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2023 PEA Q4

p=100−qp=100-q, n=10n=10 firms, ci(qi)=qic_i(q_i)=q_i. Total deadweight loss is

(The intended options correspond to 922,922,1022,10022\frac{9^2}{2},\frac{9^2}{2},\frac{10^2}{2},\frac{100^2}{2}, the first being 922\frac{9^2}{2}.)

Reveal answer and solution

Answer

A

Solution

  1. 1

    Marginal cost MC=1MC=1. Competitive output: p=MC=1⇒Qc=99p=MC=1\Rightarrow Q^c=99.

  2. 2

    Cournot output. FOC of firm ii:

  3. 3
    100−Q−i−2qi=1  ⟹  99−Q−i=2qi. 100-Q_{-i}-2q_i=1 \;\Longrightarrow\; 99-Q_{-i}=2q_i.
  4. 4

    Symmetric: 99−9q∗=2q∗⇒11q∗=99⇒q∗=999-9 q^*=2 q^*\Rightarrow 11 q^*=99\Rightarrow q^*=9.

  5. 5

    So QN=90Q^N=90, pN=100−90=10p^N=100-90=10.

  6. 6

    Deadweight Loss.

  7. 7
    DWL=12(Qc−QN)(pN−MC)=12(99−90)(10−1)=12(9)(9)=812=922. DWL=\tfrac{1}{2}(Q^c-Q^N)(p^N-MC)=\tfrac{1}{2}(99-90)(10-1)=\tfrac{1}{2}(9)(9)=\frac{81}{2}=\frac{9^2}{2}.

Answer structure / marking notes

The DWL triangle uses the gap between Cournot and competitive (not monopoly) output and price.

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Content note

Imported from public/resources/isi/msqe/solutions/pea/2023/ISI_MSQE_PEA_2023_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.