in order to maximize profits
treats P, v, and w as fixed parameters in its decisions

32. Profit Functions Envelope Results
A firm’s profit function shows its maximal profits as a function of the prices that the firm faces
Envelope Results
We can apply the envelope theorem to see how profits respond to changes in output and input prices

33. Producer Surplus in the Short Run
The profit function is nondecreasing in output prices, we know that if P2 > P1
(P2,…)  (P1,…)
The welfare gain to the firm of this price increase can be measured by
welfare gain = (P2,…) – (P1,…)
Mathematically, we can use the envelope theorem results

34. Producer Surplus in the Short Run
Market
price
SMC P1 q1
If the market price
is P1, the firm will produce q1
If the market price
rises to P2, the firm will produce q2 P2 q2
Quantity per period

35. Producer Surplus in the Short Run
Market
price
SMC
The firm’s profits rise by the shaded area P2 P1
Quantity per period q1 q2

36. Producer Surplus in the Short Run
Market
price
SMC
Suppose that the firm’s shutdown price is P0 P0 P1
Quantity per period q1

37. Producer Surplus in the Short Run
Market
price
SMC
Producer surplus at a market price of P1 is the shaded area P1 P0
Quantity per period q1

38. Producer Surplus in the Short Run
The extra profits available from facing a price of P1 are defined to be producer surplus
Producer surplus is the extra return that producers make by making transactions at the market price over and above what they would earn if nothing were produced
the area below the market price and above the supply curve

39. Producer Surplus in the Short Run
Because the firm produces no output at the shutdown price, (P0,…) = -vk1
profits at the shutdown price are equal to fixed costs
This implies that
producer surplus = (P1,…) – (P0,…)
= (P1,…) – (-vk1) = (P1,…) + vk1
producer surplus is equal to current profits plus short-run fixed costs

40. 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 –C(q) = Pf(k,l) – (vk + wl)
The FOCs for a maximum are
/k = P[f/k] – v = 0
/l = P[f/l] – w = 0

41. Profit Maximization and Input Demand
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.
MRPL = PL and MRPK = PK
These FOCs for profit maximization also imply cost minimization
they imply that MRTS = w/v

42. Profit Maximization and Input Demand
To ensure a true maximum, SOCs require that
kk = fkk < 0
ll = fll < 0
kk ll - kl2 = fkkfll – fkl2 > 0
capital and labor must exhibit sufficiently diminishing marginal productivities so that marginal costs rise as output expands

43. Input Demand Functions
In principle, the first-order conditions can be solved to yield input demand functions
Capital Demand = k(P,v,w)
Labor Demand = l(P,v,w)
These demand functions are unconditional
they implicitly allow the firm to adjust its output to changing prices

44. Single-Input Case We expect l/w  0
diminishing marginal productivity of labor
The first order condition for profit maximization was
/l = P[f/l] – w = 0

45. Two-Input Case
For the case of two (or more inputs), the story is more complex
if there is a decrease in w, there will not only be a change in l but also a change in k as a new cost-minimizing combination of inputs is chosen
when k changes, the entire fl function changes
But, even in this case, l/w  0

46. Two-Input Case When w falls, two effects occur substitution effect
if output is held constant, there will be a tendency for the firm to want to substitute l for k in the production process
output effect
a change in w will shift the firm’s expansion path
the firm’s cost curves will shift and a different output level will be chosen

47. Substitution Effect
If output is held constant at q0 and w falls, the firm will substitute L for K in the production process
k per period
Because of diminishing MRTS along an isoquant, the substitution effect will always be negative q0
l per period

48. Output Effect A decline in w will lower the firm’s MC
Price
Consequently, the firm will choose a new level of output that is higher P q0 q1
Quantity per period

49. Output Effect Output will rise to q1
k per period
Thus, the output effect also implies a negative relationship between l and w q0
l per period

50. Input Demand
For a fixed quantity, as price of labor decreases from $2 to $1, firm moves along its labor demand curve from B to A. Increase in output shifts the demand curve.

51. Cross-Price Effects
No definite statement can be made about how capital usage responds to a wage change
a fall in the wage will lead the firm to substitute away from capital
the output effect will cause more capital to be demanded as the firm expands production

52. Substitution and Output Effects
We have two concepts of demand for any input
the conditional demand for labor, lc(v,w,q)
the unconditional demand for labor, l(P,v,w)
At the profit-maximizing level of output
lc(v,w,q) = l(P,v,w)

53. Substitution and Output Effects
Differentiation with respect to w yields
substitution effect
output effect
total effect