Partial Solutions to Homework 2

Transcription

1 Partial Solutions to Homework. Carefully depict some of the indi erence curves for the following utility functions. In each case, check whether the preferences are monotonic and whether preferences are convex. Be very precise! Once you have depicted the indi erence curves it is OK to use the graphs to check monotonicity and convexity, but then you need a short explanation and show how you perform the graphical test. It is of course also possible to use some algebra and the de nitions of convexity and monotonicity. You ll have to draw these by yourself! (a) u (x ; x ) = x + p x Convex and monotonic (b) u (x ; x ) = x 3 x 3 Convex and monotonic (c) u (x ; x ) = 5 + x + p x Convex and monotonic (se prefs as u (x ; x ) = x + p x ) (d) u(x ; x ) = k min fx ; x g + 3 where k > 0 (Weakly) convex and monotonic (I ve been a bit careless in de nitions. Usually, one makes a distinction between weak and strict monotonicity and convexity, where the weak version requires a weak inequality and the strict version requires a strict inequality) (e) u(x ; x ) = p x + x Convex and monotonic (relabeling of u (x ; x ) = x + p x ; so you don t need any additional work). q (f) u(x ; x ) = (x ) + (x 4) Convex, but non-monotonic (indi erence curves are circles centered on (,4). (g) u (x ; x ) = integer (x ) + integer (x ) ; where integer (x) means the integer part of x. That is and so on. Non-convex but (weakly) monotonic. (h) u (x ; x ) = (x ) + (x ) Non-convex but (strictly) monotonic. integer (x) = 0 if 0 x < integer (x) = if x < integer (x) = if x < 3. Let preferences be represented by u(x ; x ) = x a x b and suppose that the income is m and that prices are given by and (a) Set the consumer choice problem up as a problem in a single variable (skip if you are comfortable using Lagrangian methods).

2 Answer Follow steps from lecture and write problem as 0x m x a m p x b or 0x m m p x a x b (b) Can you assume that b = a? Why/why not? Why would you want to make such an assumption? Answer Yes, provided that a; b > 0. This is because we then have that f (u) = u increasing. Hence, we can de ne ba = and note that a x a x b = (x ) a (x ) b = x ba x b is strictly This shows that for every pair a; b > 0 we can nd some ba so that x ba x b represents the se preferences as x a x b Consequently, we do not lose any generality by insisting on a + b = (c) Write down the necessary conditions for an interior solution ( rst order conditions) for the consumer choice problem. Answer in case you plugged in the budget constraint we get " # a d m p x 0 = dx = ax a x a m p x a x a a m p x (d) Solve out explicitly for a bundle (x ; x ) that satis es the rst order conditions. Which two bundles except for the one you just derived may solve the problem? Evaluate the utility function at these two bundles and at (x ; x ). What is the solution to the utility imization problem? Answer Proceed as in class and you ll eventually get that (x ; x ) = The only other bundles that can be optimal are so the bundle (x ; x ) = ; ( m ; 0 a)m and 0; mp ; but u (x ; x ) = u ; ( a)m a a ( a)m = > 0 m = u ; 0 = u 0; mp ; ; ( a)m does indeed solve the problem.

3 (e) (Optional) Use the Lagrangian to solve the problem and convince yourself that you get the se solution. 3. Now suppose that u(x ; x ) = ln (x + x ) (a) Formulate the consumer choice problem. Answer The cleanest solution is to observe that the problem is which has the se solutions as s.t. x + x m s.t. x + x m ln (x + x ) x + x (b) Suppose that the solution is interior. What must be true about and? If the solution is interior, how many solutions are there? Illustrate with a picture. Answer. Since the utility function is monotonic we can plug in the constraint just like in the previous case and write the problem as x + m x 0x m If the solution is interior (meaning here that 0 < x < m ) it must be that 0 = d x + m p x = dx i Hence, an interior solution exists only if = ; in case any x h0; mp solves the problem. (c) Under what condition does the consumer only consume x in the solution? Under what condition does the consumer only consume x in the solution? What is the most likely type of solution with this type of preferences, corner solutions or interior solutions? Answer It is relatively easy to see that if < then the consumer consumes only good and if > the consumer consumes only good. One way to realize this is to draw a graph. One can also note that d x + m p < 0 if p x > 0 = = 0 if = 0 dx > 0 if < 0 4. Now suppose that preferences are represented by u(x ; x ) = x + ln x, the income is m and that prices are given by and (a) Carefully depict the indi erence curves for these preferences. In particular, be very careful about the slopes near the x axis and the x axis. 3

4 (b) Formulate the consumer choice problem and write down the necessary conditions for an interior solution ( rst order conditions). Will the solution always be interior? If = =, what level of m is required for the solution to be interior? Can you come up with more general conditions? Write problem as m x + ln x 0x m For solution to be interior it must be that is satis ed for some x + = 0, x = x 0; mp This requires that < m Now, if m we have that + x > + = 0 for every x < p and m p Hence, the imand is strictly increasing over the whole range and (x ; x ) = 0; mp solves the problem. 5. Consider a farmer from the Eau Claire region in Wisconsin who lives a two period life and is the proud owner of an Airshyre diary cow. The cow produces 0000 pounds of milk in each period and we imagine that the farmer/consumer only consumes milk and that the milk goes bad unless it is consumed within the period. The farmer has preferences u (c ) + u (c ) where c and c denotes the milk consumption in each period. Assume that u (c) is strictly increasing. (a) Suppose that there is a bank where the farmer can borrow and lend milk. Let r be the interest rate, so that if the farmer deposits s pounds of milk in the bank he gets (+r)s pounds back in the second period. Carefully derive and depict the budget set. Answer. This is just like in class. In the end you ll get c + + r c r 0000 (b) Formulate the relevant choice problem for the farmer. Answer u (c ) + u (c ) c ;c s.t.c + + r c r 0000 c 0 c 0 (c) Derive the rst order condition (the necessary condition for optimality under the assumption that the solution is not at a corner) 4

5 Write u c + r r c + u (c ) The rst order condition is 0 u + + r r c A + r + u0 (c ) = 0 {z } Hence we can write =c u 0 (c ) u 0 (c ) = + r (d) Assume that u (c) = ln c; r = 0 and = Solve for the optimal consumption plan. Answer c = c = 0000 (e) Now, assume that there is no credit market. However, the farmer can give the cow antibiotics in period, which increases the milk production in period, but decreases it in period. Let x be the quantity antibiotics given to the animal and assume that 0 x 00 Suppose that the milk production as a function of x is m (x) = x in the rst period and m (x) = 0000 x in the second period. Carefully depict in a graph the set of feasible consumption plans (c ; c ) given that the farmer has this technology available, but has no access to any kind of market. Correct answers have a at segment until (0000; 0000) and then the slope of the "budget line" will get steeper and steeper as c increases. 5