economics 3030: intermediate microeconomic theory

Economics 3030: Intermediate Microeconomic Theory Mid-Term 2

1

Economics 3030: Intermediate Microeconomic Theory Mid-Term Examination 2: Suggested Solutions The examination contains 11 questions and is out of 100 points. Questions are divided into three sections: Section A (True/False) contains five questions at 6 points per question (Total 30 points). Section B (Multiple Choice) contains five questions at 8 points per question (Total 40 points). Section C (Short Answer Question) contains one question at 30 points per question (Total 30 points).

Section A: True / False (6 points per question, total 30 points)

1. A Stackelberg leader will necessarily make at least as much profit as he would if he acted as a Cournot oligopolist:

A. TRUE B. FALSE

Solution: The leader can always announce or produce its Cournot level of output in period one such that the optimal response in period two from the following firm would be to produce its Cournot level of output. In this way, the leading firm would make it’s Cournot level of profit.

2. A discriminating monopolist is able to charge different prices in two different markets. If when the same price is charged in both markets, the quantity demanded in market 1 is always greater than the quantity demanded in market 2, then in order to maximize profits, the monopolist must necessarily charge a higher price in market 1 than in market 2:

A. TRUE B. FALSE

Solution: The profit maximising set of prices is derived from:

where . So it will be the case that if irrespective of

.

3. A firm uses ten units of labour and twenty units of capital to produce ten units of output. The marginal product of labour is equal to 0.5. If there are constant returns to scale, then the marginal product of capital must be at least equal to 0.5:

A. TRUE B. FALSE

p1 1− 1

Ε1

⎛⎝⎜

⎞⎠⎟= MC = p2 1− 1

Ε2

⎛⎝⎜

⎞⎠⎟

Ε i = − ∂qi ∂pi( ) pi qi( ) > 0 p2 > p1 Ε2 < Ε1

q1 p( ) > q2 p( )


2

Solution: Constant returns to scale imply that a ten per cent increase in capital input together with a ten per cent increase in labour input yields a ten per cent increase in output. Thus, an additional two units of capital and one unit of labour produce one more unit of output. By definition, however, if the marginal product of labour is 0.5, then an additional unit of labour applied on its own raises output by 0.5 units. The contribution of the two units of capital is therefore to add another 0.5 units of output, implying that the marginal product of capital must be 0.25. Note: This is clearly an approximate procedure since we have averaged marginal product over two units of capital and evaluated it at a slightly higher output than 10. However, it so happens that in this instance the answer obtained is exactly correct. To see this, Euler’s theorem may be used. This states that for constant returns production function:

4. Arsenal and (somewhat surprisingly) Tottenham Hotspur fans both have excellent taste in music and are huge Rod Stewart fans. A careful analysis of demand for tickets to Rod’s concerts reveals a strange segmentation in the market. Demand for tickets by Arsenal fans is described by whilst demand by Tottenham fans is . If the marginal cost of a ticket is $3 and Rod can price discriminate, then to maximize profits he should price tickets at $3.75 for Arsenal fans and $9 for Tottenham fans:

A. TRUE B. FALSE

First, note that:

And:

MPL ⋅ L+ MPK ⋅K = Q K , L( )⇒

MPK =Q K , L( )− MPL ⋅ L

K⇒

MPK =10− 0.5 10( )

20= 0.25

qad p( ) = 500 p−3 2

qtd p( ) = 50 p−5

dqad

dp= − 3

2500 p− 3 2( )−1 = − 3

2⋅ q

p⇒

Εa ≡ −dqa

d

dp⋅ p

q= 3

2


3

Profit (third-degree) maximisation implies:

5. A remote island has two shops, one run by Ben and the other by Grace. The island’s inhabitants love chocolate eggs but only buy them at Easter through the aggregate demand function

. Before Easter, Ben and Grace each set their selling price for the eggs. If Ben’s price is below Grace’s price, then eggs will only be purchased from Ben. And vice versa, if Grace’s price is below Ben’s price, then eggs will only be purchased from Grace. If Ben and Grace set the same price, then each will supply one half of the total demand. Ben and Grace can purchase eggs at c = £2 per egg without wastage – i.e. any unsold eggs can be returned to the supplier at £2. It is the case that the equilibrium price if Ben and Grace collude is £1 higher than the equilibrium price if they do not collude:

