unit ii:firms & markets theory of the firm profit maximization perfect competition/review 7/15...
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UNIT II: Firms & Markets
• Theory of the Firm• Profit Maximization• Perfect Competition/Review• 7/15 MIDTERM
7/1
Profit Maximization
• Last Time• The Short-Run and the Long-Run• Firm and Market Supply• Perfect Competition (Part 1)
We saw last time that we can solve the firm’s cost minimization problem analogously to the consumer’s utility maximization problem. Cost minimization requires that the firm produce using a combination of inputs for which the ratios of the marginal products, or the marginal rate of technical substitution, equals the ratio of the input prices:
MRTS = w/r
Last Time
2 Provisos:
• Only in the Long-Run• Only part of the firm’s problem
Profit Maximization
The firm wants to maximize this difference:
Profit ( = Total Revenue(TR) – Total Cost(TC)
TR(Q) = PQ TC(Q) = rK + wL
P Price L LaborQ Quantity K Capital
w Wage Rater Rate on Capital
Q = f(K,L)
Revenue Cost
Last time: we considered a firm that produces output according to the following production function.
Q = 4K½L½
and w = $18 and r = $36.
How much will it cost this firm to produce 10 units of output in the long-run? Q units?
Profit Maximization
Last time: we considered a firm that produces output according to the following production function.
Q = 4K½L½
and w = $18 and r = $36.
How much will it cost this firm to produce 10 units of output in the long-run? Q units?
The long-run total cost curve (TC(Q)) represents the minimum cost to produce Q units of output.
Profit Maximization
How much will it cost this firm to produce 10 units of output in the long-run?
Q = 4K1/2L1/2
w = 18; r = 36
MRTS = MPL/MPK
MPL = 2K1/2L-1/2
MPK = 2K-1/2L1/2 MRTS = K/L.
= w/r = 18/36 = L = 2K.
Cost Minimization in the Long-Run
The firm’s optimal factor proportion
(given technology and factor prices).
Cost Minimization in the Long-Run
L* L
K
K*Q = 10
How much will it cost this firm to produce 10 units of output in the long-run?
L = 2K
Tangency between the isoquant and an isocost curve shows
the economically efficient combination
K*, L*. Hence, the condition
for optimal factor proportion is:
MRTS = w/r
Cost Minimization in the Long-Run
L* L
K
K*Q = 10
How much will it cost this firm to produce 10 units of output in the long-run?
L = 2K
The condition for optimal factor proportion is:
MRTS = w/r.
This is LR condition! Why? Because some
factors (K) are fixed in the SR.
Cost Minimization in the Long-Run
L* L
K
K*Q = 10
How much will it cost this firm to produce 10 units of output in the long-run?
L = 2K
Another way to think about this: TC is a
projection of the firm’s long-run
output expansion path: the locus of
optimal factor bundles (K,L) for
different levels of Q.
Cost Minimization in the Long-Run
K
K*Q = 10
How much will it cost this firm to produce 10 units of output in the long-run?
TC = 18L + 36K = 9Q/(2)1/2 + 9Q/(2)1/2
= 18/(2)1/2(Q)
= 12.73Q
MC = 12.73 = AC
We can solve for the firm’s long run total
cost function for any level of output.
Cost Minimization
Graphically: Q = 4K1/2L1/2
w = 18; r = 36
Q
$
In the SR, K = 16
TCsr = 576 + 18(Q/16)2
Fixed Costs = rK = $576
TClr = 12.73Q
MC = 12.73
At this point, 16 units of capital is optimal.
Profit Maximization
Profit ( = Total Revenue(TR) – Total Cost(TC)
Q1 Q2 Q3 Q
$ Consider a firm with long-run total
costs TC.
TC
Profit Maximization
Profit ( = Total Revenue(TR) – Total Cost(TC)
Q1 Q2 Q3 Q
$ To maximize profits, the firms finds Q where
distance between TC and TR is greatest.
the same slope.TC
TR = PQ
Profit Maximization
Profit ( = Total Revenue(TR) – Total Cost(TC)
Q1 Q2 Q3 Q
$ To maximize profits, the firms finds Q where
distance between TC and TR is greatest.
same slope.
