Profit Maximization

Dr. Andrew McGeeSimon Fraser University

Firm Behavior• We know what a firm can produce in terms of

output with any amount of input (production function)

• We know how much producing any amount of output will cost a firm given our knowledge of the production function (cost function)

• Now we must make a key behavioral assumption about firms:– Firms choose output levels so as to maximize

profit

Profit Maximization• Firms choose Q so as to maximize profits:

Where TR(Q)=total revenuesFirst-order condition for profit maximization is

• We have solved for MC(Q), but what is MR(Q)?– Extra revenue from selling one more unit– Depends on assumptions about market structure– If firm is a price-taker, it can sell as much as it wants at

the prevailing market price P, so its total revenue is P*Q, meaning MR(Q)=P

Intuition behind profit maximization FOC

• Imagine that you run a small business and are trying to determine how much output to produce. Do you solve for a total cost function and then for the FOC for profit maximization? Probably not. More likely, you experiment with different output levels and use a decision-rule to pick the best output level. – If you expand output by one unit and profit increases, try

expanding by another unit.– When profit declines with an additional unit of output,

you stop expanding.– This simple decision rule amounts to expanding when

MR>MC and contracting when MR<MC. In other words, the optimal Q (where you stop) occurs when MR=MC.

Revenue• Total revenue=P(Q)*Q• The price may depend on how much a firm sells• If a the firm is a price taker, it can sell as much as it

wants at the prevailing price

• Marginal revenue depends on Q unless the firm is a price taker

• Because , always (Q>0). This implies that the marginal revenue curve falls on or below the demand curve faced by the firm

Marginal revenue

• The extra revenue the firm gets for each additional unit sold

• Always equal to or less than the price

Q

P

Demand faced by firm

MR(Q)

What does this graph look like for a price taker?

For a linear demand function such as P=a-bq, it is always true that the marginal revenue is given by MR(q)=a-2bq

a/ba/(2b) Because this is a linear demand function

Profit maximization

• When the firm is a price taker, its FOC implies that MC(Q)=P

• The SOC for profit maximization is

This means that the MC curve must be increasing at the profit maximizing output

Profit Maximization

$

Q

TC(Q) TR(Q)=PQ

Assume the firm is a price-taker so that is can sell as much as it wants at the prevailing market price: P(Q)=P

Q1 Q2

MC(Q1)=MR=P MC(Q2)=MR=P

Profit

Loss

Here we see two points at which the FOC for a profit maximum are satisfied, but one clearly maximizes the firm’s losses and not its gains

SOC for a profit maximum:

Example: Linear demand function• Suppose the firm faces the demand for its

product: , so

• When Q=40, P=6 and MR=2 (MR<P)• Assume MC=4. A profit maximizing firm solves

()

A side note on cost functions• STC=FC+VC• SATC=STC/q=AFC+SAVC=FC/q+VC/q• Knowing SATC & SAVC will be important in

determining firm’s short run profit maximization behavior (i.e., output choices)

q

$

SATC

SAVCAFC

Marginal revenue and elasticity of demand

• Recall: elasticity of demand:

• (This should look familiar: it’s the same expression as the one we saw earlier for the change in consumers’ expenditures)

• A monopolist will never choose an output level in an inelastic region of the demand curve

Marginal revenue and elasticity of demand

P

$

P(Q)MR(Q)

TR(Q)

Q

Q

Elastic region

Inelastic region

Unit elastic pointP

Q

P

P’

When P goes down to P’

A

B

A=lost revenue resulting from selling output at lower priceB=additional revenue from selling more units

A>B when demand is inelastic. A<B when demand is elastic

When demand is elastic, an expansion of output leads to higher TR. When demand is inelastic, an expansion of output leads to lower TR. Given that reducing output lowers TC, clearly a firm should reduce output when it faces inelastic demand

Market power

• Market power = degree of control over the price of its product a firm exercises

• A firm’s markup = how much more than the firm’s MC the firm can charge for its product– Firm’s markup is a measure of its market power

– Profit maximizing firms set MC=MR, so

Markups & elasticity

• When demand is perfectly elastic (), , so P=MC. This is the profit maximization condition for a price taker. How can we see this? The demand faced by a price taker is:

P

P=MR

Demand curve a price-taker faces:

Q

𝜀→−∞

Punchline: firms that are price takers have no market power; they have zero markup Demand faced by firm is not

the market demand (unless the firm is a monopolist)

Firms facing a less than perfectly elastic demand curve will have positive markups (market power)

Types of Market Structures

• Markets are characterized by different degrees of competitiveness (same as saying different degrees of market power, different markups)

Perfectly competitive market-No market power-No markup

Monopoly-Most market power-Potentially large markup

Oligopoly Duopoly

Perfect Competition in the Short Run

• Assume firms are price takers:

• Determine the firm’s supply curve– Choose Q such that MR(Q)=MC(Q)

P

Q

D=MR=AR=P

Firm-level supply in SR under perfect competition

• Because MR(Q)=P, choose Q such that MC(Q)=P:

P

D=MR=P

MC(Q)MR>MC(Q)

q1 q2

Recall the SOC for profit maximization: MC must be increasing at the optimal q.q2 here is the profit maximizing output level.