A. TRUE B. FALSE

Solution: First note that:

dqtd

dp= −5×500 p−5−1 = −5⋅ q

p⇒

Εt ≡ −dqt

d

dp⋅ p

q= 5

MRa ≡ pa∗ 1− 1

Εa

⎛⎝⎜

⎞⎠⎟= MC = pt

∗ 1− 1Εt

⎛⎝⎜

⎞⎠⎟≡ MRt

⇒

pa∗ 1− 2

3⎛⎝⎜

⎞⎠⎟= 3= pt

∗ 1− 15

⎛⎝⎜

⎞⎠⎟

⇒

pa∗ 1

3⎛⎝⎜

⎞⎠⎟= 3= pt

∗ 45

⎛⎝⎜

⎞⎠⎟

⇒pa∗ = 9

pt∗ = 3.75

qd p( ) = 100 4− p( )


4

If Ben and Grace do not collude then, by engaging in Bertrand competition they will reach a Bertrand equilibrium in which:

By colluding, however, they will choose the monopoly profit maximising price vis:

Such that:

See Figure 2:

qd p( ) = 100 4− p( ) = 400−100 p

⇒100 p = 400− q⇒pd q( ) = AR = 4− 0.01q

⇒MR = 4− 0.02q

pB = pG = pb = 2 = MC = AC

MR qm( ) = 4− 0.02qm = 2 = MC qm( )⇒0.02qm = 2⇒qm = 100

pm = pd qm( ) = 4− 0.01qm

⇒

pm = pd qm( ) = 4− 0.01(100)

⇒

pm = pd qm( ) = 4−1= 3


5

Figure 2

Section B: Multiple Choice (8 points per question; total 40 points).

6. In the reclining chair industry (which is perfectly competitive), two different technologies of production exist. These technologies exhibit the following total cost functions:

Due to foreign competition, the market price of reclining chairs has fallen to £190 per unit In the short run:

A. Firms using technology 1 and firms using technology 2 will remain in business B. Firms using technology 1 will remain in business and firms using technology 2 will shut

down C. Firms using technology 1 will shut down and firms using technology 2 will remain in

business D. Firms using technology 1 and firms using technology 2 will shut down E. More information is needed to make a judgment

Solution: In the short run a perfectly competitive firm will produce providing that it is able to cover its average variable costs. Thus, for Firm 1:

p

0 100 200 400 q AR

2 MC

MR

4

3

TC1 q( ) = 1000+ 600q − 40q2 + q3

TC2 q( ) = 200+145q −10q2 + q3


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Note that the second-order condition implies that reaches a

minimum at with . Thus, Firm 1 would shut down (i.e. set ) if . For Firm 2, we have:

Note that the second-order condition implies that the reaches a

minimum at with . Thus, Firm 2 would not shut down but

would instead set [i.e. where ] if .

7. A monopolist has discovered that the inverse demand function of a person with income M for the monopolist’s product is . The monopolist is able to observe the incomes of its consumers and to practice (second-degree) price discrimination according to income. If the monopolist has a total cost function , then the price it will charge a consumer depends on the consumer’s income, M, according to the formula:

A. B. C. D. E. None of the above

Solution:

AVC1 q( ) = 600− 40q + q2

⇒dAVC1 q( )

dq= −40+ 2q∗ = 0

⇒q∗ = 20

d2 AVC1 q( ) dq2 = 2 > 0 AVC1

q∗ = 20 AVC1 20( ) = 600− 40 ⋅20+ 202 = 200

q = 0 p = 190

AVC2 q( ) = 145−10q + q2

⇒dAVC2 q( )

dq= −10+ 2q∗ = 0

⇒q∗ = 5

d2 AVC2 q( ) dq2 = 2 > 0 AVC2

q∗ = 5 AVC2 5( ) = 145−10 ⋅5+52 = 120

q∗∗ > 5

MC2 q∗∗( ) = 190 p = 190

p = 0.002M − q

c q( ) = 100q

p = 0.001Μ +50

p = 0.002Μ−100

p =Μ2

p = 0.01Μ2 +100


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Thus, profit maximisation implies:

such that:

