Economics 50: Intermediate Microeconomics

Summer 2010

Stanford University

Michael Bailey

Lecture 7: Firm Supply

Overview

• In a perfectly competitive market, the upward sloping portion of the firm’s marginal cost curve (that

is above AV C(y) in the short run, and above AC(y) in the long run) is its supply curve

• The producer’s surplus is the area above the supply curve and below the price

• The per-unit profit is py −AC(y)

• Producer’s surplus is equal to profit plus fixed costs

• The firm will shut down in the long run if py < AC(y), and will shut down in the short run if

py < AV C(y)

• Amarket is in long run equilibrium when profits are 0 and is characterized by py = MC(y) = minAC(y)

• A monopolist sets a price that is a markup above marginal cost, and produces less than the competitive

equilibrium

• A monopsonist sets a wage that is below the marginal revenue product, and hires less labor than the

competitive equilibrium

Firm Supply in a Perfectly Competitive Market

If the firm is a price taker, then it will supply y∗(w, py) to the market where y∗(w, py) = f(x∗i (w, py)). Given

the cost function, the supply function solves y = arg max pyy−C(y). Notice that the first order condition of

this problem is:

py −∂C(y)

∂y= 0

=⇒ py = MC(y)

Thus the marginal cost is the inverse supply curve. In a perfectly competitive market, the market price is

the marginal revenue to the firm, since it takes the market price as given, and thus the profit-maximizing firm

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equates the price with marginal cost. The second order condition of this problem puts another restriction

on the supply curve:

−∂2C(y)

∂y2< 0

=⇒ ∂2C(y)

∂y2=∂MC(y)

∂y> 0

The output where p = MC(y) is a maximum only ifMC(y) is upward sloping, else the firm could produce

another unit and marginal cost would be falling. So the inverse supply curve is the upward sloping region of

the marginal cost curve.

Figure 1: The firm wants to maximize the difference between revenue and cost

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Figure 2: The maximum profit is where p = MC(y) and MC(y) is upward sloping

Figure 3: The upward-sloping portion of the marginal cost curve is the inverse supply curve

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Graphical Analysis

We can rewrite profits as:

π = pyy − C(y)

= pyy − yC(y)

y

= pyy − yAC(y)

= y(py −AC(y)︸ ︷︷ ︸π per unit

)

Thus the profit per unit is equal to py −AC(y), the revenue per unit minus the cost per unit.

Revenue is the rectangle pyy

Profits are the rectangle y(py −AC(y))

Figure 4: π = y(py −AC(y) which is the rectangle shown

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Figure 5: When the price is less than AC(y), the firm incurs a loss equal to y(py −AC(y)

Variable costs are the area under MC(y)(MC(y) = ∂V C(y)

∂y

)Producer’s surplus is the area above the supply curve (MC(y)) and below the price

PS =

∫ p

0

y∗(p)dp

∆PS =

∫ p2

p1

y∗(p)dp

Total Cost is the rectangle yAC(y) = C(y)

Therefore, producer’s surplus is equal to profits plus total costs minus variable costs:

PS = π + C(y)− V C(y)

= π + FC

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Figure 6: Revenue = 1 + 2 + 3 + 4; π = 1 + 2; C(y) = y ·AC(y) = 3 + 4; Variable Costs = 2 + 4 (area underMC(y)); PS = 1 + 3; FC = C(y)− V C(y) = 3− 2; =⇒ PS = π + FC

Shut Down Decision

The firm would prefer to shut down and make 0 output if the profit from doing so were greater than the

profit from producing. The firm will shut down if:

−C(0) > max pyy − C(y)

In the long run, C(0) = 0 so we can write this condition as:

0 > pyy − C(y)

=⇒ C(y)

y= AC(y) > py

So in the long run, if the price is less than the average cost, the firm will shut down and produce no

output.

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Figure 7: The long-run supply curve is the upward-sloping portion of the marginal cost curve that is aboveAC(y). If p < AC(y), the firm will shut down in the long run

In the short run, C(0) = FC, so the shut-down condition is:

−FC > pyy − FC − V C(y)

=⇒ V C(y)

y= AV C(y) > py

In the short run, the firm will shut down if the price is less than the average variable cost. Why are the

conditions different? In the long run, the firm can avoid all fixed costs and make a profit of 0, so the firm

must make at least 0 profits in the long run. In the short-run, the firm will pay it’s fixed costs no matter

what, so it could be that firm is making negative profit, but as long as the profit is greater than its fixed

costs, it is still more profitable to produce than to shut down. Notice that if the price is greater than the

average variable cost, then each unit will make positive profit for the firm, so it is better to produce, even if

those profits are not enough to cover fixed costs, it is more profitable than shutting down.

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Figure 8: The short-run supply curve is the upward-sloping portion of the marginal cost curve that is aboveAV C(y). If p < AV C(y), the firm will shut down in the short run

Long-run Market Equilibrium

In a perfectly competitive market, firms are free to costlessly enter or leave the market (no barriers to entry).

