Solution to Homework Set 7

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Solution to Homework Set 7

Managerial Economics Fall 2011

1. An industry consists of five firms with sales of $200 000, $500 000,

$400 000, $300 000, and $100 000.

(a) (2 points) Calculate the Herfindahl-Hirschman index (HHI).

The HHI is

HHI = 10 000(2/15)²+ (5/15)²+ (4/15)²+ (3/15)²+ (1/15)²

= 2 444.

(b) (1 point) Calculate the four-firm concentration ratio (C4).

The four-firm concentration ratio is ¹⁴₁₅ = 0.933.

(c) (2 points) Based on the FTC and DOJ Horizontal Merger Guide- lines described in the text, do you think the Department of Justice would attempt to block a horizontal merger between two firms with sales of $200 000 and $400 000? Explain.

If the firms with sales of $200 000 and $400 000 were allowed to merge, the resulting HHI would increase by 712 to 3 156. Since the pre-merger HHI exceeds that under the Guidelines (1 800) and the HHI increases by more than that permitted under the Guidelines (100), the merger is likely to be challenged.

2. (1 point) Suppose the own price elasticity of market demand for re- tail gasoline is -0.8, the Rothschild index is 0.6, and a typical gasoline retailer enjoys sales of $1.45 million annually. What is the price elasticity of demand for a representative gasoline retailer’s product?

The elasticity of demand for a representative firm in the industry is

−1.33, since 0.6 = −0.8/E_F ⇒ E_F = −0.8/0.6 = −1.33.

3. Based on the information given, indicate whether the following industry is best characterized by the model of perfect competition, monopoly, monopolistic competition, or oligopoly.

(a) (1 point) Industry A has a four-firm concentration ratio of 0.005 percent and a Herfindahl-Hirschman index of 75. A representative firm has a Lerner index of 0.45 and a Rothschild index of 0.34.

Industry A is a monopolistically competitive industry.

1 pt 1 pt 1 pt

1 pt 1 pt

1 pt

= 2444,44

Let G be the smaller number between (32+(achieved points))/12 and 6.

Round G exactly to a quarter of a grade to get your grade.

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(b) (1 point) Industry B has a four-firm concentration ratio of 0.0001 percent and Herfindahl-Hirschman index of 55. A representative firm has a Lerner index of 0.0034 and Rothschild index of 0.00023.

Industry B is a perfectly competitive industry.

(c) (1 point) Industry C has a four-firm concentration ratio of 100 percent and Herfindahl-Hirschman index of 10 000. A representative firm has a Lerner index of 0.4 and Rothschild index of 1.0.

Industry C is a monopoly industry.

(d) (1 point) Industry D has a four-firm concentration index of 100 percent and Herfindahl-Hirschman index of 5 573. A representative firm has a Lerner index equal to 0.43 and Rothschild index of 0.76.

Industry D is an oligopoly industry.

4. (3 points) Suppose firm A recently entered into an agreement and plan of merger with firm B for $7.5 billion. Prior to the merger, the market for the good produced by firm A and B consisted of five firms. The market was highly concentrated, with a Herfindahl-Hirschman index of 3 621. Firm B’s share of that market was 28 percent, while firm A comprised just 15 percent of the market. If approved, by how much would the postmerger Herfindahl-Hirschman index increase? Based only on this information, do you think the U.S. Justice Department would challenge the merger? Explain.

If approved, the merger would raise the HHI by 10 000[(.28 + 0.15)²− (.28²+ .15²)] = 840 points. Since the pre-merger HHI is 3 621, which is greater than the Guidelines (1 800), and the HHI increases by 840 (which is greater than the 100 points permitted in the Guidelines), it is unlikely that the merger will receive unconditional approval.

5. (Multiple Choice; 3 points) You own two products, each of which is a substitute for the other. You raise price on the first product. What happens to marginal revenue? Assume that demand for each of the two goods can be written as a linear function of prices. To get all three points, justify your choice.

(a) MR for the first product falls but increases for the second.

(b) MR rises for both products.

(c) MR falls for both products.

(d) MR for the second product falls but increases for the first.