8. There are two types of consumer, H and L, with inverse demand curves and

. The profit maximizing second-degree packages, are:

A. (72, 4), (168, 12) B. (72, 4), (108, 12) C. (72, 6), (108, 12) D. (68, 6), (98, 12) E. None of the above

Solution: See Figure 5: The monopolist’s aim is to maximise profits by extracting as much consumer surplus (CS) from the two types of individual as possible. Note that the monopolist cannot identify individual types, but he knows that the two types exist. Thus, for example, he could offer two packages: = (B + E + F); and = (A + B + C + D + E + F + G + H).

Type-L would prefer to since the former leaves him with = 0 whilst

the latter leaves him with . The problem, however, is that Type-H would also

prefer to since the former leaves him with = (A + C + D + G) whilst

AR ≡ p = 0.002Μ− q⇒TR ≡ pq = 0.002Μq − q2

⇒

MR ≡ ∂TR∂q

= 0.002Μ− 2q

MR = 0.002Μ− 2qm = 100⇒qm = 0.001Μ−50

pm = p qm( ) = 0.002Μ− qm

⇒pm = 0.002Μ− 0.001Μ−50( )⇒pm = 0.001Μ +50

pHd = 36− 3q

pLd = 24− 3q PL ,qL( ), PH ,qH( )⎡⎣ ⎤⎦

PL ,8( ) PH ,12( ) PL ,8( ) PH ,12( ) CSL PL ,8( )

CSL PH ,12( ) < 0

PL ,8( ) PH ,12( ) CSH PL ,8( )


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the latter leaves him with = 0. Thus, the monopolist must remove (A + C + D +

G) from the package such that Type-H is indifferent between the two packages vis:

. Note that Type-L would still prefer to since

.

Figure 5

The monopolist, however, can increase its profits by reducing the size of the ‘Low’ package. As he does this, he will lose part of the area F (i.e. revenue lost from Type-L) but will gain part of the area (D + G) (i.e. revenue gained from Type-H). The monopolist will keep reducing the size of the ‘Low’ package until the revenue lost from Type-L equals the revenue gained from Type-H, that is until the height (x~y) equals the height (y~z). We may locate the size of the low package as follows:

CSL PH ,12( ) pH ,12( )

CSH PH ,12( ) = 0 = CSH PL ,8( ) PL ,8( ) PH ,12( ) CSL PH ,12( ) < 0

p

0 4 8 12 q

B C D

36

24

12

A

pH q( ) = 36− 3q

pL q( ) = 24− 3q

pH

pL

E F G H

x

y

z


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Thus:

Thus:

B+E =

And:

Thus:

:

9. On a tropical island there are 100 potential boat builders, numbered 1 through 100. Each can build up to 20 boats a year, but anyone who goes into the boat-building business has to pay a fixed cost of £19. Marginal costs differ from person to person. Letting q denote the number of boats built per year, then boat builder 1 has a total cost function , boat builder

2 has a total cost function , and so on. Generally, for each i from 1 to 100, boat

builder i has a cost function . If boats are competitively unit priced at £25, how many boats will be built in total on the island per year?

pHd qL

∗( ) = 36− 3qL∗ = 2 24− 3qL

∗( ) = 2 pLd qL

∗( )⇒36− 3qL

∗ = 48− 6qL∗

⇒3qL

∗ = 12⇒qL∗ = 4

pH

d 4( ) = 36− 3 4( ) = 24 = 2 24− 3 4( )⎡⎣ ⎤⎦ = 2 pLd 4( )

PL∗ = 6× 6+ 1

2 18− 6( )6 = 36+ 36 = 72

PH∗ = A+ B +C + D + E + F +G + H( )− A+C( )

⇒

PH∗ = 1

2 12× 36( )− 12 4×12( ) + 1

2 4×12( )⎡⎣ ⎤⎦⇒PH

∗ = 216− 24+ 24( )⇒PH

∗ = 168

PL ,qL( ), PH ,qH( )⎡⎣ ⎤⎦ = 72,4( ), 168,12( )⎡⎣ ⎤⎦

c1 q( ) = 19+1q

c2 q( ) = 19+ 2q

ci q( ) = 19+ iq


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A. 480 B. 120 C. 60 D. 720 E. Any number between 500 and 520 is possible.

