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Econ 600 Schroeter Fall 2006 Homework 1. A firm employs two inputs, and , to produce one output 1x2x(21, xxfy)=, facing fixed prices for the output and inputs: , and . The firm's problem is 1,wp2w (),,,;,max2121,...21wwpxxxxtrwπ where ()()2211212121,,,;, xwxwxxpfwwpxx−−=π. Assume that, for all vectors of strictly positive prices, optimal factor employment levels and optimal output exist as continuous functions of prices: ()21*1,, wwpx , and ()21*2,, wwpx()()( )().,,,,,,,21*221*121*wwpxwwpxfwwpy ≡ Define the value function ()()()().,,;,,,,,,,2121*221*121*wwpwwpxwwpxwwpππ≡ Also define the following function: ()()().,,,,;,,,;,21*21212121wwpwwpxxwwpxxDππ−≡ Note that for all values of the arguments but that, for all ()0≤⋅D()21,, wwp, ()()().0,,;,,,,,2121*221*1=wwpwwpxwwpxD Thus has a local maximum at every point of the form: ()⋅D ()()().,,;,,,,,2121*221*1wwpwwpxwwpx a.) Write down the first-order necessary conditions for these local maxima and show that they provide an alternative proof of the envelope theorem. b.) Write down the second-order necessary condition for these local maxima and show that it implies the following restrictions on comparative static derivatives:20,0,0*2*21*1≥∂∂≤∂∂≤∂∂pywxwx 02*11*22*21*1≥⎟⎟⎠⎞⎜⎜⎝⎛∂∂⎟⎟⎠⎞⎜⎜⎝⎛∂∂−⎟⎟⎠⎞⎜⎜⎝⎛∂∂⎟⎟⎠⎞⎜⎜⎝⎛∂∂wxwxwxwx 0*2*22**2≥⎟⎟⎠⎞⎜⎜⎝⎛∂∂⎟⎟⎠⎞⎜⎜⎝⎛∂∂−⎟⎟⎠⎞⎜⎜⎝⎛∂∂⎟⎟⎠⎞⎜⎜⎝⎛∂∂pywxwypx 0*1*11**1≥⎟⎟⎠⎞⎜⎜⎝⎛∂∂⎟⎟⎠⎞⎜⎜⎝⎛∂∂−⎟⎟⎠⎞⎜⎜⎝⎛∂∂⎟⎟⎠⎞⎜⎜⎝⎛∂∂pywxwypx Hint: If a symmetric matrix is negative semi-definite, then each of its principal sub-matrices is negative semi-definite. 2. Consider the same scenario described in problem 1 but this time, to simplify notation, fix the prices of output and at 1 so that the profit function becomes: 2x ()().,,;21121211xxwxxfxxw−−=π Input 1 is "variable," and input 2 is "fixed." The firm faces short-run and long-run optimization problems, each yielding optimal factor employment levels: ()()()1*21*1211;,...,,;max121wxwxxxwwgivenxxtrw⇒π ()().;,;max211211,;...211xwxxxwSxwgivenxtrw⇒π Define the value functions for the short- and long-run problems: ()()()2211121,;;, xxwxwxwSSππ≡ ()()()()1*21*111*,; wxwxwwππ≡ For a given value of , say, we have: 1w01w ()()()()()().,;;01*201*10101*20101*wxwxwwxwwSπππ==3However, for any , we have: 011ww ≠ ()()()()()().,;;01*201*1101*211*wxwxwwxwwSπππ≥≥ These inequalities follow from the simple proposition that the removal of a constraint cannot reduce profit. The term on the right is the level of profit at given that factor employments are fixed at levels not necessarily optimal for . The term in the middle holds fixed at the same not-necessarily-optimal level, but incorporates optimization with respect to . Finally, the term on the left incorporates optimization with respect to both and . Now imagine graphing each of these three as functions of for a fixed . Noting that 1w1w2x1x1x2x1w01w()⋅π is linear in (with slope equal to 1w()01*1wx− ), the graph will look like the one below. Use the second-order envelope property evident in the graph to establish the following restrictions on slopes of the long- and short-run demands for : 1x For any value of : 01w() ()().0;01*20111011*1≤∂∂≤∂∂wxwwxwwxS 01w 1w ()1*wπ ()()01*21; wxwSπ ()()()01*201*11,; wxwxwπ


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ISU ECON 600 - Homework-2006

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