Suppose that ranchers can buy up to a collective total of 200 cows and run them

Question

Suppose that ranchers can buy up to a collective total of 200 cows and run them on the open range. The value of each cow after it has grazed is given by V= $2000-10C, where C is the total number of cows on the range. The cost of buying each cow is $1000.

question: Suppose that the ranching industry were run by a single monopoly that could decide how many cows to buy and let graze. Would the monopoly outcome be better or worse than the competitive equilibrium? How much better or worse? Explain.

Okay, so I calculated these from previous questions:

Perfectly competitive equilibrium quantity: C=100
Monopoly quantity: C=50
Monopoly price: $1500

Explanation / Answer

We treat V like an inverse demand function. V = 2000 - 10C Let's call profit Y and marginal profit MY For a monopoly, Y = (2000 - 10C - 1000)*C Y = (1000 - 10C)*C MY = 1000 - 20C = 0 C = 1000/20 C = 50 The price is V. V = 2000 - 10C V = 2000 - 10*50 V = 1500 In perfect competition, price equals marginal cost. 2000 - 10C = 1000 1000 = 10C C = 100 V = 2000 - 10C V = 2000 - 10*100 V = 1000 Your answers are correct. To say that one outcome is better or worse, we should calculate total surplus. This is the integral of the demand curve minus the marginal cost over the quantity that is produced. First, let's calculate this for the perfectly competitive market. Int[0,100] {2000 - 10C - 1000}dC Int[0,100] {1000 - 10C}dC {1000C - 5C^2}|[0,100] 1000*100 - 5*(100^2) Total surplus is 50,000 Now, let's calculate this for the monopoly Int[0,50] {2000 - 10C - 1000}dC Int[0,50] {1000 - 10C}dC {1000C - 5C^2}|[0,50] 1000*50 - 5*(50^2) Total surplus is only 37500 This means that the perfectly competitive market results in more total surplus than the monopoly. In particular, the dead weight loss from the monopoly is equal to: 50000 - 37500 = 12500

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