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  • Undergraduate Industrial Organization

    Solution to Practice Questions

    Keigo Makino Yuta Toyama

    Last Updated: January 5, 2020

    1 Monopoly

    1.1 Monopoly with CES Demand Function

    Consider a monopolist in the market whose marginal cost is constant and denoted by c. The market

    demand is given by Q = AP−γ , where γ > 0

    (a) Calculate the price elasticity.

    (Answer) The definition of the price elasticity is ε ≡ dQdP × P Q . With CES demand function,

    ε = −γAP−(γ+1) × P AP−γ

    = −γ.

    (b) Solve the monopolist’s problem. What is the optimal price? [Hint: Consider the two cases

    when γ > 1 and γ ∈ (0, 1] separately.

    (Answer) The profit of the monopolist is π(P ) = P ·AP−γ − cAP−γ = AP−γ+1 − cAP−γ.

    1. Suppose γ > 1. The FOC is dπdp = (−γ + 1)AP −γ + cγAP−γ−1 = 0. The optimal price is

    P ∗ = cγγ−1 .

    2. Suppose γ ≤ 1. Then dπdp = AP −γ(−γ + 1 + cγP ) is always positive where P > 0. To maximize

    π(P ), P should be infinitely large, which is not realistic.

    Remember that a monopolist always operates in a price region where |ε| ≥ 1. See the lecture slide for details.

    1.2 Monopoly with Carbon Tax

    Consider a steel market in which there exists a monopolist named Nippon Steel Company (hereafter

    NSC). The market demand for a steel is given by Q = 100 − 2P , where Q is the quantity of steel production and P is the market price of steel. The NSC produces steel with constant marginal cost

    mc = 20. Note that price and cost is measured in 1 USD and the production quantity is measured

    in a ton.

    1

  • (a) Solve the profit maximization problem for the NSC.

    (Answer) The maximization problem for the NSC is

    max P,Q

    PQ−mc ·Q = P (100− 2P )− 20(100− 2P ).

    The FOC is 100− 4P + 40 = 0. The optimal price and quantity are P ∗ = 35 and Q∗ = 30.

    Steel production is associated with emissions of CO2, which has a detrimental effect on envi-

    ronment. Denote the total external cost from producing CO2 as E. Assume that one unit of steel

    production leads to an environmental damage equivalent to 2 USD.

    (b) Calculate consumer surplus CS, producer surplus PS, and the total external cost E.

    (Answer) PS = 35 · 30− 20 · 30 = 450. E = 2 · 30 = 60. CS = 30 · (50− 35) · 12 = 225.

    (c) The government now introduces a carbon tax to reduce CO2 emissions. Denote the carbon

    tax by τ . Under carbon tax regime, the NSC has to pay τ for each unit of production. Let

    τ = 2, so that the carbon tax is exactly same as the social cost of steel production. Solve the

    profit maximization problem in this case.

    (Answer)The maximization problem for the NSC is

    max P,Q

    PQ− (mc+ τ) ·Q = P (100− 2P )− (20 + 2)(100− 2P ).

    The FOC is 100− 4P + 44 = 0. The optimal price and quantity are P ∗ = 36 and Q∗ = 28.

    (d) Calculate consumer surplus CS, producer surplus PS, the total external cost E, and the tax

    revenue T = τq∗ where q∗ is the production quantity.

    (Answer) PS = 36 ·28−22 ·28 = 392. E = 2 ·28 = 56. CS = 28 ·(50−36) · 12 = 196. T = 2 ·28 = 56.

    (e) Define the total welfare by CS+TS+T −E. Can we achieve higher welfare under the carbon tax? If not, discuss why.

    (Answer) No. The welfare in (b) is PS+CS−E = 615. The welfare in (d) is PS+CS+T−E = 588, which is lower than the welfare in (b). Since the monopolist already produces less than the socially-

    optimal level, the loss of consumer surplus and producer surplus might overwhelm the gains from

    pollution mitigation. See Buchanan (1969) for theoretical argument and Fowlie, Reguant, and Ryan

    (2014) for empirical studies.1

    1James Buchanan (1969) “External Diseconomies, Corrective Taxes, and Market Structure,“ American Economic Review. Fowlie, Meredith, Mar Reguant, and Stephen P. Ryan. "Market-based emissions regulation and industry dynamics." Journal of Political Economy 124.1 (2016): 249-302.

    2

  • 2 Price Discrimination

    2.1 Price Discrimination in General

    (a) Write down an example of 2nd (self-selection) and 3rd (market segmentation) degree price

    discrimination. Explain why these examples are 2nd or 3rd degree price discrimination.

    (Answer) Skipped. See the lecture slide.