Firm’s Supply rule

P

D=MR=P’

SMC(Q)SAVC(Q)

Q

P’

PSD

q*

Firm’s supply rule: At firm chooses q such that MC(q)=P provided MC is increasing at q if the firm chooses to produce a positive level of outputAt , the firm chooses q=0 (i.e., the firm shuts down)

Shutdown price• Why is PSD=min(SAVC)?

PSMC(Q)

SAVC(Q)

Q

P’

PSD

q*

SATC(Q)

Recall that . Here, P’>ATC(q*), so at P’ this firm earns a positive profit. The area of the green shaded box represents the firm’s profits.

Shutdown price

PSMC(Q)

SAVC(Q)

Q

PSD

SATC(Q)

P’’ MR=P’’

q*

Here, P’’<ATC(q*). At price P’’, this firm earns a loss. Will this firm continue to operate? The answer is maybe. At P’’, the price exceeds the firms’ variable costs, meaning that it can pay for its variable inputs and earn enough to recoup some but not all of its fixed costs. Some firms in this position will choose to remain in the market, recoup some fixed costs, and hope for an increase in the market price (perhaps because other firms in a similar position exit the market). Other firms in this position will exit the market. This is why the firm’s output rule says “if the firm produces positive output….”

Shutdown price

PSMC(Q)

SAVC(Q)

Q

PSD

SATC(Q)

MR=P’’’

q*Here, P’’’<AVC(q*)<ATC(q*). At P’’’, this firm earns a loss, but it fails to even recoup the costs of its variable inputs. Recall that the cost per unit of output of only the variable inputs is AVC. If P<AVC, the costs of variable inputs exceeds revenue, but you can always drive variable costs to zero by simply producing nothing. This also drives revenues to zero, but this is better than throwing money away on variable costs that you can’t recoup. At prices below PSD, all firms should shutdown. Because the P=MC and MC (P) is below AVC everywhere to the left of min(SAVC) (where the SAVC curve intersects the MC curve), the firm never operates at prices below min(AVC).

Intuition behind the shutdown price

• Suppose you were holding a bake sale to raise money for your student organization. You choose to sell brownies for $1.00/brownie. You buy eggs, cocoa, flour, vanilla, sugar, and butter. You choose the output level at which MC=$1, but at this output level you AVC=$2.00. This means that for every brownie you sell, you are paying $2.00 for ingredients (eggs, cocoa, flour, vanilla, sugar, and butter) but only recouping $1. This is a BAD bake sale.

Intuition behind the shutdown price

• Now suppose the price of brownies is $2.50. Here, you recoup the costs of ingredients for each brownie ($2.00 still) and earn $0.50 on top of that. Suppose, however, that Student Life charges $50/day for a table on the quad. Your SATC at this output level is $2.75. This means that the $0.50 you earn on top of variable costs goes part but not all of the way towards helping you recoup your fixed costs (here, AFC=$0.75, so you are $0.25 short of breaking even).

Breakeven price

• Because , the price at which the firm earns zero profit (“breaks even”) is P*=ATC(Q*). Because P=MC(Q*), the firm’s breakeven price occurs at the Q where SATC(Q) is minimized because this is where MC(Q)=SATC(Q).

• In the bake sale example, if MC(Q*)=2.60=SATC(Q*), then PBE=$2.60

Breakeven price

PSMC(Q)

SAVC(Q)

Q

SATC(Q)

PBE

q*

Firm’s output rule

PSMC(Q)

SAVC(Q)

Q

SATC(Q)

PBE

q*

PSD

Firm shuts down

Firm earns positive profit

Firm earns short run losses and may or may not continue to operate

P*<PBE, P*=PBE, , . P*>PBE, , .In the long run if there is free entry/exit from the industry, positive profits will attract new firms (entrants). Losses will cause some firms to exit the industry. What is an individual firm’s supply curve?

The Firm’s Supply Curve

PSMC(Q)

Q

SATC(Q)

PBE

PSD

Consider the firm’s output at a variety of different prices. Recall that the firm’s decision rule when it is a price taker sets MC(Q)=P. As a result, its supply curve is its marginal cost curve above the SAVC curve. At prices below PSD, the firm supplies zero units of output.

P1

P2

P3

P4

SAVC(Q)

P5

0 q1 q2 q3 q4

Firm’s supply curve

P

Q

PSD

SMC

Supply curve is in green

Why do some firms stay in the market when earning losses in the SR?

