Econ 600 Schroeter Practice Problem Set #4 1. Let be a differentiable function of a vector of arguments ℜ→ℜℜℜ1: xxFmn()θ,, zx, where x and z are vectors of choice variables of dimensions and , respectively, and 1≥n 1≥mθ is a scalar parameter. Consider the problem: ()θθ,,max;,...zxFgivenforzxtrw. Assume that the problem has a strict global solution for each value of θ. Denote the optimal values of x and z by and ()θ*x()θ*z (assumed differentiable) and define the value function: () ()()().,,***θθθθzxFF ≡ Now consider the problem: ()θθ,,max,;...zxFzgivenforxtrw. Again assume that there is a strict global solution for all z and θ denoted ()θ,ˆzx (again assumed differentiable) and define the value function: ()()()θθθ,,,ˆ,ˆzzxFzF ≡. It is obvious that () ()()θθθ,ˆ**zxx = and ()()()θθθ,ˆ**zFF =. a. Use the envelope theorem to show () ()()θθθθθ,ˆ**zFddF∂∂=. b. Show that () ()()θθθθθ,ˆ*222*2zFdFd∂∂≥. (Hint: For any 0θ and 1θ, we have ()()()()()10*11*1*,ˆ,ˆθθθθθzFzFF ≥=. Viewing and as functions of ()1*θF()(10*,ˆθθzF)1θ for fixed 0θ, write down second-order Taylor series expansions for them taking 0θ as the expansion point.) c. Use the results of part b and Hotelling's lemma (see, for example, Simon and Blume, Theorem 22.11) to prove that a competitive firm's "long-run" supply curve is at least as elastic as its "short-run" supply curve.22. This problem is closely related to the previous one. Let and be differentiable functions of a vector of arguments ℜ→ℜℜℜ1: xxFmnℜ→ℜℜℜ1: xxgmn()θ,, zx, where x and z are vectors of choice variables of dimensions and , respectively, and 1≥n 1≥mθ is a scalar parameter. We can think of the elements of x as choice variables that can be adjusted in the "long-run" or the "short-run," while the elements of z are choice variables that can be adjusted only in the long-run and are fixed in the short-run. First consider the long-run, equality-constrained, maximization problem: ()()bzxgtosubjectzxFgivenforzxtrw=θθθ,,,,max;,..., where b is a scalar constraint constant. Assume that the problem has a strict global solution for each value of θ. Denote the optimal values of x and z by and ()θ*x()θ*z (assumed differentiable) and define the long-run value function: () () ()().,,***θθθθzxFF ≡ Now consider the short-run, equality-constrained, maximization problem: ()()bzxgtosubjectzxFzgivenforxtrw=θθθ,,,,max,;.... Again assume that there is a strict global solution for all z and θ denoted ()θ,ˆzx (again assumed differentiable) and define the short-run value function: . It is obvious that () ()(θθθ,,,ˆ,ˆzzxFzF ≡)()()()θθθ,ˆ**zxx = and . () ()()θθθ,ˆ**zFF = a. Use the envelope theorem to establish the "first-order envelope property": () ()()θθθθθ,ˆ**zFddF∂∂=. b. Review the proof of the "second-order envelope property," () ()()θθθθθ,ˆ*222*2zFdFd∂∂≥, given in part b of the previous problem. This proof carries over to the present case of an equality-constrained maximization problem except that we have to add the assumption that, for any z and θ, we can find an x that satisfies the constraint. Where is this additional assumption used in the proof? (Note: The first-order envelope property says that the long-run and short-run value functions are tangent at the point they have in common. The second-order envelope property says that, at this tangency point, the long-run value function is at least as convex as the short-run value function.)3c. Use the result of part b, the expenditure function from consumer theory, and its relation to Hicksian (compensated) demand functions (discussed in Mas-Colell, Whinston, and Green, section 3.E and Proposition 3.G.1, for example) to prove that a consumer's Hicksian demand for a good is at least as responsive to changes in own-price in the long-run (when quantities of all goods can be chosen freely) as in the short-run (when the quantities of some goods are fixed). (Note: This result is called the LeChatelier-Samuelson principle.) 3. This problem is closely related to the previous two. Let and be differentiable functions of a vector of arguments ℜ→ℜℜℜ1: xxFmnℜ→ℜℜℜ1: xxgmn()θ,, zx, where x and z are vectors of choice variables of dimensions and , respectively, and 1≥n 1≥mθ is a scalar parameter. Consider the problem: ()()0,,,,min;,...=θθθzxgtosubjectzxFgivenforzxtrw, Assume that the problem has a strict global solution for each value of θ. Denote the optimal values of x and z by and ()θ*x()θ*z (assumed differentiable) and define the value function: () ()()().,,***θθθθzxFF ≡ Now consider the problem: ()()0,,,,min,;...=θθθzxgtosubjectzxFzgivenforxtrw. Again assume that there is a strict global solution for all z and θ denoted ()θ,ˆzx (again assumed differentiable) and define the value function: ()()()θθθ,,,ˆ,ˆzzxFzF ≡. State and prove the second-order envelope property relevant to these two problems. Every principles of microeconomics textbook contains a graph of a family of U-shaped short-run average cost curves and the associated long-run average cost curve. How is the second-order envelope property evident in the appearance of these graphs? 4. Consider the following expenditure minimization problem for a consumer with utility function : ()21, xxu ().,min212211,...21Uxxutosubjectxpxpxxtrw=+ Assume that is differentiable with strictly positive partial derivatives throughout . Further assume that; given initial values for prices, and , and utility, U; we have strictly positive solutions, and at which the second-order sufficient conditions for the problem are satisfied. (These assumptions mean that it is safe to ignore the non-()⋅u2+ℜ1p2p*1x*2x4negativity restrictions on and , and to treat the utility constraint as an equality, as we have done in the statement of the problem above. Denote the solutions as functions (assumed differentiable) of the parameters: 1x2x()Uppx ;,21*1 and ()Uppx ;,21*2. Define the expenditure function: ()()()UppxpUppxpUppE ;,;,;,21*2221*1121+≡ . a. Prove that 1*22*1pxpx ∂∂=∂∂ with an approach that uses the envelope theorem and Young's theorem. (Young's theorem says that the second-order cross partial derivatives of differentiable functions are independent of the