TC
TR = PQ
Profit Maximization
Profit ( = Total Revenue(TR) – Total Cost(TC)
Q1 Q2 Q3 Q
$ To maximize profits, the firms finds Q where
distance between TC and TR is greatest.
same slope.
TC
TR = PQ
Profit Maximization
Profit ( = Total Revenue(TR) – Total Cost(TC)
$ To maximize profits, the firm finds Q where
distance between TC and TR is greatest.
This will be where they have the same
slope.. slope.
TC
TR = PQ
max
Q1 Q2 Q3 Q* Q
Profit Maximization
Profit ( = Total Revenue(TR) – Total Cost(TC)
$ Marginal Analysis:
If TC is rising faster than TR, reduce Q.
If TR is rising faster than TC, increase Q.
TC
TR = PQ
max
Q1 Q2 Q3 Q* Q
Profit Maximization
Profit ( = Total Revenue(TR) – Total Cost(TC)
$ Marginal Analysis:
Recall: slope TR = MRslope TC = MC
Hence, to maximize profits:
MR = MC
TC
TR = PQ
max
Q1 Q2 Q3 Q* Q
Profit Maximization
The firm wants to maximize this difference:
Profit ( = Total Revenue(TR) – Total Cost(TC)
TR(Q) = PQ TC(Q) = rK + wL
P Price L LaborQ Quantity K Capital
w Wage Rater Rate on Capital
Q = f(K,L)
Profit Maximization
The firm wants to maximize this difference:
Profit ( = Total Revenue(TR) – Total Cost(TC)
TR(Q) = PQ TC(Q) = rK + wL
P Price L LaborQ Quantity K Capital
w Wage Rater Rate on Capital
Q = f(K,L)
Revenue Cost
Profit Maximization
Demand for the firm’s output is given by Q = 100 – 2P. Find the firm’s profit maximizing level of output.
Q = 4K1/2L1/2
w = 18; r = 36Q = 100 – 2P => P = 50 – 1/2Q
TR = PQ = (50 – 1/2Q)Q= 50Q – 1/2 Q2
MR = 50 – Q = MC = 12.73
=> Q* = 37.27; P* = 31.37
Profit Maximization
Demand for the firm’s output is given by Q = 100 – 2P. Find the firm’s profit maximizing level of output.
Q = 4K1/2L1/2
w = 18; r = 36Q = 100 – 2P => P = 50 – 1/2Q
TR = PQ = (50 – 1/2Q)Q = TR – TC = 50Q – 1/2 Q2 – 12.73Q
FOC: d/dQ = 50 – Q – 12.73 = 0
=> Q* = 37.27; P* = 31.37
Profit Maximization in the Long-Run
We solved the firm’s optimization problem focusing on the profit output level, Q*, but it is important to emphasize that the optimization principle also tell us about input choices.
When the firm chooses an output level Q* that maximizes for given factor prices (w, r), the firm has simultaneously solved for L* and K*.
To produce Q* = 37.27 (given the production function, Q = 4K1/2L1/2, and optimal factor proportion, L = 2K), we find: L* = 6.58; K* = 3.29.
Finally, = TR–TC = PQ–12.73Q = (31.37–12.73)37.27 = $694.71.
Profit Maximization in the Short-Run
We saw that the firm maximizes profit by choosing a level of output such that marginal revenue equals marginal cost MR = MC. In the long-run, this implies that the firm will be utilizing its optimal factor proportion, such that MRTS = w/r.
In the short-run, however, K is fixed, so the firm will not be able to optimally adjust factor proportions.
Does the firm maximize profits by setting MR = MC in the short-run? How?
Profit Maximization in the Short-Run
We solved the firm’s optimization problem focusing on the output level, but it is important to emphasize that the optimization principle also tell us about input choices.
When the firm chooses an output level (Q*) that maximizes profit for given factor prices (w, r), the firm has simultaneously solved for L* and K*.
In the short-run, the firm also maximizes by setting MR = MC. But because K is fixed, the problem is simply one of optimal utilization of labor. In other words, the firm asks, “On the margin, can I increase my profit by adding (or subtracting) another unit of labor from the production process?”
