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# handout%205 - Department of Economics M Majumdar Cornell...

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Unformatted text preview: Department of Economics M. Majumdar Cornell University Handout #5 Suppose that we are producing an output level 41* at a minimum cost with prices (w,r) and the cost-minimizing bundle is given by (L*,K*); this means : q*=F(L*,K*) (i) and wL*+rK* s wL+rKfor all (L,K) satisfying q*=F(K,L) (ii) Suppose that the wage rate changes to w+ A w and the ﬁrm chooses a new cost- minimizing bundle (L*+ AD“, K*+AK*) for producing the same level 61*; this means: q*=F(L*+AL*,K*+AK*) (iii) and \ (w+Aw)(L*+ AU) + r(K*+AI(*) s (w+Aw)L +rKfor all (L,K) satisﬁJing q*=F(L,K) (iv) From (ii) and (iii) : wL*+rK* 5 w(L*+A L*) +r (K*+AK*) (v) From (i) and (iv): (w+Aw)(L*+AL*) +r(K*+AK*) s (w+Aw)L*+rK* (vi) From (v) wAL* + rAK* a 0 (vii) From (vi) wAL* + AwAL* +rAK* s 0 (viii) From (vii) and (viii) we get: AwAL*S 0 homogeneous production function: F(;.LL,;.LK)=pm F(L,K) for n>0 [homogeneous ofdegree m ] 13‘ F is homogenous of degree m, then partials are homogeneous of degree m-I Euler’s Theorem: If F is homogeneous of degree m, FKK+ FLL sz(L,K) Supply curve ofaﬁrm (very short run, short run) ...
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