Question

2. (18 marks total) In this exercise, we will derive the famous "envelope theorem". Suppose you wish to (unconditionally) maximize some objective function f(x,y; a), where r and y are two variables you can choose, while a is some variable that is given exogenously. Note that, even though we don't get to choose a, it may still affect the optimal choice of r. An example of a variable like this would be the wage in the household problem we discussed in class: w affects the household's choice of c and I, but the household doesn't get to choose w. (a) (2 marks) Let * and y* denote the optimal choices of x and y, respectively. Write down the FOCs that implicitly determine r* and y*. (b) (5 marks) Given a, let Qa) = f(x*,y*:a) be the maximized value of the objective function (i.e., the value of f at the optimal choices r* and y*). Note that x* and y' here are implicit functions of a: if a changes, the optimal values of x and y will generally change in response. Totally differentiate the equation Q(a) = f(x*,y*; a), then solve to get an expression for Q'(a) that depends on dx*/da and dy* /da. Briefly interpret this expression in words. (c) (3 marks) Substitute the FOC's you got in part (a) into the expression from part (b). The resulting expression is effectively a statement of the envelope theorem. Interpret that theorem in your own words. (d) (8 marks) Suppose instead we have a constrained maximization problem. That is, sup- pose we want to maximize f(r, y; a) subject to the equality constraint g(x,y; a) = 0. Re-do (a)-(c) for this case in order to obtain the envelope theorem for constrained max- imization. (HINT: Remember that the maximization of f subject to a constraint is

Answer 1

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