Profit Maximization in the Short-RunQ = 4K1/2L1/2 w = 18; r = 36K = 16 P = 10
MR = MC
TR = PQ = 10Q TC = rK + wL = 36(16) + 18L [ Q = 16L½ => L = (Q/16)2 ] TC = 36(16) + 18(Q/16)2 = 576 + 18(Q/16)2
MR = 10 = MC = (36/256)Q
=> Q = 71.1; L = 19.8
Revenue Cost
Profit Maximization in the Short-Run
In the short-run, the firm’s profit maximizing calculus weighs the benefit of hiring an additional unit of labor versus its cost. If the firm can add (subtract) another unit of labor and increase revenue by more than it increases cost, it should add (subtract) it, and it should keep on adding (subtracting) until MR = MC.
Profit Maximization in the Short-Run
What does an additional unit of L contribute to ?
But hiring an additional unit of L also increases the firm’s total costs.
MFCL = wMarginal Factor
Cost (MFC)
The dollar amount added
to total cost from an additional unit of labor
(capital).
NOTE (for fixed K):
MC = dTC/dQ = w (dL/dQ) MC = w (1/MPl)
Profit Maximization in the Short-Run
So the firm’s short-run optimality condition can be rewritten as:
MRPL = MFCL
If the firm can add (subtract) another unit of labor and increase revenue by more than it increases cost, it should add (subtract) it, and it should keep on adding (subtracting) until MRPL = MFCL.
This, in turn, is what determines the firm’s (short-run) demand for labor:
P (MPL) = w
Profit Maximization in the Short-Run
Consider a price-taking firm that is currently producing 450 units of output at a price of $2.50 per unit. In the short run, the firm’s capital stock is fixed at 16 machine-hours. The firm is currently employing 100 hours of labor at a wage of $10/hour, and the at a rental rate (r) is $10/hour. With this mix of inputs MPK=2 and MPL= 4.
Is the firm maximizing profits in the short run?
Profit Maximization in the Short-Run
Is the firm maximizing profits in the short run?Yes.
MRPL = P(MPL) = 2.50(4) = 10 = w = MFCL MR = P = $2.50; MC = w/MPL = $10/4 = $2.50
Does it matter that MPL/MPK = w/r (4/2 = 10/10) ?Does it matter that = TR-TC
= 1125 – 1160 = -35 ?
Shut down rule: If P < AVC, Q = 0. The firm would lose its fixed costs ($160) if it shut down, or $35 if it produces Q = 450. What matters is variable costs, because fixed costs are spent (“sunk”) in the short run. Profit max = Loss min.
$
$
= TR – TC = 0
PRICE TAKER P = MR
TC
MCQ
AC
Supply in the Short-Run
P = MR
TR
Q* Q
At price Po,
the firm earns 0 profit.
Q Q
$
$
= TR – TC > 0
PRICE TAKER P = MR
TC
MCQ
AC
Supply in the Short-Run
If the price rises …
Output rises …
TR
P = MR
Q Q
$
$
= TR – TC > 0
PRICE TAKER P = MR
TC
MCQ
AC
Supply in the Short-Run
If the price rises …
And the firm earns profit
TR
P = MR
$
$
= TR – TC < 0
PRICE TAKER P = MR
TC
MCQ
AC
Supply in the Short-Run
P = MR
TRIf the price decreases…
Output falls
Q Q
$
$
= TR – TC < 0
PRICE TAKER P = MR
TC
MCQ
AC
Supply in the Short-Run
P = MR
TRIf the price decreases…
And the firm has losses
Q Q
Q
$
$
= TR – TC < 0
PRICE TAKER P = MR
TC
MCQ
AC
Supply in the Short-Run
P = MR
TRThe firm will continue to produce (< Q*)
iff P > AVC …
Losses are less than fixed costs. AVC
Q
$
$
= TR – TC < 0
PRICE TAKER P = MR
TC
MCQ
AC
Supply in the Short-RunTR
The firm’s short-run supply curve is in orange
Where
MR = MC and P > AVC.
AVC
Supply in the Long-Run
In the short run, firms adjust to price signals by varying their utilization of labor (variable factors).
In the long-run, firms adjust to profit signals by – varying plant size (fixed factors); and – entering or exiting the market.
What determines the number of firms in the long-run?
