• Consider the following market of a homogeneous product, which has de- mand function: Q 105 P. Each firm has a total cost function: C (qi) = 5qi. = 1. Let n = {1,2,...} be the number of firms. For n ≥ 2 the firm engage in Cournot competition. Let q; be the output of firm i, for i = 1, . . ., n, and Q = ½ q₁ be the total output. Write down the inverse demand function for the market. Calculate each firm's marginal cost MC₁. Define each firm's best response function. Solve for the Nash equilibrium point, and calculate the P-MCi market price P, output Q, and each firm's Lerner index L₁ = P Show how these results change with parameter n and explain their economics meaning. dqi Q 2. Use your results from part 1 to show for n → ∞, the market share of firm i: s₁ = 2 qi → 0, and firm i's marginal revenue: MR₁ = d(P(Q)qi) → P(Q), and thus firm i tends to behave like a price taker, and P(Q) → MC. Consequently, for n → ∞, the market tends to be perfectly competitive. Use this result to calculate the perfectly competitive equilibrium price, and total output. The calculate Lerner index, consumer surplus, producer surplus, and social surplus under perfect competition. 3. Apply your results from part 1 to a monopoly market with the same demand and cost function. Calculate the monopoly price, and out- put. Then calculate the Lerner index, consumer surplus, producer surplus, and social surplus of the monopoly market. Calculate the monopoly deadweight loss (relative to perfect competition). 4. Use your results from part 1 to calculate the consumer surplus, pro- ducer surplus and deadweight loss as a function of n, and show how they change with n, including their limits as n → ∞. Explain the economics meanings of these results. 5. Apply your results from part 4 to a free entry problem: each firm has to incur a fixed cost f = 100 in order to enter the market, and in the long-run equilibrium the net profit of each firm is zero. Calculate the number of firms in the long-run equilibrium.
• Consider the following market of a homogeneous product, which has de- mand function: Q 105 P. Each firm has a total cost function: C (qi) = 5qi. = 1. Let n = {1,2,...} be the number of firms. For n ≥ 2 the firm engage in Cournot competition. Let q; be the output of firm i, for i = 1, . . ., n, and Q = ½ q₁ be the total output. Write down the inverse demand function for the market. Calculate each firm's marginal cost MC₁. Define each firm's best response function. Solve for the Nash equilibrium point, and calculate the P-MCi market price P, output Q, and each firm's Lerner index L₁ = P Show how these results change with parameter n and explain their economics meaning. dqi Q 2. Use your results from part 1 to show for n → ∞, the market share of firm i: s₁ = 2 qi → 0, and firm i's marginal revenue: MR₁ = d(P(Q)qi) → P(Q), and thus firm i tends to behave like a price taker, and P(Q) → MC. Consequently, for n → ∞, the market tends to be perfectly competitive. Use this result to calculate the perfectly competitive equilibrium price, and total output. The calculate Lerner index, consumer surplus, producer surplus, and social surplus under perfect competition. 3. Apply your results from part 1 to a monopoly market with the same demand and cost function. Calculate the monopoly price, and out- put. Then calculate the Lerner index, consumer surplus, producer surplus, and social surplus of the monopoly market. Calculate the monopoly deadweight loss (relative to perfect competition). 4. Use your results from part 1 to calculate the consumer surplus, pro- ducer surplus and deadweight loss as a function of n, and show how they change with n, including their limits as n → ∞. Explain the economics meanings of these results. 5. Apply your results from part 4 to a free entry problem: each firm has to incur a fixed cost f = 100 in order to enter the market, and in the long-run equilibrium the net profit of each firm is zero. Calculate the number of firms in the long-run equilibrium.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
Transcribed Image Text:• Consider the following market of a homogeneous product, which has de-
mand function: Q 105 P. Each firm has a total cost function:
C (qi) = 5qi.
=
1. Let n = {1,2,...} be the number of firms. For n ≥ 2 the firm
engage in Cournot competition. Let q; be the output of firm i, for
i = 1, . . ., n, and Q = ½ q₁ be the total output.
Write down the inverse demand function for the market. Calculate
each firm's marginal cost MC₁. Define each firm's best response
function. Solve for the Nash equilibrium point, and calculate the
P-MCi
market price P, output Q, and each firm's Lerner index L₁ = P
Show how these results change with parameter n and explain their
economics meaning.
dqi
Q
2. Use your results from part 1 to show for n → ∞, the market share
of firm i: s₁ = 2 qi
→ 0, and firm i's marginal revenue: MR₁ =
d(P(Q)qi) → P(Q), and thus firm i tends to behave like a price taker,
and P(Q)
→ MC. Consequently, for n → ∞, the market tends
to be perfectly competitive. Use this result to calculate the perfectly
competitive equilibrium price, and total output. The calculate Lerner
index, consumer surplus, producer surplus, and social surplus under
perfect competition.
3. Apply your results from part 1 to a monopoly market with the same
demand and cost function. Calculate the monopoly price, and out-
put. Then calculate the Lerner index, consumer surplus, producer
surplus, and social surplus of the monopoly market. Calculate the
monopoly deadweight loss (relative to perfect competition).
4. Use your results from part 1 to calculate the consumer surplus, pro-
ducer surplus and deadweight loss as a function of n, and show how
they change with n, including their limits as n → ∞. Explain the
economics meanings of these results.
5. Apply your results from part 4 to a free entry problem: each firm
has to incur a fixed cost f = 100 in order to enter the market, and in
the long-run equilibrium the net profit of each firm is zero. Calculate
the number of firms in the long-run equilibrium.
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