Solution: The marginal (i.e. least efficient) boat-builder who is able to produce boats without making a loss at is boat builder 24 since:

Notice that boat-builder 25 would not produce at since:

So each of the 24 boat-builders will build 20 boats, making a total of 480 boats.

10. A monopolist finds that a person’s demand for its product depends on the person’s age. The inverse demand function of someone of age y is , where . The product can be produced at a constant marginal cost of and cannot be resold from one buyer to another. If the monopolist knows the ages of its consumers and is allowed to price discriminate on age, then profit maximization implies that:

A. older people will pay higher prices and purchase less of this product B. older people will pay higher prices and purchase more of this product C. older people will pay lower prices and purchase more of this product D. everyone will pay the same price, but older people will consume more E. None of the above

Solution:

Profit maximisation implies:

p = £25

AC24 q( ) = 24+ 19

q≤ 25 ∀q ∈ 19,20⎡⎣ ⎤⎦

p = $25

AC25 q( ) = 25+ 19

q> 25 ∀q ∈ 1,20⎡⎣ ⎤⎦

p =α y( )− βq ′α y( ) > 0

c > 0

AR ≡ p =α y( )− βq

⇒TR ≡ p ⋅q =α y( )q − βq2

⇒

MR ≡ ∂TR∂q

=α y( )− 2βq


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Such that:

It is apparent that:

And:

Section 3: Short Answer Question (30 points).

Answer Question 11 and/or Question 12 (best answered question will count)

11. Considerable attention has been focused in New Hampshire on the Northeast Dairy Compact. The Northeast Dairy Compact places a tax on top of the federal minimum price of drinkable milk. In this question, we will explore some of the issues associated with this Compact. (Total 30 points). Let milk producers have a cost function of the form:

Where m is the hundredweight of milk produced by a farm.

(a) By the short run, we mean a situation where (i) a given number of farms are in the market and new ones cannot enter, (ii) a farm in the market cannot avoid its fixed cost even if it produces 0, and (iii) each farm can vary its output to maximize profit. What is the minimum price at which a farm will produce milk in the short run? (2 points)

In the short run, farmers produce if they can recoup their variable costs. Thus, they produce at or above a price that equals average variable costs. In this case, average variable costs are

α y( )− 2βqm = c

⇒

qm =α y( )− c

2β

pm =α y( )− βqm

⇒

pm =α y( ) + c

2

∂qm

∂y=

′α y( )2β

> 0

∂pm

∂y=

′α y( )2

> 0

C = 800+10m+ 0.5m2


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. Thus, the lowest price at which a farm would produce a positive quantity is 10.

(b) By the long run, we mean a situation where (i) each farm can avoid its fixed cost by producing 0 and (ii) farms can enter or exit freely. What is the minimum price at which a farm will produce a positive quantity of milk in the long run? (2 points)

In the long run, farmers produce if they can recover all of their costs. Thus, the minimum price at which farmers will produce in the long run is where price equals average cost.

Note that:

And:

Thus, average cost is minimized where . Average cost at is:

Thus, the lowest price at which a firm would produce in the long run is 50.

(c) For the long run case, find each farm’s supply curve and the industry supply curve for this situation. Be careful to specify the ranges of prices where the quantity is zero, positive but finite, and infinite. (2 points)

Firms supply where p=MC, but they only supply positive quantities if it is profitable to do so. MC = 10 + m so Ms = -10 +p. Hence each firm’s supply is:

And industry supply is

AVC = 10+ 0.5m

AC = 800m

+10+ 0.5m

dACdm

= − 800

m∗( )2+ 0.5= 0

⇒m∗ = 40

d 2ACdm2

= 1600

m∗( )3> 0

m∗ = 40 m∗ = 40

AC m∗( ) = 80040 +10+ 0.5 40( ) = 50

0 if 5010 if 50p

mp p

<⎧= ⎨ − ≥⎩


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.

