Understanding Consumer Preferences: Solving Utility and Demand
School
Columbia University**We aren't endorsed by this school
Course
ECON 3213
Subject
Economics
Date
Dec 10, 2024
Pages
2
Uploaded by aaa122121
Intermediate Microeconomics (Spring 2023)Problem Set 4Due February 23, 11AMWrite your answers with your name on separate sheets of paper, and upload a scan to Courseworks.1. Consider a consumer with preferences represented by the utility functionu(x) =x3/41+x3/42.(a) Solve for the Marginal Rate of Substitution.MRS=MU1MU2=34x−1/4134x−1/42=x2x11/4(b) Under what economic conditions (values of prices and income) does this consumer choose interior bundles?To answer this, we only need to study the marginal rate of substitution. Note that the MRS approaches 0 as x2approaches 0 and it approaches infinity as x1approaches 0. Thus we definitely have convex indifference curves,and interior solutions will arise at points of tangency between the indifference curves and budget constraints.Corner solutions could arise in some cases even with convex preferences, as we have seen with quasi-linear utilityfunctions. However, in this case, this cannot hold. Consider the corner where x1=0. Near this corner, the slopeof the indifference curve is nearly infinite, much steeper than the budget constraint. Thus the consumer wouldmuch rather give up some units of good 2 in exchange for some units of good 1. The same logic applies for theopposite corner. There will always be a bundle between these corners where tangency is achieved between theindifference curve and the budget line, and thus there will always be an interior solution to the consumer problemfor all economic conditions.(c) Solve for their Marshallian Demand functions.MRS=x2x11/4=p1p2⇒x2=p1p24x1p1x1+p2p1p24x1=I⇒p1p32+p41p32x1=I⇒xm1(p,I) =p32p1(p31+p32)·I,xm2(p,I) =p31p2(p31+p32)·I(d) Isx1a normal good or an inferior good?Since prices are all positive, we can see that the Marshallian Demands are both positively linear with respect toincome, and thus both goods are normal goods because consumption increases with income.(e) (Challenging) Isx1a gross substitute forx2?To answer this we must calculate the cross-price derivative of the Marshallian Demand for good 1. Since p2appearsin both the numerator and denominator of the Marshallian Demand function, we can apply the quotient rule.∂∂p2xm1(p,I) =p1(p31+p32)·3p22I−3p1p22·p32Ip21(p31+p32)2=3p41p22Ip21(p31+p32)2>0Thus, since this derivative is positive, good 1 is a gross substitute for good 2: as the price of good 2 rises, thedemand of good 2 falls, and in it’s place the demand of good 1 rises.1
2. Revisit the general perfect complement utility from the previous homework,u(x) =min{αx1,βx2}. Feel free touse your previous results for this exercise as well.(a) Verify Roy’s identity for good 1.∂∂p1V(p,I) =−αβ2I(βp1+αp2)2∂∂IV(p,I) =αββp1+αp2⇒xm1(p,I) =αβ2I(βp1+αp2)2×βp1+αp2αβ=βIβp1+αp2(b) Solve for their Hicksian Demands and their expenditure function.At the cost-minimizing Hicksian Demands, we know that once again the consumer will not be wasteful andαx1=βx2, but instead of a binding budget constraint this time we will have a binding utility constraintmin{αx1,βx2}=u. Putting these two equations together, we can saymin{αx1,βx2}=αx1=βx2=u⇒xh1(p,u) =u/α,xh2(p,u) =u/βThe expenditure function is the cost of the Hicksian Demand.e(p,u) =p1xh1(p,u) +p2xh2(p,u) =p1uα+p2uβ=(βp1+αp2)uαβ=up1α+p2β(c) Verify Shephard’s Lemma for good 1.∂∂p1e(p,u) =uα3. For a symmetric Cobb-Douglas consumer with utility functionu(x) =x1x2, verify both of the Woodchuck Identities.That is, calculate the indirect utility function, the expenditure function, and show that they are indeed inverses ofone another (holding prices constant).While I expect any student to be able to solve both the utility max an cost min problems, for this solution I will recyclesome generic formulas for Marshallian and Hicksian Demands for consumers with generic Cobb-Douglas preferences.xm1(p,I) =I/2p1,xm2(p,I) =I/2p2xh1(p,u) =˘p2u/p1,xh2(p,u) =˘p1u/p2V(p,I) =I24p1p2,e(p,u) =2pp1p2u⇒V(p,e(p,u)) =e(p,u)24p1p2=u⇒e(p,u) =p4p1p2u=2pp1p2ue(p,V(p,I)) =2˘p1p2V(p,I) =I⇒p1p2V(p,I) =I2/4⇒V(p,I) =I24p1p22