Homework 2 Solution (pdf)

Problem 4. Creative assignment for extra-credit:Suppose that in a 2 by 2 economy the endowment of labor increases. Can you show thatas the PPF expands it will stretch towards the labor-intensive good as illustrated in Figure1.10 on page 21 in Feenstra? Try to show this without relying on specific functionalforms but just employing the three assumptions imposed on the production technology:(1) higher inputs lead to higher outputs, (2) balanced inputs lead, on average, to higheroutputs, and (3) constant returns to scale.Proof: There are two steps. First, we use the assumption that industry 1 is laborintensiveto see what it tells us about the production functions. Second, we use thatinformation to compare the percentage changes in the maximal outputs of each good theeconomy can produce for given changes in the factor endowments.Step 1. Note that the optimal factor requirements satisfy (see the definition of the unitcost function) the following optimality condition:wrf1L( a1L, a1K) f2L( a2L,a2K)= =, orf ( a , a ) f ( a , a )1K1L1K2K2L2Kff1L1K( a( a1L1L/ a/ a1K1K,1),1)f2L( a2L/ a2K,1)= .f ( a / a ,1)2K2L2KBy assumption, a1L ( w,r)/ a1K( w,r)> a2L(w,r)/ a2K( w,r)for all factor prices. Then, byconcavity and homogeneity of the production functions, it follows thatff1L1K( a( a1L1L/ a/ a1K1K,1),1)f2L( a2L/ a2K,1) f2L( a1L/ a1K,1)= >, orf ( a / a ,1) f ( a / a ,1)2K2L2K2K1L1Kf1L ( x,1)f2L( x,1)> ∀x .f ( x,1)f ( x,1)1K2KStep 2. The endpoints of the PPF are y ( 1= f1 L,K)and y ( 2= f2L,K). The PPFstretches towards good 1 as the labor endowment increases when d ( y1 / y2) / dL > 0 .d1 dy1dy2y1 / y2) / dL = ( y2− y ) >2( y ) dL dL(120 , or1y1dy1dL1 dy2> .y dL2To show the last inequality, rewrite it asf1L ( L,K)f2L( L,K)> , orf ( L,K)f ( L,K)12