(d) The market demand function linking the total quantity of drinkable milk M demanded by consumers and the price of milk p is given by:

M = 2000 - 8p

Suppose the industry is in long-run equilibrium. Calculate the price, the total quantity demanded, the quantity produced by each farm, the number of farms, and the profit of each farm. (2 points)

The long run equilibrium price is 50. Plugging this in, M=1600. At a price of 50, firms produce 40 each. 1600/40=40. Thus, there are 40 firms. When p=AC, profits are zero.

(e) Let’s say the federal government mandates a minimum price for milk of $100. The market price of milk can go higher than the federal minimum, but it cannot go below the federal minimum. The federal government prohibits new farms from entering the market [i.e. the number of firms is fixed at your answer to (d)]. The federal government buys the excess supply (if there is any) at the federal minimum and dumps the extra milk into space (the Milky Way). What is the market demand and market supply of milk with this price support? (2 points)

At a price of 100, plug into the equation for market demand. Demand=1200. Firms want to supply 100-10=90. There are 40 firms, so market supply is 3600.

(f) What is the change in consumer surplus associated with this price support relative to perfect competition? (2 points)

CS=1200*(100-50) + .5(1600-1200)*(100-50)=70,000

(g) On top of the price support, let’s throw on the Northeast Dairy compact. A tax of $4 dollars per hundredweight is paid by consumers in addition to the federal price support. Still, no new farms are allowed to enter the market. What is the market supply of milk with this tax? (2 point)

It doesn’t change, 3600.

(h) The Compact does not work exactly this way. Instead it transfers the revenue of the tax back to farmers. Each farmer gets back the amount it produces times the $4 per hundredweight for each hundredweight of milk it produces. Write the farm’s new cost function with the Northeast Dairy Compact in operation. (2 points)

(i) What is the new market supply of milk (with both this tax rebate and the price floor)? (2 points)

Produce where p=MC. MC=6+m so each firm supplies 94. There are 40 firms. 40*94=3760.

0 if 5040 if 50 where n is the number of firms

if 50

pM n p

p

<⎧⎪= =⎨⎪ ∞ >⎩

21800 62

C m m= + +


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(j) Now let’s say the government is sick of this Northeast Dairy Compact so it decides to just buy all the farms and produce the milk itself. It creates a new company: We Milk You (WMY). The government eliminates all subsidies, taxes, and price floors on milk. It prohibits anyone but WMY from producing milk. Thus, WMY faces the demand curve described in (d) vis. M = 2000 - 8p and the cost function described before vis. . (Total 6 points)

(i) Setup the Monopolist’s profit maximization problem as choosing how much milk to produce. (4 points)

(ii) What is the quantity of milk produced by WMY? (1 point)

You maximize the above function.

or m=192.

(iii) What price does WMY charge for its milk? (1 point)

Just plug into the (inverse) demand function. p=226

(k) Let’s say milk consumers can be easily split into two groups: skim milk drinkers and whole milk drinkers. It costs WMY the same to produce both types of milk (so WMY’s total cost is a function of the sum of the two types of milk produced). Let’s say skim milk drinkers have the following (inverse) demand function (Total 6 points)

and whole milk drinkers have the following (inverse) demand function:

(i) Set up the monopolist’s profit maximization problem as choosing how much of each type of milk to produce. (4 points)

For the next two parts, you need to maximize the above function with respect to and . In order to get a closed solution, you have to plug one first order condition

C = 800+10m+ 0.5m2

maxπ

m= max

m250 − 1

8m

⎛⎝⎜

⎞⎠⎟

m− 800 +10m+ 12

m2⎛⎝⎜

⎞⎠⎟

∂π∂m

= 250 − 14

m−10 − m = 240 − 54

m = 0

11002skim skimP M= −

110010whole wholeP M= −

[ ] [ ]

, ,

2

1 1max max 100 1002 10

1800 102

skim whole skim wholeskim skim whole wholeM M M M

skim whole skim whole

M M M M

M M M M

⎛ ⎞ ⎛ ⎞Π = − + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞− + + + +⎜ ⎟⎝ ⎠

SkimM

WholeM


15

into the other. Then you get and . Plug these into the (inverse) demand functions to find the price.

