Econ808H3ans.02

Econ808H3ans.02 - The Ohio State University Department of...

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The Ohio State University Department of Economics Econ 808–Problem Set #3 Answers Spring 2002 Levin and Peck 1. (a) Di¤erentiating the Lagrangean expression with respect to w ( ¼ ) , NOT with respect to ¼ ,wehave : ¡ f ( ¼ j e )+ ° f ( ¼ j e ) w ( ¼ ) ¡ g ( e ) =0 : Simplifying, we have ° = w ( ¼ ) ¡ g ( e ) for all ¼: That is, the wage is a constant function of output. The other …rst-order condition is the individual rationality constraint, holding as an equality. (b) Letting w ¤ e denote the (constant) wage corresponding to e¤ort e ,the individual rationality constraint simpli…es to log( w ¤ e ¡ g ( e )) = 0 ; or w ¤ e ¡ g ( e )=1 : Thus, we have w ¤ L = 1 4 +1= 5 4 ,a n d w ¤ H = 1 3 4 3 . (c) Since wages are independent of ¼ , the principal’s objective can be sim- pli…ed to Z 1 0 ¼f ( ¼ j e ) ¡ w ¤ e : (1) If high e¤ort is speci…ed, equation (1) becomes Z 1 0 ¼d¼ ¡ 4 3 = ¡ 5 6 : If low e¤ort is speci…ed, equation (1) becomes Z 1 0 ¼ 2(1 ¡ ¼ ) ¡ 5 4 = ¡ 11 12 : Since the principal’s objective is higher (less negative) with high e¤ort than low
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Econ808H3ans.02 - The Ohio State University Department of...

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