Short-run equilibrium with three firms. Firm A is making positive profits, Firm B is making zero profits, and Firm C is making negative profits (losses).
Firm A Firm B Firm C q q q
$
P
AC AC
MC MCMC
AC
q: firm
Q: market
Supply in the Long-Run
Supply in the Long-Run
Short-run equilibrium with three firms. Firm A is making positive profits, Firm B is making zero profits, and Firm C is making negative profits (losses).
Firm A Firm B Firm C q q q
$
P
MC MC MCAC
AVCACAC
Supply in the Long-Run
Short-run equilibrium with three firms. Firm A is making positive profits, Firm B is making zero profits, and Firm C is making negative profits (losses).
Firm A Firm B Firm C q q q
$
P
In the long run,Firm C will exit the market.
MC
AC
MC
AC
Supply in the Long-Run
In the long-run, inefficient firms will exit, and new firms will enter, as long as some firms are making positive economic profits.
Firm A Firm B Firm D q q q
$
P
MC
AC
MC
AC
MC
AC
Supply in the Long-RunIn the long-run, if there are no barriers to entry, then new firms have access to the most efficient production technology. We call this the efficient scale.
Firm A Firm D Firm E q* q q* q q* q
$
P*
MC MC
AC
MC
ACAC
Supply in the Long-Run
The long-run supply curve. P* is the lowest possible price associated with non-negative profits, P* = ACmin.
$
P*
Q
s1=(mc1) s2
s3
s4
Sn, where n is the number of firms in the market.
Supply in the Long-Run
The long-run supply curve. We can eliminate portions of the individual firms’ supply curves below P* (firms will exit).
$
P*
Q
s1 s2
s3
s4
Supply in the Long-Run
The long-run supply curve. We can also eliminate portions of the individual firms’ supply curves above P* (demand will be met by another firm’s supply curve).
$
P*
Q
s3
s4
s1 s2
Supply in the Long-Run
The long-run supply curve. As the number of firms increases, the long run supply curve approximates a straight line at P* = ACmin.
$
ACmin = P*
Q
LRS
Supply in the Long-Run
Long-run equilibrium. Firms are producing at the efficient scale. P* = ACmin; = 0.
$
P*
q* q Q* Q
$
LRS
MC
AC
D
Perfect Competition
• Perfect Competition • Equilibrium and Efficiency• General Equilibrium• Welfare Analysis
Perfect Competition
Profit ( = Total Revenue(TR) – Total Cost(TC)
$ To maximize profits, the firm finds Q where
distance between TC and TR is greatest.
This will be where they have the same slope:
MR = MC.
TC
TR = PQ
max
Q1 Q2 Q3 Q* Q
Price-Taker
P is given; can sell any amount at P.
Later we will look at other possible assumptions.
Perfect Competition
Profit ( = Total Revenue(TR) – Total Cost(TC)
$ To maximize profits, the firm finds Q where
distance between TC and TR is greatest.
This will be where they have the same slope:
MR = MC.
TC
TR = PQ
max
Q1 Q2 Q3 Q* Q
Price-Taker
P is given; can sell any amount at P.
Later we will look at other possible assumptions, e.g., monopoly.
Perfect Competition
Assumptions
• Firms are price-takers: can sell all the output they want at P*; can sell nothing at any price > P*.
• Homogenous product: e.g., wheat, t-shirts, long-distance phone minutes.
• Perfect factor mobility: in the long run, factors can move costlessly to where they are most productive (highest w, r).
• Perfect information: firms know everything about costs, consumer demand, other profitable opportunities, etc.
Perfect Competition
Implications
1) Firms earn zero (economic) profits.
2) Price is equal to marginal cost.
3) Firms produce at minimum average cost, i.e., “efficient scale.”
4) Market outcome is Pareto-efficient.Now we want to consider the welfare implications
Perfect Competition
The Short-Run & the Long-Run
In the short-run, firms adjust to price signals by varying their utilization of labor (variable factors).
In the long-run, firms adjust to profit signals by – varying plant size (fixed factors); and – entering or exiting the market.
We can use this info to solve for the long-run competitive equilibrium.
Perfect Competition
Consider a perfectly competitive industry characterized by the following total cost and demand functions:
TC = 100 + q2 QD = 1000 – 20P
Find the market equilibrium in the long-run. How many firms are in the market?
