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Sample Exam 3 (practice) · Ido Eisdorfer · worked-solution

Sample Exam 3 — Worked Solutions

Original paper ↗

The setup (read this first)

One paper, the whole second half of Micro 3

Six MC questions and two long open questions sweep monopoly pricing, insurance, and oligopoly/vertical structure. The framing that unlocks it:

  1. Q1, Q4 — monopoly price tools: bundling and price discrimination. Always start from MR = MC in each channel.
  2. Q2, Q3 — adverse selection vs moral hazard: deductibles trade one off against the other; low switching is soft evidence on distortion.
  3. Q5, Open Q1, Open Q2-F — merger analysis: compare merged-monopoly output/profit against the pre-merger benchmark, and the benchmark matters (Cournot < Stackelberg < Bertrand).
  4. Open Q2-A→E — one demand curve, four market structures: monopoly, Stackelberg, vertical separation, and the two-part tariff that restores efficiency.
No official answer key was distributed with the question paper

An official solutions document was provided, and every number in this note has been checked against it and re-derived programmatically (all pass). The one genuine ambiguity — Open Q1's contradictory cost line — is flagged in that question and resolved the way the official key does (constant MC=cMC = c).

  1. Q1 — Bundling vs. separate selling

    A monopolist sells two products, XX and YY, with zero production costs initially. There are 40 Type-1 consumers, 40 Type-2, and 20 Type-3. Willingness-to-pay:

    Type XX YY
    1 (×40) 100 20
    2 (×40) 20 100
    3 (×20) 60 60

    Now assume each unit of XX and each unit of YY costs the firm $30 to produce. Which statement is correct?
  2. Q2 — Deductibles, moral hazard & adverse selection

    Consider the statement: "Insurance with a deductible may reduce the moral hazard problem but worsen the adverse selection problem." Which best explains it?
  3. Q3 — Low switching ⇒ weak benefit-package distortion?

    Consider: "The fact that there are very few switches between health funds in Israel suggests that adverse selection through distortion of the benefit package is not a significant problem." Best evaluation?
  4. Q4 — Price discrimination with a fixed cost

    A monopolist sells QQ in two markets. Demands: p1=22−q1p_1 = 22 - q_1 and p2=8−q2p_2 = 8 - q_2. Marginal cost MC=2MC = 2, fixed cost F=99F = 99 (paid only if output is positive). The firm may charge different prices across markets. Which statement is correct?
  5. Q5 — Cournot merger & consumer surplus

    Two identical Cournot firms, each with marginal cost kk, face demand p=100−Qp = 100 - Q. If they merge into a monopoly, marginal cost falls to zero. Consumer surplus increases after the merger as long as approximately:
  6. Q6 — The Lerner Index

    The Lerner Index is L=p−MCpL = \dfrac{p - MC}{p}. Which statements are correct?

    1. If a firm's marginal cost decreases, its Lerner Index decreases.
    2. Lower demand elasticity (in absolute value) increases the Lerner Index.
    3. The Lerner Index of a competitive firm is zero.
  7. Open 1a — Cournot equilibrium

    Two identical firms compete in quantities (Cournot), each with marginal cost cc (the paper writes "TC=12QC2TC = \tfrac12 Q_C^2", but see the note in the solution). Demand: P(Q)=120−QP(Q) = 120 - Q, Q=q1+q2Q = q_1 + q_2.

    (a) Find the Cournot equilibrium as a function of cc.
  8. Open 1b — Merger approved on an output test

    (b) The two firms merge into a monopolist whose marginal cost falls to zero. The antitrust authority approves only if the merged firm's output exceeds the aggregate pre-merger Cournot output. Find the maximum cc for which this holds. Show your calculations.
  9. Open 1c — Merger approved on a profit test

    (c) Instead, the authority approves only if the profit of the merged firm exceeds the aggregate Cournot profit of the two firms. Find the minimum condition on cc. Show your calculations.
  10. Open 1d — Bertrand benchmark

    (d) How would the answer to (b) change if, before the merger, the firms competed in prices (Bertrand) rather than quantities? Explain.
  11. Open 2a — Integrated monopoly

    Inverse demand P=120−QP = 120 - Q; constant MC=cMC = c with 0<c<1200 < c < 120. (Let a≡120−ca \equiv 120 - c.)