Clearly, raising price of the first good reduces demand for it but increases demand for the second good. So we can write demand for the

2 pts

1 pt 1 pt

1 pt

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first good as Q₁ = A₁− B₁P₁+ C₁P₂ and it follows that

⇒ P₁ = A1+ C1P2− Q₁ B₁

⇒ M R₁ = A1+ C1P2− 2Q₁

B₁ .

Analogously, M R₂= ^A²^+C²_B^P¹^−2Q²

2 . This gives

∂M R₁

∂P1

= − 2 B1

∂Q₁

∂P1

> 0, and

∂M R₂

∂P1

= C₂ B2

− 2 B2

∂Q₂

∂P1

= C₂ B2

−2C₂ B2

< 0 (d is right).

6. (2 points) Suppose elasticity of demand for you parking lot spaces is

−2, and price is $8 per day. If your marginal cost is zero, and your capacity is 80% full at 9am over the last month, are you optimizing?

Why (not)? If not, should you increase or decrease the price?

No, you are not optimizing. If your marginal cost is zero, revenues are profits. To maximize revenue, you should choose a level of supply where the price elasticity of demand is −1. Because the elasticity currently is −2, increasing demand by decreasing the price would increase revenue.

7. (2 points) Suppose elasticity of demand for you parking lot spaces is

−0.5, and price is $20 per day. If your marginal cost is zero, and your capacity is 96% full at 9am over the last month, are you optimizing?

Why (not)? If not, should you increase or decrease the price?

No, you are not optimizing. If your marginal cost is zero, the revenues are profits. To maximize revenues, you should choose a level of supply where the price elasticity of demand is −1. Because the elasticity currently is −0.5, decreasing demand by increasing the price would increase revenue.

Vertical Integration

The following exercise analyzes the possibility of a vertical merger between an online CD-shop and a CD producer and its impact industry profits and social surplus.

VI1: CD-Shop (4 points) Assume that the CD-shop can buy CDs a unit price of P_buy. The cost of distributing a number Q_shop of CDs be

1 pt

1 pt 1 pt

1 pt 1 pt A sentence saying that Q_1 goes down and

Q_2 goes up if P_1 goes up also gives 1 point.

Any argument that uses the elasticity to arrive at the concusion that your not

otpimizing

gives 1 pt

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c · Q_shop. The total cost of selling Q_shop CDs is the cost of CDs plus distribution cost. The shop can sell an amount Qshop of CDs according to the inverse demand function P_shop= A−BQ_shop. What is the profit function π_shop(Q_shop) of the CD-shop? What is the profit maximizing amount Qshop that the CD-shop should try to sell? What is the price P_shop it should set?

The profit function is

π_shop(Q_shop) = (A − BQ_shop)Q_shop− (c + P_buy)Q_shop. The FOC

∂π_shop(Q_shop)

∂Qshop

= A − 2BQ_shop− c − P_buy = 0 implies

Q_shop= A − c − P_buy 2B

P_shop= A − BQ_shop= A −A − c − P_buy 2

= A + c + Pbuy

2 .

VI2: CD producer (4 points) Assume that the CD-producer has consid- erable fix cost F for producing an album (artist wages, promotional spending, etc.) but the printing of CDs is virtually free. The cost function C(Q_prod) simply is C(Q_prod) = F . If the CD-producer can sell CDs according to the inverse demand function P_prod= α − βQ_prod, what is the profit function π_prod(Q_prod)? What is the profit maximizing amount Q_prod that the CD producer should try to sell? What is the price P_prod it should set?

The profit function is

πprod(Qprod) = (α − βQprod)Qprod− F.

The FOC

∂π_prod(Q_prod)

∂Q_prod = α − 2βQ_prod= 0 implies

Qprod= α 2β

P_prod= α − βQ_prod= α − α 2 = α

2.

1 pt

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VI3: Market Equilibrium (7 points) Assume that the producer is the only source for CDs of certain artists for the CD-shop. Therefore, Q_shop= Q_prod and P_buy = P_prod. In VI1 you determined Q_shop if P_buy is given. Use this relationship to derive α and β of the inverse demand function that the CD producer is faced with! α and β should depend on A, B, and c. What are the maximized profits π_prod? What are the maximized profits π_shop, if the shop buys CDs at the price P_buy = P_prod that the producer has set? How many CDs will the CD shop sell at what price? Your results should not contain α and β any more.

