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Firm's Product Elastic

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1. The Johnson Robot Company's making officials report to the company's chief executive officer that the demand curve for the company's robots in 2001 is P=3,000-40Q, where P is the price of a robot, and Q is the number sold per month.
a. Derive the marginal revenue curve for the firm.
b. At what price is the demand for the firm's product price elastic?
c. If the firm wants to maximize its dollar sales volume, what price should it change?

2. Since the Hawkins Company's costs (other than advertising) are essentially all fixed costs, it wants to maximize its total revenue (net of advertising expenses). According to a regression analysis (based on 124 observations) carried out by a consultant hired by the Hawkins Company, Q=-23-4.1P+4.2I+3.1A, where Q is the quantity demanded of the firm's product (in dozens),, P is the price if the firm's Product (in dollars per dozen), I is the per capita income (in dollars) and A is advertising expenditure (in dollars).
a. If the price of the product is $ 10 per dozen, should the firm increase it advertising?
b. If the advertising budget is fixed at $ 10,000, and per capita income equals $8,000 what is the firm's marginal revenue curve and what price should the Hawkins Company charge?

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This solution finds the firm's product elastic.

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a. Derive the marginal revenue curve for the firm.

The total revenue will be TR = P*Q = (3,000-40Q)*Q = 3,000 Q -40Q^2
And marginal revenue is the first derivative of TR:
MR = dTR/dQ = 3,000 - 80 Q

b. At what price is the demand for the firm's product elastic?

It's not clear if you want the demand to be unit elastic or just elastic.
Because given the demand function P=3,000-40Q, which is downward sloping, the demand is always elastic along the demand curve.

c. If ...

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