Help with the following questions

This Problem set tests the knowledge that you accumulated in the first 4 lectures. It is mostly on the mathematical techniques that we developed, but there is also some application to consumer decisions, as introduced in Lecture 4. General rules for problem sets: show your work, write down the steps that you use to get a solution (no credit for right solutions without explanation), write legibly. If you cannot solve a problem fully, write down a partial solution. We give partial credit for partial solutions that are correct. Do not forget to write your name on the problem set! Problem 1. Univariate unconstrained maximization. (10 points) Consider the following maxi- mization problem: max f(1; 20) = exp(-(x - 10)") 1. Write down the first order conditions for this problem with respect to r (notice that ro is a parameter, you should not maximize with respect to it). (1 point) 2. Solve explicitely for r* that satisfies the first order conditions. (1 point) 3. Compute the second order conditions. Is the stationary point that you found in point 2 a maxi- mum? Why (or why not)? (2 points) 4. As a comparative statics exercise, compute the change in r* as ro varies. In other words, compute der* /dro. (2 points) 5. We are interested in how the value function f(r*(To); To) varies as ro varies. We do it two ways. First, plug in r*(zo) from point 2 and then take the derivative with respect to ro- Second, use the envelope theorem. You should get the same result! (2 points) 6. Is the function f concave in z? (2 points) Problem 2. Multivariate unconstrained maximization. (13 points) Consider the following maxi- mization problem: max f (x, y; a, b) = ar' - x + by' - y 1. Write down the first order conditions for this problem with respect to r and y (notice that a and b are parameters, you do not need to maximize with respect to them). (1 point) 2. Solve explicitely for r* and y* that satisfy the first order conditions. (1 point) 3. Compute the second order conditions. Under what conditions for a and b is the stationary point that you found in point 2 a maximum? (2 points) 4. Asume that the conditions for a and b that you found in point 3 are met. As a comparative statics exercise, compute the change in y* as a varies. In other words, compute dy* / da. Compute it both lirectly using the solution that you obtained in point 2 and using the general method presented in class that makes use of the implicit function theorem. The two results should coincide! (3 points) 5. We are interested in how the value function f(x*(a, b);y*(a, b)) varies as a varies. We do it two ways. First, plug in r*(a, b) and y*(a, b) from point 2 into f and then take the derivative of f(x*(a, b); y*(a, b)) with respect to a. Second, use the envelope theorem. You should get the same result! Which method is faster? (3 points) 6. Under what conditions on a and b is the function f concave in a and y? When is it convex in r and y? (3 points)Problem 3. Multivariate constrained maximization. (19 points) Consider the following maxi- mization problem: max u(x, y) = roy a.t. pax + pyy = M. with 0 0,y > 0. Does your solution for r* and y* satisfies these constraints? What assumptions you need to make about Pr, Py and M so that r* > 0 and y* > 0? (1 point) 5. Write down the bordered Hessian. Compute the determinant of this 3x3 matrix and check that it is positive (this is the condition that you need to check for a constrained maximum) (3 points) 6. As a comparative statics exercise, compute the change in a* as p, varies. In order to do so, use directly the expressions that you obtained in point 3, and differentiate r* with respect to py. Does your result make sense? That is, what happens to the quantity of good a* consumed as the price of good z increases? (2 points) 7. Similarly, compute the change in r* as py varies. Does this result make sense? What happens to the quantity of good a* consumed as the price of good y increases? (2 points) 8. Finally, compute the change in r* as M varies. Does this result make sense? What happens to the quantity of good r* consumed as the total income M increases? (2 points) 9. We have so far looked at the effect of changes in pr, py, and M on the quantities of goods consumed. We now want to look at the effects on the utility of the consumer at the optimum. Use the envelope theorem to calculate du(x* (Pz, Py; M) , y* (Pz, Py: M))/dpr. What happens to utility at the optimum as the price of good r increases? Is this result surprising? (2 points) 10. Use the envelope theorem to calculate du(r* (pr. py: M) , y* (Pr, Py; M))/OM. What happens to utility at the optimum as total income M increases? Is this result surprising? (2 points) Problem 4. Rationality of preferences (5 points) Prove the following statements: . if _ is rational, then ~ is transitive, that is, a ~y and y ~ z implies a ~ z (3 points) . if > is rational, then > has the reflexive property, that is, 2 _ I for all z. (2 points)1. "Liking more of everything" Consider a set X = R", and define that, for every two bundles r, y e X, x _ y, if and only if I* 2 yx for every component k, that is, bundle r is at least as good as bundle y if the former contains more units than the latter in each of its components. Check if this preference relation satisfies (a) completeness, (b) transitivity, (c) strong monotonicity, and (d) strict convexity. 2. Checking properties of a preference relation. Consider a consumer with the following preference relation: he weakly prefers (71, 12) to (31, 32), ie., (21, 12) _ (31, y2), if and only if max {21, x2} > min (31, 32}. (a) Provide a verbal description of his preference relation. (b) Check whether this preference relation is complete, transitive, monotone, convex, and locally nonsatiated. 3. Strictly Convex Preferences. Consider preferences defined on the consumption set X = RY. (a) Suppose Alex has a utility function U(x) = (1 + 21)(1 + 12). Show that his preferences are convex. Are his preferences strictly convex? (b) Barbara has a utility function U(r) = $172. Are her preferences convex or strictly convex? 4. Quasi-Linear Preference. Write the (n + 1) -dimensional consumption vector as (y, z) where y is a scalar and z is an n-dimensional consumption vector. A utility function U(x) is quasi-linear if it can be written as follows U(x) = ay + V(=). Consider that the consumption set is X = R4+, where y e R, and z c RY. (a) Show that if V is concave, U is quasi-concave. (b) Show that if U is quasi-concave, V is concave. . Hint: Suppose that for some a", r], s', concavity fails; that is, V(2) )