unit ii:firms & markets theory of the firm profit maximization perfect competition/review 7/15...

UNIT II: Firms & Markets

• Theory of the Firm• Profit Maximization• Perfect Competition/Review• 7/15 MIDTERM

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).

K*Q = 10

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

K*Q = 10

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.

K*Q = 10

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.

K*Q = 10

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

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

Q1 Q2 Q3 Q

$ Consider a firm with long-run total

costs TC.

Profit Maximization

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

Q1 Q2 Q3 Q

same slope.

TR = PQ

Profit Maximization

Q1 Q2 Q3 Q

same slope.

TR = PQ

Profit Maximization

$ To maximize profits, the firm finds Q where

This will be where they have the same

slope.. slope.

TR = PQ

Q1 Q2 Q3 Q* Q

Profit Maximization

$ Marginal Analysis:

If TC is rising faster than TR, reduce Q.

If TR is rising faster than TC, increase Q.

TR = PQ

Q1 Q2 Q3 Q* Q

Profit Maximization

$ Marginal Analysis:

Recall: slope TR = MRslope TC = MC

Hence, to maximize profits:

MR = MC

TR = PQ

Q1 Q2 Q3 Q* Q

Profit Maximization

Q = f(K,L)

Profit Maximization

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?

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

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.

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)

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

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?

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

Supply in the Short-Run

P = MR

At price Po,

the firm earns 0 profit.

= TR – TC > 0

PRICE TAKER P = MR

If the price rises …

Output rises …

P = MR

= TR – TC > 0

PRICE TAKER P = MR

If the price rises …

And the firm earns profit

P = MR

= TR – TC < 0

PRICE TAKER P = MR

P = MR

TRIf the price decreases…

Output falls

= TR – TC < 0

PRICE TAKER P = MR

P = MR

TRIf the price decreases…

And the firm has losses

= TR – TC < 0

PRICE TAKER P = MR

P = MR

TRThe firm will continue to produce (< Q*)

iff P > AVC …

Losses are less than fixed costs. AVC

= TR – TC < 0

PRICE TAKER P = MR

Supply in the Short-RunTR

The firm’s short-run supply curve is in orange

MR = MC and P > 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

MC MCMC

q: firm

Q: market

MC MC MCAC

AVCACAC

In the long run,Firm C will exit the market.

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

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

The long-run supply curve. P* is the lowest possible price associated with non-negative profits, P* = ACmin.

s1=(mc1) s2

Sn, where n is the number of firms in the market.

The long-run supply curve. We can eliminate portions of the individual firms’ supply curves below P* (firms will exit).

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).

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*

Long-run equilibrium. Firms are producing at the efficient scale. P* = ACmin; = 0.

q* q Q* Q

Perfect Competition

• Perfect Competition • Equilibrium and Efficiency• General Equilibrium• Welfare Analysis

Perfect Competition

This will be where they have the same slope:

MR = MC.

TR = PQ

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

This will be where they have the same slope:

MR = MC.

TR = PQ

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).

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

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

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

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

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

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

MC = 2q

AVC = q

AC = 100/q + q

n = 60

Perfect Competition

TC = 100 + q2 QD = 1000 – 20P

P* = 20

q* = 10 q Q* = 600 Q

MC = 2q

AVC = q

AC = 100/q + q

n = 60

Perfect Competition

TC = 100 + q2 QD = 1500 – 20P

P* = 20

q* = 10 q Q* = 600 Q

MC = 2q

AVC = q

AC = 100/q + q

n = 60

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

MC = 2q

AVC = q

AC = 100/q + q

n = 60

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

MC = 2q

AVC = q

AC = 100/q + q

n = 60

P’ = 30

Perfect Competition

TC = 100 + q2 QD’ = 1500 – 20P

P* = 20

q* = 10 q Q** = 1100 Q

MC = 2q

AVC = q

AC = 100/q + q

n = 110

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).

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

unit ii:firms & markets theory of the firm profit maximization perfect competition/review 7/15...

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