Solving

Q_shop= A − c − P_buy 2B for P_buy gives

P_buy = A − c − 2BQ_shop

⇒P_prod = A − c − 2BQ_prod. Comparing with P_prod = α − βQ_prod gives

α = A − c β = 2B.

Inserting the profit maximizing choices P_prod = A − c

2 Q_prod = A − c

4B

P_shop= A + c + P_prod

2 = 3A + c

4 Q_shop= Q_prod

into the two firms’ profit functions yields π_prod = P_prodQ_prod−F = (A − c)²

8B −F π_shop= (P_shop−c − P_prod)Q_shop= 3A + c

4 −c − A − c 2

A − c 4B

= 3(A − c)

4 −A − c 2

A − c

4B = A − c 4

A − c 4B

= (A − c)² 16B .

1 pt

P_prod & Q_prod: 1 pt P_shop : 1 pt

1 pt 1 pt

1 pt

If neither pi_prod nor pi_shop were found, but

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VI4: Merged firm (5 points) The two firms now merge. The merged firm will sell a new quantity Qm of CDs at a new unit price of Pm. If CD production still requires fix cost F , and distribution of CDs to customers is the only variable cost and amounts to cQ_m, what is the merged firm’s profit function π_m(Q_m) (if consumer demand remains the same)? What quantity will it sell at what price? How high are the firm’s maximized profits?

The merged firm’s profit function is

π_m(Q_m) = (A − BQ_m)Q_m−F − cQ_m. The FOC

∂π_m(Q_m)

∂Q_m = A − c − 2BQ_m = 0 implies that the merged firm should choose

Q_m = A − c 2B . The price at which it can sell this quantity is

Pm = A + c 2 . The maximized profits are

π_m= A − c 2

A − c

2B −F = (A − c)² 4B −F.

VI5: Comparison (6 points) Compare the profits of the firms in VI3 with the profits in VI4. Is there an incentive for the firms to merge and share the after-merger-profits? Compare the quantities of CDs that are sold to the consumers in the two cases. Compare prices charged to consumers as well. Which situation comes closer to the social op- timum? Does this model motivate government intervention to reduce or increase the number of vertical mergers (assume that firms always merge if this increases joint profits, but never merge if it does not in- crease profits)?

Industry profits in the two firm equilibrium are π_tot,2= π_prod+ π_shop= (A − c)²

16B + (A − c)² 8B −F

= 3(A − c)² 16B −F

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while the profits of the merged firm are π_m = (A − c)²

4B −F

= 4(A − c)² 16B −F.

As the profits of the merged firm are bigger than the profits of the two separate firms, the owners of the two firms have an incentive to merge and share profits. In terms of approximating the social optimum (P = M C = c), comparing

P_shop= 3A + c 4 with

P_m = A + c 2

shows that as long as c < A, P_shop> P_m > c. Therefore the case of the merged firm comes closer to the social optimum. c ≥ A can be ruled out, as in this case, neither the shop, the producer, nor the merged firm would optimally produce positive output. Because P_shop > P_m, marked demand implies Q_shop< Qm. The consumers get to buy more CDs at a lower price if the two firms merge. For all plausible choices of A and c, the owners of the two firms should agree to a merger.

This increases social surplus, so government intervention should be unnecessary.

How to grade VI5 (you don’t need to have found the correct results in VI3 and VI4, to get points):

If you compare πprod+ πshop (VI3) with πm (VI4): 1 pt If you compare P_shop (VI3) with P_m (VI4): 1 pt If you compare Q_shop= Q_prod (VI3) with Q_m (VI4): 1 pt If you compare to the social optimum (P = M C = c): 1 pt If your results only contain A, B, c, and F : 1 pt If you conclude that in VI4, profit and quantity are

bigger and consumer price is smaller than in VI3: 1 pt

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CES production function

The constant elasticity of substitution ( ces) production function is defined as

Y (K, L) = γ (αK^ρ+ (1 − α)L^ρ)^1/ρ.

