# Finding the optimal output level

Suppose a firm in an oligopolistic industry faces the following demand curve:

P = 4,800 - 0.4 Q for Q < 3,000

P = 20,000 - 2 Q for Q > 3,000

Suppose further its Cost function is as follows:

TC = 1,000 + 100Q + 0.5 Q2

a. What price do you suppose they will charge in the short run?

b. What do you suppose will happen in the long-run?

The McCauley Company hires a marketing consultant to estimate the demand function for its product. The consultant concludes that this demand function is Q = 100P2.91I -2.3A0.4

Where Q is the quantity demanded per capita per month, P is the product's price (in dollars), I is per capita disposable income (in dollars), and A is the firm's advertising expenditures (in thousands of dollars).

a. What is the price elasticity of demand?

b. Will price increases result in increases or decreases in the amount spent on McCauley's product?

c. If the t-statistic to Part A was -1.5, would you rely on this estimate?

d. What is the advertising elasticity of demand?

#### Solution Preview

I. The McCauley Company hires a marketing consultant to estimate the demand function for its product. The consultant concludes that this demand function is Q = 100P2.91I -2.3A0.4

Where Q is the quantity demanded per capita per month, P is the product's price (in dollars), I is per capita disposable income (in dollars), and A is the firm's advertising expenditures (in thousands of dollars).

a. What is the price elasticity of demand?

Q=100P^2.91*I^(-2.3)*A^0.4

Differentiate with respect to P, we get

dQ/dP=2.91*100P^(2.91-1)*I^(-2.3)*A^0.4=2.91*100P^1.91*I^(-2.3)*A^0.4

Price elasticity of demand=(dQ/dP)*(P/Q)

= 2.91*100P^1.91*I^(-2.3)*A^0.4*P/[100P^(-2.91)*I^(-2.3)*A^0.4]

= 2.91*P^1.91*P/P^2.91

= 2.91

It's a surprising result. Price elasticity of demand should be negative.

There might be some error in typing demand function, it could be

Q=100P^(-2.91)*I^(-2.3)*A^0.4

Differentiate with respect to P, we get

dQ/dP=-2.91*100P^(-2.91-1)*I^(-2.3)*A^0.4=-2.91*100P^(-3.91)*I^(-2.3)*A^0.4

Price elasticity of demand=(dQ/dP)*(P/Q)

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#### Solution Summary

Solution depicts the steps to find the optimal output level in the given case.