Perfect Competition
TC = 100 + q2 QD = 1000 – 20P
$
q* = 10 q 600 Q
$
LRS
MC = 2q
AVC = q
AC = 100/q + q
1) = TR – TC = 0
= Pq – (100 + q2) = 0
= (2q)q – (100 + q2) = 0
2q2 – 100 - q2 = 0
q2 = 100; q* = 10
Perfect Competition
TC = 100 + q2 QD = 1000 – 20P
$
q* = 10 q 600 Q
$
LRS
MC = 2q
AVC = q
AC = 100/q + q
1) = TR – TC = 0
= Pq – (100 + q2) = 0
= (2q)q – (100 + q2) = 0
2q2 – 100 - q2 = 0
q2 = 100; q* = 10
Perfect Competition
TC = 100 + q2 QD = 1000 – 20P
$
q* = 10 q 600 Q
$
LRS
MC = 2q
AVC = q
AC = 100/q + q
1) = TR – TC = 0
= Pq – (100 + q2) = 0
= (2q)q – (100 + q2) = 0
2q2 – 100 - q2 = 0
q2 = 100; q* = 10
2) P = MC = 2q
Perfect Competition
TC = 100 + q2 QD = 1000 – 20P
q* = 10 q 600 Q
$
LRS
MC = 2q
AVC = q
AC = 100/q + q
1) = TR – TC = 0
= Pq – (100 + q2) = 0
= (2q)q – (100 + q2) = 0
2q2 – 100 - q2 = 0
q2 = 100; q* = 10
$
P* = 20
Perfect Competition
TC = 100 + q2 QD = 1000 – 20P
q* = 10 q 600 Q
$
LRS
MC = 2q
AVC = q
AC = 100/q + q
3)Firms produce at ACmin
i) AC = MC 100/q + q = 2q 100 + q2 = 2q2
q2 = 100; q* = 10
ii) AC’ = 0 -100/q2 + 1 = 0 q2 = 100; q* = 10
$
P* = 20
q* is the efficient scale
Perfect Competition
TC = 100 + q2 QD = 1000 – 20P
$
P* = 20
q* = 10 q Q* = 600 Q
$
LRS
MC = 2q
AVC = q
AC = 100/q + q
D
n = 60
Perfect Competition
TC = 100 + q2 QD = 1000 – 20P
$
P* = 20
q* = 10 q Q* = 600 Q
$
LRS
MC = 2q
AVC = q
AC = 100/q + q
D
n = 60
Perfect Competition
TC = 100 + q2 QD = 1500 – 20P
$
P* = 20
q* = 10 q Q* = 600 Q
$
LRS
MC = 2q
AVC = q
AC = 100/q + q
D
n = 60
D’
Perfect Competition
TC = 100 + q2 QD’ = 1500 – 20P
q.n = (MC/2)n = (P/2)n => QS = ??? SRS$
P* = 20
q’ = 15 q Q’ = 900 Q
$
LRS
MC = 2q
AVC = q
AC = 100/q + q
n = 60
SRS
D
D’
P’= MR
Perfect Competition
TC = 100 + q2 QD’ = 1500 – 20P
q.n = (MC/2)n = (P/2)n => QS = 30P SRS$
P* = 20
q’ = 15 q Q’ = 900 Q
$
LRS
MC = 2q
AVC = q
AC = 100/q + q
n = 60
SRS
D
D’
P’ = 30
Perfect Competition
TC = 100 + q2 QD’ = 1500 – 20P
$
P* = 20
q* = 10 q Q** = 1100 Q
$
LRS
MC = 2q
AVC = q
AC = 100/q + q
n = 110
D
D’
Perfect Competition
The Short-Run & the Long-Run
In the short-run, firms adjust to price signals by varying their utilization of labor (variable factors).
In the long-run, firms adjust to profit signals by – varying plant size (fixed factors); and – entering or exiting the market.
What determines the number of firms in long run equilibrium?
Perfect Competition
In the Long-Run …
1) Firms earn zero (economic) profits.
2) Price is equal to marginal cost.
3) Firms produce at minimum average cost, i.e., “efficient scale.”
4) Market outcome is Pareto-efficient.
Next Time
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