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# cournot competition

#### Solution Preview

1. Recall that to maximize profit, the firm choose to produce to the point such that marginal revenue = marginal cost.

Revenue = price X quantity = (80 - 0.5Q) X Q = 80Q - 0.5Q^2
Marginal revenue = 80 - Q

Cost = 0.125Q^2
MC = 0.25Q

Equating the two gives 80 - Q = 0.25Q => Q = 64. P = 80 - 0.5Q = 48.

Profit = 48 X 64 - 0.125(64)^2 = 2560

In this case, the consumer surplus is CS = 0.5 X Qmarket X (Demand Curve Y intercept - Pmarket) = 0.5 X 64 X (80 - 48) = 1024
The producer surplus is PS = 0.5 X Qmarket X (Pmarket - Supply Curve Y intercept) = 0.5 X 64 X (48 - 0) = 1536.

Total surplus is 2560

If P=AC instead of MR = MC, then P = 80 - 0.5Q = Cost/Q = 0.125Q^2/Q = 0.125Q, then we would get Q = 128 and P = 16.

Demand Curve in this case is P = 80 - 0.5Q and supply curve (or cost curve) is C(q) = 0.125Q

Profit = 16 X 128 - 0.125 X 128 = 2032

CS = 0.5 X 128 X (80 - 16) = 4096
PS = 0.5 X 128 X (16 - 0) = 1024

Total surplus is 5120.

If the firm were to operate competitively (i.e. setting price equals cost and makes no economic profit, then total surplus is much bigger).

2.

a) Let's consider the two markets separately.

In market 1, revenue = price X quantity = 100Q - Q^2, thus MR = 100 - 2Q

MC = 3. So MR = MC leads to Q = 48.5 and P = 51.5. Profit = 100(48.5) - 48.5^2 - 3(48.5) = 2352.25

In market 2, revenue = 120Q - 2Q^2, so MR = 120 - 4Q

MC = 3 still, so MR = MC leads to Q = 29.25 and P = 61.5. Profit = 125(29.25) - 2(29.25)^2 - 3(29.25) = 1857.38

Total profit = 4209.63

b) The two prices from part a) are 51.5 and 61.5. So suppose that there is a threshold n, and if people from either markets purchase n units or more, they could buy at 51.5 per unit, and if they buy less than n ...

#### Solution Summary

cournot competition

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