PSMC(Q)

SAVC(Q)

Q

SATC(Q)

PBE

PSD

Q

P

PSD

D

S

S’

P*

If the price is between PSD and PBE, this firm earns a loss in the short run. Presumably other firms in the same position (if all firms have the same production technologies, they will have the same cost curves). Some choose to exit. This causes the market supply (the horizontal sum of individual firm supply curves) to shift to the left. The equilibrium price goes up. Eventually we expect the equilibrium price P* to equal the breakeven price because at this price there will be neither entry nor exit of firms. In the long run, any firm earning a loss would shut down.

Finding shutdown & breakeven prices analytically

• Suppose and we set so that • The fixed costs (i.e., costs when q=0) are 16• Variable costs are • Marginal costs: • Average variable costs: • Average costs: • Minimum value of SATC: (take derivative of

SATC, set equal to 0 and solve)

Finding shutdown & breakeven prices analytically

SMC=q/50=firm’s supply curve

SAVC=q/100

SATC

P

Q

40

PBE=40

PSD=0

Benefit to absorbing a loss in the SR?

• When the price is below PBE but above PSD, the firm earns enough to cover its variable costs and some of its fixed costs. Given that the firm pays its fixed costs regardless of whether it produces in the short run, it makes sense to produce positive output. In the long run, of course, a firm earning a loss should exit.

• In the previous example, FC=16.• At q=30, P=$0.60, & ATC(30)=0.83, so • This firm loses $7 instead of $16 (which it would have

lost in the short run by producing zero)

Effects of input price changes• How do changes in r affect the supply curve?– No effect. Short run supply curve reflects short run

marginal costs. These costs are incurred through increased labor usage, which has nothing to do with r. ()

• How do changes in w affect the supply curve?– Supply curve gets steeper (see SMC expression

above)P

Q

S

S’

w increases

Input choices & supply decisions in the long run

• Firm: maximize profit by choices of capital and labor:

• FOC: • These are the marginal revenue product of

capital and labor, the value of the extra output than an extra unit of the input will produce

• From these expressions we derive the firm’s demand for inputs

Input choices in the long run

• Use the FOC to solve for and • Output is given by and & are the choices of K

and L that maximize profits given w, r, and P, so is the output level that maximizes profit:

This is the firm’s supply function in the long run

Example: LR Supply function

• Suppose . This is a CRS production function, meaning that MC is constant. As such, the supply function is not well-defined. If P* happens to be equal to MC, it is unclear how much the firm will produce:

MC

Q

P

Example: LR Supply function• Suppose , so this function exhibits DRS.

Divide (1) by (2): (note: same condition as for cost minimization)

Example: LR input demand functions

• Re-write (1) & (2):

Now substitute using :

Example: LR firm supply function

• Now plug & into the production function to get the firm’s LR supply function:

P

Q

S

S’

W ↑ or r ↑Notice that in the LR, r does affect the supply curve because K is a variable input, so its price (r) affects marginal costs. Show that the supply curve equals the firm’s marginal cost curve.

Producer’s surplus• Producer’s surplus refers to the gains accruing

to the producer from producing and selling q units of output at price P

• Producer’s surplus is a short-run concept• Is producer’s surplus the same as the firm’s

profit? No.P

S

D

Qq*

P*Producer’s surplus: the gains accruing to the firm from selling units of output 0 through q*

When the supply and demand functions are linear, the producer’s surplus is given by the area of a triangle:

𝑃=0

Intercept is not always 0

Producer’s surplus

• Producer’s surplus is not the same as the firm’s profits because of fixed costs in the short run. The producer’s surplus is the benefit to the firm from selling each unit of output 0 through q* relative to producing nothing

Example: Calculating Producer’s Surplus

• Suppose

• Suppose P=$10 →

Alternative behavioral hypothesis• Suppose for the sake of argument we assumed that

firms maximize their revenues instead of their profits. • Why might we think firms maximize revenues?• How would this affect a firm’s output level?

• The profit maximization FOC imply that the firm sets MR(q)=MC(q). Given that generically MC(q)>0 and the MR(q) curve is generally decreasing (for firms which are not price takers),

• In principle this is a testable hypothesis• How much would a price-taking firm produce if it were

maximizing revenue?

Example: Revenue maximization

• Suppose AC=MC=4 & (this firm is not a price taker)

• If the firm is revenue maximizer, it chooses q such that

• If this firm is a profit maximizer, it chooses q such that

• How would you actually test the hypothesis of revenue maximization against that of profit maximization? Think about what you actually observe.

Deriving industry supply from firm-level supply

P

Q1

P

Q2

P

Qm

+ =

S1 S2

4 6

P1

P2

0 10

Sm

4 16

Firm 1 Firm 2 Market supply

Suppose the industry consist of 2 firms. Each firm supplies a given amount of output at each market price P. The sum of each firms’ quantities supplied at market price P is the market supply. That is, (if there are n-firms), .

