To prove the Le chatelier principle in the two input world ( K and L). Note that the long run comparative statics are <0 and PflX?Pf 70 fuxK+EKl PfLLfKK?PfL
To prove the LeChatelier principle in the two input world (K and L). Note that the long run comparative statics are: /L /W = fKK PfLLfKK?PfKL 2 < 0 and /K /W = fKL PfLLfKK?PfKL 2 /Q /W = fL fKK+fKfKL PfLLfKK?PfKL 2
In respose to a price increase we have that Pfil d pjKL /w fi and /L-+PkK/W PfKl/w Which gives us /L=KKt/KL and uuL+ifL PfLLfxx? PL PfLX?PfKl tKK? LL 2fifxfKl /P PfilfKx PfkL a(/k3??f3?/k PfiLfKx? PfkL +2/V+3?/kpa>
In respose to a price increase we have that PfLL /L /W + PfKL /K /W = ?fL and PfKL /L /W + PfKK /K /W = ?fK Which gives us: /L /P = ?fL fKK+fKfKL PfLLfKK?PfKL 2 and /K /W = ?fKfLL+fL fKL PfLLfKK?PfKL 2 /Q /P = ?fL 2 fKK ? fK 2 fLL + 2fLfKfKL PfLLfKK ? PfKL 2 = 1 PfLLfKK ? PfKL 2 á fK 2 ?fLL ? fL 2 ?fKK 2 + 2fLfKÝfKL + 2 ?fLLfKK Þâ > 0
The short run supply responses are just come from differentiating: PfLYL, Kp= w This yields /w Pf T r so and /W PfiL and/( To show that the long response is bigger for wages show that: PfiL PfilfKk Pfkl or fLLjKK Well that's got to be true
The short run supply responses are: just come from differentiating: PfLÝL,KÞ = W This yields: /L /W = 1 PfLL < 0 and /Q /W = fL PfLL < 0 /L /P = ?fL PfLL and /Q /P = ?fL 2 PfLL To show that the long response is bigger for wages show that: 1 PfLL > fKK PfLLfKK ? PfKL 2 or fLLfKK ? fKL 2 >fKKfLL Well that’s got to be true
For the long run quantity response to be bigger than the short run response fifKK xkL- or rearranging Pfulfkk? pfkL ?fifa,>fkfiufKl If fkl <o this is clearly false- so the long run supply elasticity is smaller than the short run(because you can use more capital and the goods are substitutes) If fkl>0 then the condition becomes: WfKL >?RfLL which might or might not hold
For the long run quantity response to be bigger than the short run response: fL PfLL > fLfKK + fKfKL PfLLfKK ? PfKL 2 or rearranging ? fLfKL 2 > fKfLLfKL If fKL < 0 this is clearly false– so the long run supply elasticity is smaller than the short run (because you can use more capital and the goods are substitutes). If fKL > 0 then the condition becomes: WfKL > ?RfLL which might or might not hold