Econ 101A — Solution to Midterm 1 Problem 1. Utility ...

1 Econ 101A Solution to Midterm 1 Problem 1. Utility maximization .(52 points) In this exercise, we consider a standard maximizationproblem with an unusual Utility function. The Utility function isu(x, y)= x+ price of goodxispxand the price of denote income byM,as usual, withM> is well-defined forx>0and fory> now on, assumex>0andy>0unless Compute u/ xand 2u/ the Utility function increasing inx? Is the Utility function concaveinx?(3 points)2. The consumer maximizes Utility subject to a budget constraint. Write down the maximization problemof the consumer with respect the budget constraint is satisfied withequality. (Hint: you can use the answer in point 1) (5 points)3.

2 Write down the Lagrangean function. (2 points)4. Write down thefirst order conditions for this Problem with respect tox,y,and .(4points)5. Solve explicitly forx andy as a function ofpx,py,andM.(8points)6. Do the solutions forx andy satisfy the positivity constraint, that is,x >0andy >0?.(2 points)7. Are these points maxima of the Problem above? Check that the determinant of the bordered Hessianis positive atx ,y ,and .(8 points)8. Use the expression forx that you obtained in point 6. Differentiate it with respect toM,that is,compute x / this good a normal good? (4 points)9. We are now interested in the sign of x / is, we would like to know if the demand functionis downward sloping.

3 (at higher prices, a lower quantity of the good is demanded). Argue, using theanswer that you gave in point 9 and something you learnt in class, that we know the sign of x / is this sign? (8 points)10. Can you guess the solutions forx andy for the following maximization Problem ? Develop yourargument. (8 point)max x+ y 2(1) +pyy MSolution to Problem u(x, y)/ x=(1/2) x 1/2>0and 2u(x, y)/ 2x= (1/4) x 3/2<0forx> derivativetells us that the Utility function is increasing inxfor all second derivative tell us thatthe Utility function is concave inx,that is, the marginal Utility from consumption of goodxdecreaseswith the consumption The budget constraint ispxx+pyy M.

4 The maximization Problem ismaxx,y x+ +pyy can write down the budget constraint with equality because the Utility function is strictly increasingboth inxandy.(Now follows the long explanation, you were not required to give this) As we proved inpoint 1, u(x, y)/ x >0, and we can by symmetry show u(x, y)/ y > that Utility is strictlyincreasing in both goods, the consumer will never choose a point on the interior of the budget set, ,apoint( x, y)such thatpx x+py y< reason is that the consumer could choose a point( x0, y0)1with x0> xand y0> ythat still satisfies the budget constraint, , such thatpx x0+py y0 M.(justpick( x0, y0)sufficiently close to( x, y)) But, given the monotonicity ofu,the bundle( x0, y0)provides ahigher Utility than the bundle( x, y).

5 Therefore the consumer in the optimum will never choose a bundle( x, y)such thatpx x0+py y0< can therefore limit ourselves to the points withpxx+pyy= Lagrangean isL(x, y, )= x+ y (pxx+pyy M).4. First order conditions: L x=(1/2) (x ) 1/2 px=0 L y=(1/2) (y ) 1/2 py=0 L = (pxx+pyy M)=05. Using thefirst twofirst order conditions, wefind(y )1/2(x )1/2=pxpyor(y )1/2=(px/py)(x )1 can raise this expression to the square power to gety =p2xp2yx .(2)We substitute this into the budget constraint to getpxx +pyp2xp2yx =Morpxx +pxpxpyx =Morx px 1+pxpy =Morx =Mpx11+pxpy.(3)We substitute (3) into (2) to gety =p2xp2yMpx11+pxpy=Mpypxpy11+pxpy=Mpy11+p ypx.

6 (4)6. Usingexpressions(3)and(4)itiseasytoseeth atx andy are both positive as long as The determinant of the bordered Hessian isdetH= 0 px py px 2L 2x 2L x y py 2L x y 2L 2y = 0 px py px 14(x ) 3/20 py0 14(y ) 3/2 ==0 ( px) 14 px(y ) 3/2 +( py) 14 py(x ) 3/2 ==14( px)2(y ) 3/2+14( py)2(x ) 3 >0andy >0(see point 7) followsdetH> The derivative ofx with respect to incomeMis x M=1px11+pxpy> good is a normal good, since its demand is increasing in In order to determine the sign of x / px,we do not need to do any derivation. We can use the , x px= h px x x Mwhereh is the Hicksian or compensated demand function.

7 We know that h / px<0,that is,compensated demand functions are always downward sloping. From point 8, we also know x / M >0,that is, goodxis a normal good. It follows, since both right-hand side terms are negative, that theleft-hand side, x / pxis also negative. The intuitive explanation: We know thatxis a normal normal goods, the income and substitution effect go in the same direction: an increase in theprice of goodxmakes goodymore desirable. In addition, the consumer is poorer, which leads her toconsume less of Problem (1) has one very important similarity to the initial Problem : the Utility function in the newproblem is the square of the Utility function in the old Problem .

8 Since the Utility function in the oldproblem was always positive (forx>0andy>0),it follows that the Utility function in the newproblem is an increasing function of the Utility function in the old Problem . (if the Utility functionin the old Problem could take on negative values, this argument would not apply, since the squarefunction would not be an increasing function over this larger range) Since the Utility function is just anincreasing function of the old one, it represents the same preferences. The Solution to the new Problem ,therefore, has to coincide with the Solution of the old Problem . One way to see this is as Problem 1maxu(x, y) +pyy Mand Problem 2:maxf(u(x, y)) +pyy Mwheref:R Ris an increasing function.

9 Consider the of Problem 2:f0 u0x px=0f0 u0y py=0 (pxx+pyy M)= we transform thefirst 2 to get the usual condition on the MRS, we obtainu0xu0y=pxpywhere the functionfdisappeared! This is the same condition that we would have obtained in Problem1. Since the budget constraint is the same in both problems, it follows that the solutions are the samein the two problems. Notice that, in order to do the above simplification, we needf06=0,that is, thefunction has to be strictly increasing, which was true forx>0andy>0.[We should also check thesecond order conditions].Short problems.(26 points) In this part, you are required to provide short answers to the problemsbelow.

10 Provide the steps in the derivation of your Problem 1.(16 points) Consider the Utility functionU(x)= xif0 x 10(20 x)if10<x 20defined for0 x Plot the Utility function as a function ofx.(1 point)2. IsU(10)>U(5)?IsU(20)>U(5)?Are the preferences represented by this Utility function monotonic(that is, ify x,theny x)?(5points)3. (Harder) Write down the set of preferences over the numbers between 0 and 20 that this Utility functionrepresents. Try to be precise, but you can help yourself using words. [Hint: for eachx, y [0,20],when isx y?](10points)Short Problem 2.(10 points)1. Consider the implicit functiong(x, y)=y 1 ln(x y)=0forx>0,y>0.