    (b) To make the price discrimination successful, firms need to prohibit resale of products. Using

    the example, discuss why firms have to prohibit (or hinder) resale.

    (Answer) Skipped. See the lecture slide.

    2.2 2nd Degree Price Discrimination

    As a marketing manager of a software company, you have to set the price of a new product. The

    market is divided into two equally sized segments: Professional users willing to pay $500 and non-

    professional users willing to pay $200 for the full version of the software. The production cost of

    the soft ware is zero.

    (Notice) I did not make explicit this above, but the most important assumption in this question

    is that the seller cannot distinguish who is professional or non-professional. If the seller can dis-

    tinguish them, the seller can offer a separate price menu for each consumer type, which is the case

    of the third-degree price discrimination (discrimination by indicator). Here, we consider the case

    of the second-degree price discrimination (discrimination by self-selection).

    (a) Suppose there is a scaled down version of the product, which is worth $100 to non-professionals

    but worthless to professionals. Assume the production cost of this version is also zero. What

    is the optimal price of each version?

    (Answer)

    # of consumers Full (product 1) Scale-down (product 2) Intermediate (product 3)

    Professionals 1 wp1 = 500 wP2 = 0 wp3 = 250

    Non-professional 1 wn1 = 200 wn2 = 100 wn3 = 150

    Let pjbe the price of product j. The consumer chooses product j that gives the highest net-utility

    given by wi,j − pj for i = p and n. As explained in the lecture, p1 = 500 and p2 = 100. The profit is 500 + 100 = 600.

    (b) The company also has an intermediate version of the product, for which professionals are

    willing to pay $250 and non-professional $150. Again, assume that the production cost is 0.

    Which versions of the product should the firm sell to maximize profits?

    (Answer) Consider the case when a firm sells both product 1 and 3. In this case the price is

    p1 = 400 and p3 = 150. The seller needs to lower the price for product 1 so that professionals

    3

  • prefer to buy product 1 rather than product 3. The keywords are incentive compatibility constraint

    and participation constraint. See the lecture slide 4 for the details. The profit is 450 + 100 = 550.

    Therefore, the firm prefers to sell product 1 and 2 rather than product 1 and 3.

    2.3 3rd Degree Price Discrimination

    A market consists of two population segments, A and B. An individual in segment A has demand

    for your product q = 50−p. An individual in segment B has demand for your product q = 120−2p. Segment A has 1000 people in it. Segment B has 1200 people in it. Total cost of producing q units

    is C = 20q.

    (a) What is total market demand for your product?

    (Answer) Total market demand is

    D(p) =

     1000 · (50− p) + 1200 · (120− 2p) = 194000− 3400p if p < 50

    1200 · (120− 2p) if p ∈ [50, 60]

    0 if p > 60

    (b) Assume that you must charge the same price to both segments. What is the profit-maximizing

    price? What are your profits?

    (Answer) If p < 50, the profit is π(p) = (194000− 3400p)(p− 20). From the FOC is −400(−655 + 17p) = 0. The optimal price under p < 50 is 65517 ≈ 38.5 and the profit is

    19845000 17 ≈ 1167352.94.

    If p ∈ [50, 60], the profit is π(p) = 1200 · (120 − 2p)(p − 20). The optimal price under p ∈ [50, 60] is p = 50 and the profit is 720000. If p > 60, the profit is always zero. Thus the optimal price is

    p∗ = 65517 and the maximized profit is π(p ∗)1984500017 ≈ 1167352.94.

    (c) Imagine now that members of segment A all wear a scarlet “A” on their shirts or blouses and

    that you can legally charge different prices to these people. What price do you charge to the

    scarlet “A” people? What price do you charge to those without the scarlet “A”?

    (Answer) Let the former price be pA and the latter one be pB. The profit maximization problem is

    max pA,pB

    1000 · (50− pA)(pA − 20) + 1200 · (120− 2pB)(pB − 20).

    The FOCs are 70− 2pA = 0 and 160− 4pB = 0. The optimal pricing is p∗A = 35 and p∗B = 40. The profit is 1185000. The profit becomes higher if you can use the third degree price discrimination.

    3 Oligopoly

    4

  • 3.1 Bertrand Competition

    Two firms produce a specialized microchip (Trium X406) used in certain home appliances. Demand

    is given by Q = 100 − 2p (Q in thousands of units, p in $). Whichever firm sets the lowest price gets all of the demand, and there are no capacity constraints. Both producers have a marginal cost

    of $30. A cost-reducing innovation would allow Firm 1 to decrease its cost down to $20. The cost

    of the innovation is $520 per period.

    (a) Should Firm 1 go ahead and acquire the innovation?

    (Answer) No.