Problem Set 1 Answer Key. I. Short Problems 1. Check whether the following three functions represent the same underlying preferences

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1 Problem Set Answer Key I. Short Problems. Check whether the following three functions represent the same underlying preferences u (q ; q ) = q = + q = u (q ; q ) = q + q u (q ; q ) = ln q + ln q All three represent di erent underlying preferences. In each of these cases we get di erent shapes for indi erence curves: while the second utility function represents perfect substitutes with M RS =, the rst qand third utility functions represent strictly convex preferences with MRS = q q and MRS = q q ; respectively. Hence there is no positive monotonic transformation that can transform one of these utility functions in either of the other.. Consider the following cost function c (u; p) = u (p + ) : List two properties of the cost function and verify these properties with the given cost function. The two properties I will verify are () homogeneity of degree, and () increasing in u: To verify () C (u; p) = u (p + ) = u (p + ) = c (u; p) : To verify (u; p) = (p + ) > II. Longer Problems. Assume a person has monotone preferences. Given are two price-income combinations p ; p ; x = (; ; ) p ; p ; x = (; 5; 4) : a) Draw both budget sets in a two-good diagram. Can you tell just from this information whether the individual is worse o or better o with p ; p ; x compared to p ; p ; x?

2 Good Good Without any further information we cannot say whether the person is better o or worse o than before, because neither budget set is a subset of the other. b) Suppose the consumption bundle under p ; p ; x = (; ; ) was (=3; =3) : Can you tell just from this information whether the individual is worse o or better o with p ; p ; x compared to p ; p ; x? We can say that the person cannot be worse o under the new budget set than under the old one, because the same consumption bundle that was chosen previously is still available and lies on both the old and the new budget line. To see this, evaluating expenditure on the old consumption bundle with the new prices yields = ; and evaluating expenditure on the old consumption bundle with the new prices yields = 4: Thus the old consumption bundle lies on both the old and the new budget line. c) Suppose the consumption bundle under p ; p ; x = (; ; ) was (=; =) and the consumer s preferences can be represented by u (q ; q ) = q + (q ) where (q ) is a strictly concave function of q : What type of preferences is represented in c)? Explain. These are quasi-linear preferences, because the indi erence curves are paralell o sets of each other with respect to the horizontal axis (good ). We know this because for any two points on the same indi erence curve (q; q) and (q ; q ),

3 if we add the same amount of good to both of these points we end up with two points on another indi erence curve. That is, if then q + (q ) = q + (q ) (q + ) + (q ) = (q + ) + (q ) d) Given the preferences in c) where on the new budget line will the consumer s new consumption bundle lie? Mark all the candidates for optimal consumption bundles. The consumer solves the problem max + (q ) q s:t: p q + q = x With (q ) strictly concave, the utility function is strictly quasi-concave and the rst order conditions are necessary and su cient. Since we know that the optimal consumption bundle in case of p ; p ; x = (; ; ) was (=; =) ; we know that with this budget constraint we have an interior solution, i.e. the tangency condition holds or MRS = = (q ) (q ) = : Comparing the two budget constraints, the price of good went up relatively less than the price of good, and for an interior solution after the change in the budget constraint we must have (q ) = :5 This implies that as the relative price of good two increases, the marginal utility of this good increases. This also means that less of this good is demanded. Hence the individual will demand less than = of the second good and therefore demand more than 4 5 = :75 units of the rst good. See thick line on budget line. 3

4 Good Good e) Solve for the Marshallian demand using the utility function given in c) with (q ) = p q Which are the bundles chosen by the consumer at p ; p ; x = (; 5; 4)? First solve for the Marshallian demand: max + p q q s:t: p q + q = x p ; p ; x = (; ; ) and Because we have a strictly quasi-concave function, we can use the tangency condition to solve for an interior solution for q > : We nd p q = p g (x; p) = p From this interior solution for q ; we can see that q will always be positive for any price-total outlay combination. To nd the solution with respect to q we need to use the budget constraint. The demand for good is given by : g = x g (x; p) p g (x; p) = x p p 4

5 From this "demand" we can see that there is a possibility that it could be negative if g (x; p) x p x x: Therefore, the Marshallian demand is given by g (x; p) = ; g (x; p) = x if p x g (x; p) = x p p ; g (x; p) = p if p < x Next we nd the optimal consumption bundles under our two budget sets by plugging prices and income into the Marshallian demand we just found. For the original budget set p g = = = For the new budget set g = g = u = : p = :5 = :6 g = 4 5 :6 = :6 u = :6 + p :6 = :4:. Suppose a consumer s preferences can be represented by the following utility function de ned on R + q u (q ; q ) = (q + q ): a) Draw the consumer s indi erence curves. First of all, we can look at a much simpler utility function by applying the positive monotonic transformation f (u) = u we then can write u (q ; q ) = q + q : 5

6 The best way to nd out how the indi erence curve looks like, is by solving for an indi erence curve and its shape. We can also see from our utility function that more of any good will yield a higher utility, that is, higher indi erence curves are to the northeast of the lower indi erence curves. u = + q q q = u q dq dq = q p u q d q dq = p u q + p x u q u q = u (u q ) 3 This means that the indi erence curves are downward sloping and that they are strictly concave. < Good Good b) Are the preferences monotone? Yes, = q = u > ; u > : Are they convex? No, because the indi erence curves are strictly concave and thus the better sets are nonconvex. This should be clear from the graph of indi erence curves. c) Find the Marshallian demand. Since preferences are not convex, not even locally, focussing on an interior solution to solve for the Marshallian demand will not give us the correct answer. 6

7 If we solve for the interior solution, we in fact utility minimize! Hence we need to focus on corner solutions. If p > ; g (x; p) = ; g (x; p) = x ; if p = ; either g (x; p) = x p or g (x; p) = x ; but not both; if p < ; g (x; p) = x p ; g (x; p) = : d) Find the Hicksian demand. As in the UMP, focussing on an interior solution to solve for the Hicksian demand will not give us the correct answer. If we solve for the interior solution, we in fact cost maximize! Hence we need to focus on corner solutions. If p > ; h (u; p) = ; h (u; p) = p u; if p = ; either h (u; p) = p u or h (u; p) = p u; but not both; if p < ; h (u; p) = p u; h (u; p) = : Note that for the Hicksian if you went with the original utility representation rather than the positive monotonic transformation, you would nd p > ; h (u; p) = ; h (u; p) = u; if p = ; either h (u; p) = u or h (u; p) = u; but not both; if p < ; h (u; p) = u; h (u; p) = : 7