Example: Deriving market supply

• Suppose there are 100 firms in the industry and that (not unreasonably) all have the same cost functions (because they have the same production technologies). Suppose that the firm-level supply curve for any firm is given by. Then the market supply is given by

Profits in LR for price-takersP

D

S S’MC

AC

P1

P*

Q1 Q*QMES

Q Qfirm-level

Firm-levelMarket level

𝜋>0

When firms earn positive profits, new firms enter. As we just saw, new entrants will shift the market level supply curve to the right. This will continue to happen until there are no profits to be had. When does this occur? In the long run, the price can only fall until P=P*. At P* in the LR, P=AC so profit equals zero. The market supply curve shifts from S to S’. This has an important implication. In the long run, all firms will produce at their minimum efficient scale. That is, all firms produce an output level that minimizes their costs-per-unit of output of production. (Note that S and S’ are short run industry supply curves.)

q1

Conclusions about markets consisting of price-taking firms

1. In the SR, price taking firms can earn , , or 2. In the LR, firm entry/exit leads to firms

earning 3. LR market supply curve is horizontal in most

cases. The LR market supply curve in these cases is given by P*=min(AC).

Long run market supply curvesP

MCAC

P*LR

QMES

Q Qfirm-level

D D’

S’sr

Ssr

P’

𝜋>0

Suppose there is an increase in market demand from D to D’. In the short run, the price will go up to P’. At P’, firms will enjoy positive profits. As such, new firms will enter. Assuming that the entry of new firms has no effect on existing firms’ cost structures, the price will eventually fall back down to P*LR at a higher equilibrium output level. This implies that the long run market supply curve is horizontal and given by Slr above.

Slr

Q1* Q2*

Long run market supply curvesP

P*LR

QMES

Q Qfirm-level

D D’

S’srSsr AC0

AC1

Slr

Suppose instead that the industry is a significant consumer of an important input. If new firms enter, they bid up the price of this important input for all firms. As we have seen, this changes the firms’ cost structures and shifts the average cost curves upward. This, however, implies higher long run equilibrium prices as market output expands. That is, the market supply curve in the long-run is upward sloping. There may be other reasons why new entrants drive up the costs for all firms other than the demand for important inputs. Likewise, new entrants might also lower the costs for all firms. Why?

Example: Finding the LR equilibrium price for a given cost structure

• Suppose a firm’s total costs are given by . Its average costs are given by . The minimum value of the AC curve is achieved where or q=10. Because AC=MC=P at this minimum value, we know that the long run equilibrium price in this market will be .

Perfectly competitive markets

• The market supply we have described to this point is the market supply in perfectly competitive markets, but we have not fully characterized perfectly competitive markets.

• Perfect competition is a model of market behavior. Because all models are just sets of assumptions, we must make explicit the assumptions characterizing perfectly competitive markets.

Assumptions of perfectly competitive markets

1. Large number of firms each selling a homogeneous product (i.e., selling an identical product)

2. Firms are price-takers: individual firm output choices have no effect on prices

3. Perfect information: prices are known to all firms and consumers

4. Transactions are costless: guarantees free entry and exit of firms and inputs

• Maintained assumption in all models of market structure is that firms are profit maximizers

Implications of PC assumptions

1. Firms earn in the LR2. P*LR=min(AC) and each firm produces at its MES

3. Firms earn , , or in the SR4. P=MC5. PC is allocatively efficient insofar as there are no more

gains from trade to be had insofar as the consumer and producer surpluses are maximized

6. PC is “productively” efficient insofar as firms produce at their MES, the lowest cost per unit of output, meaning society is producing the market output in the least costly manner possible—freeing resources to be devoted to other purposes

Allocative Efficiency

• A market equilibrium is allocatively efficient if the marginal cost to society from consuming output level q is equal to the marginal benefit to society of producing output level q.

S=MC

D=MB

Gains from trade of each unit of output

P

Q

Allocative efficiencyP

Q

S=MC

D=MB

Consumer surplus

Producer surplus

Another way to say that the market equilibrium is allocatively efficient is to say that the welfare gains to society from exchange, the consumer and producer surplus, are maximized. This is the same as Pareto efficiency in this context.

Inefficient allocationP

Q

S=MC

D=MB

P’

Q’

PS

CS

Suppose for some reason output was restricted to Q’. At that output level, consumers are willing to pay P’ for the good, so the price will be P’. The gains accruing to producers are the producer’s surplus. These gains are large relative to those enjoyed by consumers (CS), who must pay the higher price. Note that there are gains from trade (either to consumers or producers) that are not realized. This is known as the deadweight loss, and it is given by the green triangle above.

Deadweight loss