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E_StaticLD - Static Labor Demand Theory 1 Simplest Case...

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Static Labor Demand Theory 1. Simplest Case: Single Competitive Firm, One Factor of Production (Labor). Choose L to maximize: wL L pF ) ( FOC: 0 ) ( w L F p dL d ; or, VMP=wage SOC: 0 ) ( 2 2 L F p dL d , or, diminishing returns to labor Totally differentiating the FOC to get comparative statics: 0 ) ( dw dp F dL L F p Rearranging: (holding dw=0): 0 neg neg F p F dp dL (higher output prices lead to increased labor demand). (holding dp=0): 0 1 1 neg F p dw dL Therefore, labor demand curves are unambiguously downward-sloping. 2. Single Competitive Firm, Multifactor Labor Demand Now, choose x 1 , … x n to maximize: i i i n x w x x pF ) ,..., ( 1 According to Varian’s graduate micro text, the solution to this problem can be represented by the profit function , ) ,..., ; ( 1 n w w p which gives the maximized level of profits as a function of all the exogenous parameters. Varian also shows that: y p , where , i.e. output supplied ) ,..., ( 1 n x x F y i i x w , i.e. the demand for input i . These results are sometimes known as Hotelling’s lemma .
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2 Finally, note that (purely because it represents the maximized value of a function) that the profit function must be convex in its arguments ( p and the vector of w ’s). Applying Hotelling’s lemma to the Hessian (i.e. the matrix of second derivatives) of the profit function yields: n n n n n n nn n np n p pn p pp dw dx dw dx dp dx dw dx dw dx dp dx dw dy dw dy dp dy ... ... ... ... ... ... ... ... ... ... ... ... ... ... 1 1 1 1 1 1 1 1 11 1 1  Convexity of Π implies positive definiteness of the above matrices, which in turn implies: 1. dy / dp >0. An upward-sloping supply curve. 2. dx i / dw i <0, for all i. Downward-sloping “own” demand curves for every factor i . In contrast to the static labor supply case, there is no ambiguity here. 3. “Cross-demand” effects, dx i / dw j , can in general be either positive or negative. These elasticities sometimes matter a lot for policy purposes; a great deal of empirical work in labor economics has been devoted to estimating them in various contexts. 4. The most surprising result derives not from convexity but from the fact that the matrix of supply and
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E_StaticLD - Static Labor Demand Theory 1 Simplest Case...

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