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ARE 100 PS6-10A - University of California Davis Department...

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University of California, Davis Department of Agricultural and Resource Economics ARE 100B Winter 2009 Dr. Larson Problem Set 6 Answers 1. Consider a firm that uses a single variable input, x, to produce output q, with production function 3 ln( 1). q x = + (a) Derive the firm's inverse demand for the input x. The firm's inverse demand for the input x comes directly from its first order condition for choosing it. The firm’s short-run profit is 3 ln( 1) , PS p q w x p x w x = = + and its FOC for optimal choice of x is, using the chain rule, 1 / 3 1 0. 1 PS x p w x = ⋅ − = + Rearranging this to solve for w , the firm's inverse demand for x is 3 . 1 p w x = + (b) Derive the firm's direct demand for x. The firm's direct demand for x comes from solving the inverse demand for x , so it is 3 ( / ) 1. x p w = (c) Derive the firm's short-run supply function. The firm's short-run supply function comes from inserting the input demand for x into the production function: 3 ln(3 / ). q p w =⋅ 2. For an industry that uses one variable input, illustrate the general equilibrium supply function, as contrasted with the partial equilibrium supply function, when the output price changes. See the graph below. Suppose the output price falls, to 1 p from 0 , p because output demand shifts in from 0 D to 1 D . The partial equilibrium supply curve 0 ( , ) pe q p w predicts the change in quantity produced if the price of input x is fixed at 0 . w In looking at the input market, the demand for x shifts in, from 0 ( , ) pe x w p to 1 ( , ), pe x w p because output price has fallen. This has an effect on the input price w , which falls from 0 w to 1 , w because the supply of the input x is not perfectly elastic. The change in input price to 1 w in turn causes the product supply curve to shift, from 0 ( , ) pe q p w to 1 ( , ). pe q p w When all of these adjustments in price between inter-related markets cease, the general equilibrium supply curve ( ) ( , ( )), ge ge q p q p w p = which accounts for changes in the input market price, describes the actual change in quantity that will be supplied when the output price changes.
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