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Price Elasticity of Demand and Profit Maximizing Level of Output

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Question: In 1991, Brazil and Columbia united to form a coffee cartel and reduce coffee output. Suppose total costs for the cartel are: TC = 12 + 5Q + Q2

Here Q is millions of pounds of coffee.

The market demand curve for coffee is: P = 17 - Q

Here P is millions of dollars per million pounds. Suppose before the cartel was formed, output was 11 million pounds.

In the Wall Street Journal a Columbian delegate to the cartel said that he believed that if the cartel reduced coffee output by 10%, the price would rise by 20%.

Questions:
a. Before the cartel was formed, What did the Columbian delegate believe was the price elasticity? What was the actual price elasticity before the cartel was formed?
b. Compute the profit maximizing level of coffee output, the price the cartel should charge, the maximum cartel profits, and the price elasticity at the optimal output.

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Solution:

a)
According to the Columbian delegate:
% change in price=20%
% change in quantity=-10%
Price elasticity of demand=% change in quantity/ % change in price = -10%/20% = -0.5

Calculating actual Price elasticity:
Demand is given by relation
P = 17-Q
Or Q = 17-P
dQ/dP = -1
Price elasticity of ...

$2.19
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Demand curve of a monopolist

Suppose the demand curve faced by a monopolist is:

q=p^-a where a>0

The total cost for the monopolist are C=sq

a) Find the profit maximizing level of output.
b) Specify the first and second order condition for profit maximization.
c) What is the price elasticity of demand faced by this monopolist
d) What happens to the profit maximizing level of output as s increases
e) What happens to the profit maximizing level of output as a increases.
f)Explain results

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