The elasticity of substitution is σ = _1−ρ¹ . A firm faced with prices r and w for the inputs L and K has an incentive to maximize output from given ex- penditure M . The constrained nonlinear program that describes this problem is

maxK,L γ (αK^ρ+ (1 − α)L^ρ)^1/ρ s.t. wL + rK ≤ M.

CES1 (0 points; the solution was presented in lecture ’5a’) Solve for the output maximizing levels of inputs K and L as functions of w, r, and M . Express K and L in terms of σ rather than ρ. (Hint: K and L take the form K = α^σr^−σX and L = (1 − α)^σw^−σX. Show this using the focs of the Lagrangian method and find X.)

The Lagrangian is L = −γ (αK^ρ+ (1 − α)L^ρ)^1/ρ+λ·(M − wL − rK).

The first order conditions are λr = 1

ρ[Y (K, L)/γ]^ρ(1/ρ−1)αρK^ρ−1= λ[Y (K, L)/γ]^1−ραK^ρ−1 λw = 1

ρ[Y (K, L)/γ]^ρ(1/ρ−1)(1 − α)ρL^ρ−1= λ[Y (K, L)/γ]^1−ρ(1 − α)L^ρ−1 M = rK + wL.

The first two imply _w^r = _(1−α)L^αK^ρ−1_ρ−1 and thus ^K_L =

(1−α)r αw

_−σ

. Insert- ing K =

(1−α)r αw

_−σ

L into the last foc gives

M = r (1 − α)r αw

_−σ + w

! L.

This yields

L = M

r

(1−α)r αw

_−σ + w

=

w 1−α

_−σ M r _α^r_−σ

+ w

w 1−α

_−σ

=

w 1−α

_−σ M α^σr^1−σ+ (1 − α)^σw^1−σ

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and

K = (1 − α)r αw

_−σ L =

r α

_−σ M

α^σr^1−σ+ (1 − α)^σw^1−σ. CES2 (7 points) The above exercise yields

L =

w 1−α

_−σ M α^σr^1−σ + (1 − α)^σw^1−σ K =

r α

_−σ M

α^σr^1−σ + (1 − α)^σw^1−σ.

What is the maximized production level given prices r and w and ex- penditures M ? Solve this expression for M to get the expenditures as a function of prices and the production level. Make sure to replace ρ with expressions of σ.

Y = γM

α^σr^1−σ+ (1 − α)^σw^1−σ αr α

_−σρ

+ (1 − α)

w

1 − α

_−σρ!1/ρ

.

Because ρ = 1 − _σ¹, ρσ equals σ − 1, and

Y = γM

α^σr^1−σ+ (1 − α)^σw^1−σ α^σr^1−σ+ (1 − α)^σw^1−σ1/ρ

= γM α^σr^1−σ + (1 − α)^σw^1−σ1/ρ−1

= γM α^σr^1−σ + (1 − α)^σw^1−σ_1/(σ−1) . Solving for M, we get

M = Y

γ α^σr^1−σ + (1 − α)^σw^1−σ1/(1−σ)

.

If Y is the maximum production given wealth M , M is the minimized expenditure to achieve production Y . The “unit cost function”—the minimum cost to produce one unit of output—therefore is

c(r, w) = 1

γ α^σr^1−σ+ (1 − α)^σw^1−σ1/(1−σ)

.

CES3 (3 points) Make a convincing argument that if Y is the maximum production we can get with the money M , then M on the other hand is the minimized cost for producing at least Y units of output.

Assume there was an alternative M^′ < M that is enough to produce

Inserting above solutions: 1 pt factoring out M: 1 pt

factoring out yellow stuff: 1 pt 1 pt

2 pts

1 pt

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Y . According to this assumption, there are amounts K^′ and L^′ that can produce Y . If we were endowed with M > M^′ however, we could afford some combination (K^′′, L^′′) that satisfies K^′′ > K^′and L^′′> L^′. With this, we could produce Y^′′ = Y (K^′′, L^′′) > Y (K^′, L^′) = Y . But if we can produce Y^′′ > Y from money M , Y was not the maximum we can produce from M in the first place. If Y really is the maximum production, our assumption that M^′ < M is enough to produce Y must have been wrong.