If all firms are identical, then if firms are making positive profit, then firms will enter the market which will

increase supply and lower the price. If firms are making negative profit, then firms will leave the market,

reducing supply and raising the price. The market is in equilibrium when no firms want to enter or exit the

industry and is characterized by 0 profit, or p = AC(y). Therefore, the market is long-run equilibrium when:

p = MC(y) = minAC(y)

Since MC(y) = AC(y) at the minimum of AC(y), we can just write:

p = MC(y) = AC(y)

Remember that this means that economic profit is 0, the firm could still be earning a large accounting

profit, but if we implicitly paid all factors according to their opportunity cost, it would just equal our profit.

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Figure 9: Initially, firms in this industry make positive profits. In the long run, firms enter the industryincreasing supply. Firms will continue to enter until p = minAC(y)

Figure 10: Initially, firms in this industry make negative profits. In the long run, firms exit the industrydecreasing supply. Firms will continue to exit until p = minAC(y)

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Example 1 The short-run cost function of the firm is C(F,w,K, y) = F + wy3

3K, if all firms are identical,

what will be the long-run quantity and price in the market when w = 16, F = 32, and K = 9?

At the long-run equilibrium, p = MC = AC.

p =wy2

K=F

y+wy2

3K

=⇒ wy3

K= F +

wy2

3K

=⇒ 2

3

wy3

K= F

=⇒ y3 =3

2

FK

w

=⇒ y =

(3

2

FK

w

) 13

= 3

So for a firm in this industry to make 0 profits, each must be producing 3 units of output, and the price

must be:

p =wy2

K

=3w

K

= 16

If there are N firms in the industry, then the market quantity will be QS = 3N.

Example 2 The supply function of the 10 identical firms in the industry is y∗(p) = 49p2. Each firm has a

cost function C(y) = 500 + y32 =⇒MC(y) = 3

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√y and AC(y) = 500

y +√y. The market demand is given by

QD = 9000 − 509 p

2. What is the short-run equilibrium? How many firms will exist in this industry in the

long run?

π = pyy − 500− y 32

QS =∑

y∗(p) = 10 · 4

9p2 =

40

9p2

QD = 9000− 50

9p2

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A short-run equilibrium is where QD = QS :

40

9p2 = 9000− 50

9p2

=⇒ p = 30

Q = 4000

y∗ = 400

π = $3500

Because there is positive profit, in the long run firms enter the industry. Suppose that there are N firms

in the industry. Then the market supply curve will be given by:

QS =∑

y∗(p) = N · 4

9p2 =

4N

9p2

At the equilibrium, we must have supply equal to demand:

QS =4N

9p2 = 9000− 50

9p2

=⇒ p =90√

10√50 + 4N

Q =36000N

50 + 4N

y =36000

50 + 4N

We could solve for where the profit equals 0, but it is easier to find the output that minimizes the average

cost:

AC(y) =500

y+√y

=⇒ −500y−2 +1

2y−

12 = 0

=⇒ y32 = 1000

=⇒ y = 100

=⇒ 36000

50 + 4N= 100

=⇒ 36000 = 5000 + 400N

=⇒ N =31000

400= 77.5

In the long run, there are 77 firms in the industry, so 67 firms must have entered the industry.

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Monopoly

We now begin to relax some of the market constraints and consider a case where the firm is not a price

taker. A monopoly is a market with only one seller, and as the sole seller has the power to set the price in

the market. However, the monopolist is constrained by the inverse demand curve, p(y), so if it sets a price

of p(y), it will be able to sell at most y units of output. We can write the profits of the monopolist as:

maxπ = p(y)y − C(y)

The first-order condition of this problem is:

∂p(y)

∂yy + p(y)︸ ︷︷ ︸

Marginal Revenue

=∂C(y)

∂y︸ ︷︷ ︸Marginal Cost

As expected, the monopolist equates marginal revenue with marginal cost. The difference between the

monopolist and the perfectly competitive firm is the marginal revenue term. In a perfectly competitive

market, the firm can sell all of its units of output at the market price, py, and thus has MR(y) = py. The

monopolist also has a p(y) term in its marginal revenue, if it increases output by one unit, revenue will

increase by the price, p(y). However, the monopolist also has a ∂p(y)∂y y term in it’s marginal revenue, this is

because it is constrained by the demand curve. If the firm increases its output by one unit, it must lower

the price on all of its output so the market clears, so the marginal revenue also changes by the change in

price times quantity, or ∂p(y)∂y y.

We can predict from the marginal revenue term that the monopolist will produce less than the perfectly

competitive firm. For the marginal unit of output, the firm has to lower the price on all of its units of outputs

for the market to clear. Due to this "negative" term dragging marginal revenue down, marginal revenue will

equal cost at a much lower output then the firm that has MR = p.