(ii) What price does WMY charge for skim milk? (1 point)

(ii) What price does WMY charge for whole milk? (1 point)

12. The demand for Fab-4 ice cream is given by . The supply of Fab-4 ice cream is provided by John, Paul, George and Ringo, each of whom faces an identical cost function

, John and Paul are early risers and produce their ice cream at 7.00am. George and Ringo wake up a little later and produce their ice cream at 8.00am. The market opens at 9.00am:

(a) Calculate the equilibrium total output of Fab-4 ice cream. (28 marks)

(b) Calculate the equilibrium outputs and profits for John, Paul, George and Ringo. (2 marks)

Let John, Paul, George and Ringo denote Firms 1, 2, 3, and 4 respectively. Using backward induction from Firm 3 implies:

Thus:

SkimM WholeM

1100 (12.852) 93.572skimP = − =

1100 (64.29) 93.5710wholeP = − =

p =α − βq

i ic cq= i = 1,2,3,4.

π 3 = p − c( )q3 = α − c( )q3 − β q1 + q2 + q4( )q3 − βq32


16

where and where by symmetry. Firm 1 and Firm 2 will thus engage in Cournot duopoly competition with one another taking into account the reaction functions of Firms 3 and 4. Firm 1 will thus maximise the constrained profit function:

Note:

Thus:

∂π 3

∂q3

=α − c − β q1 + q2 + q4( )− 2βq3∗ = 0

⇒

q3* = 1

2θ − q1 + q2 + q4( )⎡⎣ ⎤⎦

⇒2q3

∗ = θ − q1 − q2 − q3∗

⇒

q3∗ = 1

3θ − q1 + q2( )⎡⎣ ⎤⎦ = q4

∗

θ = α − c( ) β q3∗ = q4

∗

Maxq1

π1 = α − βq( )− c⎡⎣ ⎤⎦q1 = α − c( )q1 − β q2 + q3 + q4( )q1 − βq12

s.t

qi = qi* = 1

3θ − q1 + q2( )⎡⎣ ⎤⎦ ∀i = 3,4

dπ1

dq1

=∂π1

∂q1

+∂π1

∂qi

⋅dqi

dq1

∀i = 3,4


17

Thus:

Thus:

And:

dπ1

dq1

=α − c − 2βq1∗ − β q2 + q3 + q4( ) + 2

3βq1

∗ = 0

⇒

q1∗ = 3

4θ − q2 + q3 + q4( )⎡⎣ ⎤⎦

⇒4q1

s = 3θ − 3q1s − 3 q3 + q4( )

⇒

7q1∗ = 3θ − 6q3

∗ = 3θ − 63

θ − q1 + q2( )⎡⎣ ⎤⎦

⇒7q1

s = 3θ − 2θ + 4q1s

⇒

q1s = θ

3= q2

s

q3

s = q4s = 1

3θ − θ

3+ θ

3⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ =

θ9

ps =α − β 23+ 2

9⎛⎝⎜

⎞⎠⎟θ =α − β 2

3+ 2

9⎛⎝⎜

⎞⎠⎟

α − cβ

⎛⎝⎜

⎞⎠⎟=α − 8

9⎛⎝⎜

⎞⎠⎟α − c( )

⇒

ps = α +8c9

π1s = ps − c( )q1

s = α +8c9

⎛⎝⎜

⎞⎠⎟− c

⎡

⎣⎢

⎤

⎦⎥θ3= α − c

9⎛⎝⎜

⎞⎠⎟

α − c3β

⎛⎝⎜

⎞⎠⎟

⇒

π1s = 1

27⎛⎝⎜

⎞⎠⎟βθ 2 = π 2

s


18

And:

π 3s = ps − c( )q1

s = α +8c9

⎛⎝⎜

⎞⎠⎟− c

⎡

⎣⎢

⎤

⎦⎥θ9= α − c

9⎛⎝⎜

⎞⎠⎟

α − c9β

⎛⎝⎜

⎞⎠⎟

⇒

π 3s = 1

81⎛⎝⎜

⎞⎠⎟βθ 2 = π 4

s

economics 3030: intermediate microeconomic theory

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