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WELLESLEY COLLEGE DEPARTMENT OF ECONOMICS ECONOMICS 201-03 JOHNSON Answers to Problem Set #2 1. In a Cobb/Douglas world, expenditure (price*quantity) is a constant fraction ofincome for each good. x* = I 1 2Px in one ofour examples implies that the consumer always SPENDS half ofhis/her income on the X good. Looking at the table, therefore, I should expect that the share ofincome spent on food to be constant if food preferences are Cobb/Douglas. Indeed, they are not. an example, using food expenditure as a fraction ofincome - we go from 4196/25,000 for the lower group to 5671145,000 for the upper group (we used the average ofthe income ranges for both). These denominators represent an 8.0% increase and clearly the numerators do NOT represent such a large % increase. Thus, the expenditure shaJ;'e on food clearly DROPS with income (Engel's Law) and-food preferences are not Cobb/Douglas. Entertainment is closer (likely because it's closer to being a luxury good), but it still doesn't exhibit constant expenditure shares. Remember that Cobb/Douglas preferences imply income elasticities of 1, so neither food nor entertainment exhibit that large an income elasticity. (S'f"-I-h"C,~~'l ~~e. ~) <; &, ~e. Il-"'~~<:~'''~ tht~'\"J ~~ >rhe. "O'S"'G-:~"S~~' ~~~S ,If, . f\)C)'h. ~~e~~~'-ts. \\~~ l;"£~ 11 S;~t1:> ~ \<c G \, ~ /b\)~<:>"S.i. ... ~ -('{'\"'Is, '\-' St,"'~ -t\'~ \), S. 'yl:>"~~ ~\"'~ o;."i",-e \'\51- ~~ Do~$. (I ) -""'-""'-- a)No ) ..rht. ," Xe'(' -ki"m ~~'ts·} f'l()~-t\'f\e~" \f\ ba~. MRS = MU x = Y + 1 . This shows that the MRS is not constant along a horizontal or X,Y MU x Y vertical ray, which implies that the indifference curves are NOT parallel displacements of each other. Remember that to be quasilinear it has to be linear in only one of the variables. Actually, perfect substitutes are also quasilinear but to me they're linear. For our purposes, quasi- linear means linear in only one ofthe two arguments. The multiplicative term insures it is not so. b) The equation for a particular indifference arises from setting U(X,Y)= constant K. So U(x,y)=X*Y + X=K implies k y=--1 x
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Y °l------.,;~~~ -1 - From the indifference curves we can see that preferences are strictly convex (MRS diminishes). Also, ¢.ey are monotonic, strictly so ifwe ignore the possibility ofX=O. b)Since preferences are strictly convex but the indifference curves touch the x-axis, we may have an interior solution but should also beware ofcorners. Using the Lagrangean gives us the potential interior solution (FOCS = First Order Conditions): L = XY +X -A(PxX + PyY -I) FOeS => y + 1 = P x x P y y = G...otherwise I +P y x = ..... .for . .I> P y ; 2P x I h . x =- ...ot erwzse P x c) Plug in the prices and income: the optimal quantities consumed are Y=5 and X=12. Recall that the marginal utility
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