Question

2. Cournot competition Aa Aa Consider a town in which only two residents, Sam and Amy, own wells that produce water safe for drinking. Sam and Amy can pump and sell as much water as they want at no cost. Assume that outside water cannot be transported into the town for sale. The following questions will walk you through how to compute the Cournot quantity competition outcome for these duopolists. Consider the market demand curve for water and the marginal cost for collecting water on the following graph Suppose Sam believes that Amy is going to collect 24 gallons of water to sell. On the graph, use the purple points (diamond symbols) to plot the demand curve (D1) Sam faces; then use the red points (cross symbols) to plot the marginal revenue curve (MR1) Sam faces. Finally, use the black point (X symbol) to indicate the profit-maximizing price and quantity (Profit Max 1) in this case. Dashed drop lines will automatically extend to both axes. PRICE (Dollars per gallon) 12 D1 Market Demand 10 MR1 Profit Max 1 6 X MC 8 16 24 32 40 48 QUANTITY (Gallons of water) Help Clear All Instead, now suppose Sam believes that Amy is going to collect 32 gallons of water to sell, rather than 24. On the following graph, use the purple points (diamond symbols) to plot the demand curve (D2) Sam faces; then use the red points (cross symbols) to plot the marginal revenue curve (MR2) Sam faces. Finally, use the black point (X symbol) to indicate the profit-maximizing price and quantity (Profit Max 2) in this case. PRICE (Dollars per gallon) 12 D2 Market Demand 10 MR2 Profit Max 2 b 2 MC 16 40 24 32 48 QUANTITY (Gallons of water) Help Clear All Fill in the following table with the quantity of water Sam produces, given various production choices by Amy. Amy's Water Production Sam's Water Production (Gallons of water) (Gallons of water) 24 32 Given the information in this table, use the green points (triangle symbols) to plot Sam's best-response function (BRF) on the following graph. Since Sam and Amy face the same costs for producing water, Amy's best-response function is simply the reverse of Sam's; that is, the curve has the same shape, but the horizontal and vertical intercept values are switched. Therefore, you can derive Amy's best-response function by following the same analysis above, but from Amy's perspective. Use the purple points (diamond symbols) to plot her best-response function on the graph. Finally, use the black point (X symbol) to indicate the unique Nash equilibrium under Cournot quantity competition AMY'S QUANTITY (Gallons of water) 48 Sam's BRF A- 40 Amy's BRF 32 Nash Eqm --X 16 8 16 24 32 40 48 SAM'S QUANTITY (Gallons of water) Help Clear All True or False: According to her best-response function, Amy will always want to decrease her output as Sam increases his. False True 24

Answer 1

market demand: P = 24 - 0.5Q
Q = 48 - 2P
Demand faced by Sam: Q = 48 - 2P - 24 = 24-2P
P = 12 - 0.5Q
Revenue = P*Q = 12Q - 0.5Q^2
MR = dRevenue/dQ = 12 - Q

Demand - purple, MR - green
Profit maximization is where MR = 0, Q=12, P = 6

(0, 24) 20 (0, 12) 6 (24,0) (48, 0) 60 0 (12, 0) 40 20

market demand: P = 24 - 0.5Q
Q = 48 - 2P
Demand faced by Sam: Q = 48 - 2P - 32 = 16-2P
P = 8 - 0.5Q
Revenue = P*Q = 8Q - 0.5Q^2
MR = dRevenue/dQ = 8 - Q

Demand - purple, MR - green
Profit maximization is where MR = 0, Q=8, P = 4

20 4 (8, 0) 20 40 0

Nash equilibrium is the intersection of the above 2 lines

(8, 32) 30+ (12, 24) 20+ (24, 12) 10+ (32, 8) 10 20 0 30 40

24, 12
32, 8

True