Econ501aH2ansS07

Econ501aH2ansS07 - The Ohio State University Department of...

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The Ohio State University Department of Economics Econ 501.02—Prof. James Peck Homework #2 Answers 1. Suppose that the demand for good x is given by the equation x =10 , 000 100 p x . (a) Derive an equation for the inverse demand function, p x ( x ) . (b) Derive formulas for the total revenue and marginal revenue functions (asfunct ionso fx) . (c) Find the price and quantity combination that maximizes total revenue. (d) Calculate the price elasticity of demand for the price-quantity combina- tion you found in part (c). Answer: (a) The inverse demand function expresses p x as a function of x (rather than quantity as a function of price, as above). We simply solve the above equation for p x : x =1 0 , 000 100 p x becomes 100 p x =1 0 , 000 x ,or p x = 100 x 100 . (b) Total revenue is price multiplied by quantity. To get total revenue as a function of x, use the inverse demand function: TR = p x x = (100 x 100 ) x = 100 x x 2 100 . Marginal revenue is the derivative of total revenue with respect to x, so MR = ∂TR ∂x = 100 x 50 . (c) To maximize total revenue, we f nd the value of x where marginal revenue is zero. Setting marginal revenue equal to zero, we have 100 x 50 =0 . Solving, we have x = 5000 , and plugging into the inverse demand function, we have p x =50 . (d) The formula for the elasticity of demand is ε d = dx dp x ³ p x
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Econ501aH2ansS07 - The Ohio State University Department of...

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