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It is instructional to rewrite the firm’s optimal condition in terms of elasticities:

∂p(y)

∂yy + p(y) = MC(y)

=⇒ ∂p(y)

∂y

y

p(y)p(y) + p(y) = MC(y)

=⇒ p(y)

εy,p+ p(y) = MC(y)

=⇒ p(y)

(1

εy,p+ 1

)= MC(y)

=⇒ p(y)

(εy,p + 1

εy,p

)= MC(y)

=⇒ p(y) =

(εy,p

εy,p + 1

)MC(y)

Thus the price the monopolist sets is a "markup" above marginal cost, and is referred to as "markup

pricing". Notice that the monopolist will never operate on the inelastic portion of the demand curve. If

−1 < εy,p < 0, then(

εy,pεy,p+1

)is negative, and the optimal price would be negative, so the optimality condition

cannot hold. This is an intuitive result, on the inelastic portion of the demand curve, the monopolist could

decrease output and increase revenue (as we discussed in detail in the first part of the course). The firm

would move to a lower cost because less output means it uses fewer inputs. More revenue and less cost

means profits would be higher. We can’t use this logic to rule out the firm operating on the elastic portion

of the demand curve. On the elastic portion, the firm could decrease its price and increase revenue, but it

would have to produce more output, so costs would icnrease and the change in profits is indeterminate. The

maximum profit is attained when MR = MC.

Example 3 Linear Demand p(y) = a− bP

Revenue = p(y)y = ay− by2 and Marginal Revenue = a−2by. So marginal revenue is demand with twice

the slope. Notice that the inelastic portion of the demand curve corresponds to a negative marginal revenue.

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Figure 11: The monopolist produces where MR(y) = MC(y)

Figure 12: The monopolist produces less than the competitive output (where p = MC(y)), capturing someof the consumer’s surplus as an increased producer’s surplus, but creating deadweight loss (DWL)

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Example 4 Constant Elasticity Demand Function p(y) = ay−b

Because the elasticity is constant along the curve, we can plot the curve(

εy,pεy,p+1

)MC(y) which is a

constant times marginal cost, or a marginal cost shifter. The monopolist will produce where this curve

intersects demand.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.00

1

2

3

y

p

Figure 13: Constant Elasticity Demand Function plotted with MC(y) and MC(y) · εε+1 . The monopolist

produces where p(y) = MC(y) · εε+1

Monopsony

A monopsony is a market with only one buyer, in contrast to a monopoly with only one seller. The monop-

sonist cannot take the prices of the good it is buying as constant because it is constrained by the supply

curve. For example, consider a firm that is the only buyer of labor services. The firm is constrained by the

labor supply curve and must offer a higher wage to attract more labor. We can write the supply function

in the labor market as w(L) which tells us how much labor, L, will be supplied at a given wage w(L). The

objective function of the firm that only uses labor to produce an output is:

max pyf(L)− w(L)L

As always, the firm sets marginal revenue equal to marginal cost:

py∂f(L)

∂L︸ ︷︷ ︸MRPL

=∂w(L)

∂LL+ w(L)︸ ︷︷ ︸

Marginal Cost of Labor

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The marginal revenue term (marginal revenue product) is the same for the monopsonist, MRPL =

pyMPL, the marginal cost term is different. In the perfectly competitive case, this term is equal to w since

the firm can hire all the labor at wants at a constant wage. The monopsonist is constrained by the supply

curve however, and if it wants to higher more labor, it must offer a higher wage, thus the marginal cost has

a w(L) term indicating it must pay a wage of w to the marginal worker, but it also has a ∂w(L)∂L L term,

indicating it must also offer this higher wage to all its labor.

From this equation we can already see that the monopsonist will hire less labor than in the competitive

equilibrium. When deciding to hire the marginal worker, the firm must not only see if the MRPL is greater

than the wage, but also if the MRPL is greater than the wage plus the effect of raising the wages for all

other workers. This additional productivity requirement on the marginal worker induces the firm to set a

lower wage and higher fewer units of labor. Rewriting the optimal condition in terms of elasticities:

MRPL =∂w(L)

∂LL+ w(L)

=∂w(L)

∂L

L

w(L)w(L) + w(L)

= w(L)

(1

εL,w+ 1

)= w(L)

(εL,w + 1

εL,w

)=⇒ MRPL

(εL,wεL,w+1

)= w(L)

Because the elasticity of supply is positive, this means the firm will set a wage that is less than the

MRPL, as expected.

Example 5 Linear Supply w(L) = a+ bL =⇒ C(L) = aL+ bL2, MC(y) = a+ 2bL.

Notice that the marginal cost curve is the supply curve with twice the slope.

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Figure 14: The monopsonist sets a wage where MC(y) = MRPL

Figure 15: The monopsonist sets a wage that is less than the competitive one, hiring too little labor thanis effi cient. The monopsonist captures some of the producer’s surplus (the surplus that would have goneto labor suppliers) thus increasing consumer’s (i.e. the monopsonist’s) surplus, but creates deadweight loss(DWL)

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