    Part A. A single integrated monopolist serves the market. Find the monopoly quantity, price and profit as functions of cc.
  12. Open 2b — Stackelberg competition

    Part B. Two identical firms compete in quantities; Firm 1 moves first and commits, Firm 2 observes and responds. Find each firm's quantity, total output, market price and each firm's profit.
  13. Open 2c — Vertical separation & double marginalisation

    Part C. One upstream producer (MC=cMC = c) sets a wholesale price ww; one downstream retailer observes ww and chooses quantity. Find ww, quantity, final price, and each party's profit.
  14. Open 2d — Ranking the three structures

    Part D. Compare integrated monopoly, Stackelberg, and vertical separation. Rank them by total quantity, final price, and total industry profit, and explain the intuition.
  15. Open 2e — Two-part tariff fixes double marginalisation

    Part E. The producer offers a two-part tariff (w,F)(w, F) — per-unit wholesale price ww plus fixed franchise fee FF. What ww maximises total industry profit? What outcome results? What range of FF makes both parties agree? Compare to Part A.
  16. Open 2f — Merger from a Stackelberg benchmark

    Part F. Before the merger the market is the Stackelberg equilibrium of Part B. A merger creates a monopolist whose marginal cost falls to zero. The authority approves only if the merged firm produces more than total Stackelberg output. Find the range of cc for approval. How does the answer change under pre-merger Bertrand competition?

One-page recap

Q Topic Tool Answer
Q1 Bundling Bundle WTP all = 120 B — separate 5,600, bundle 6,000
Q2 Insurance Deductible: MH ↓, AS ↑ B
Q3 Adverse selection Low switching = soft evidence C
Q4 Price discrimination PD opens Market 2 (F=99F=99) A — CS lower without PD (54.5 → 50)
Q5 Cournot merger CSCS rises iff k>25k>25 E — condition not listed
Q6 Lerner index L=−1/εL = -1/\varepsilon B — statements 2 & 3
Open 1a Cournot MR=MCMR=MC, symmetry q=(120−c)/3q = (120-c)/3
Open 1b Merger (output) 60>2(120−c)/360 > 2(120-c)/3 c>30c > 30
Open 1c Merger (profit) 3600>2(120−c)2/93600 > 2(120-c)^2/9 c>120−902c > 120-90\sqrt2 (always, c≥0c\ge0)
Open 1d Bertrand benchmark 60>120−c60 > 120-c c>60c > 60
Open 2a Monopoly MR=MCMR=MC Q=a2, P=60+c2, π=a24Q=\tfrac{a}{2},\,P=60+\tfrac{c}{2},\,\pi=\tfrac{a^2}{4}
Open 2b Stackelberg Backward induction QS=3a4, ΠS=3a216Q_S=\tfrac{3a}{4},\,\Pi_S=\tfrac{3a^2}{16}
Open 2c Vertical sep. Double marginalisation QV=a4, PV=90+c4Q_V=\tfrac{a}{4},\,P_V=90+\tfrac{c}{4}
Open 2d Ranking — QS>QM>QVQ_S>Q_M>Q_V; ΠM>ΠS=ΠV\Pi_M>\Pi_S=\Pi_V
Open 2e Two-part tariff w=cw=c, fee FF restores monopoly; a28≤F≤3a216\tfrac{a^2}{8}\le F\le\tfrac{3a^2}{16}
Open 2f Merger (Stackelberg) 60>3(120−c)/460 > 3(120-c)/4 c>40c>40; Bertrand c>60c>60
Exam reflexes
  • Bundling → bundle WTP = sum of values; negatively-correlated tastes ⇒ bundling beats separate selling.
  • Deductible → reduces moral hazard, worsens adverse selection (makes coverage unattractive to high risks).
  • Price discrimination + fixed cost → PD can raise welfare by making a small market worth serving.
  • Merger output test → compare merged QQ to the pre-merger benchmark; threshold rises Cournot → Stackelberg → Bertrand.
  • Vertical separation → double marginalisation halves output; a two-part tariff with w=MCw=MC restores the integrated outcome.
  • Lerner index → L=(p−MC)/p=−1/εL = (p-MC)/p = -1/\varepsilon; everything follows from this identity.