Economics 201bSpring 2010Problem Set 5Due Thursday April 22You will need to know the following trigonometric identities:cos(α − β) = cos(α) cos(β) + sin(α) sin(β)cos2(θ) + sin2(θ) = 1cos(θ) = sin(θ +π2)cos(θ + π) = − cos(θ) sin(θ + π) = − sin(θ)limθ→0sin(θ)θ= 11. Give a description of an exchange economy that satisfies all the conditions of the one in Theorem1 Lecture 6 except the condition ¯ω  0 (which is implicitly assumed in the Theorem), such thatthe Second Welfare Theorem now no longer holds - i.e. there is now a Pareto Optimal allocationthat can’t be supported as a Walrasian Equilibrium with Transfers.2. Assume the function F : RL−1++× RLI+−→ RL−1from Definition 1 Lecture 10 is continuous. ProveProposition 2. (Recall, f : U → V is continuous if for every closed set D ⊂ V , f−1(D) is closed.)3. Given a price p = (p1, p2) ∈ ∆o, we can define θ(p) (measured in radians) to be the angle betweenp and the horizontal axis. Suppose we have a function z : ∆o→ R2z(p) =f(θ(p)) cos(6θ(p)) cos(θ(p) +π2), f(θ(p)) cos(6θ(p)) sin(θ(p) +π2)where f is a positive differentiable function defined on (0,π2) such thatlimθ↓0f(θ) = limθ↑π2f(θ) = ∞andf(θ) =≤ M1θθ < 1 θ ∈ [,π2− ]≤ M1π2−θθ >π2− (1)for some constant M > 0.(a) Draw a picture of z(p).(b) Express p1and p2as functions of θ(p).(c) Prove z(p) · p = 0.1(d) Prove z(p) satisfies the conditions of D-G-K-N Lemma. (Hint: z(p) can be rewritten asZ(θ(p)). To prove that z(p) is bounded below on ∆o, it suffices to show Z(θ) is boundedbelow on (0,π2) (notice I wrote Z(θ) not Z(θ(p))). First write down Z(θ). Then to proveZ(θ) is bounded below, break (0,π2) into three pieces: the compact interval [,π2− ] andthe two pieces (0, ) and (π2− ,π2) . Then consider each piece separately. To show Z(θ) isbounded on (0, ), it suffices to show limθ→0Z(θ) is bounded below. A similar method maybe applied to the interval (π2− ,π2).)(e) Suppose z(p) was the excess demand function of some economy. What are the equilibriumprices p∗i= (p∗1i, p∗2i) ∈ ∆o, i = 1, 2, . . . N (where N is the number of equilibrium prices youfind)? Then re-express each such price p∗iin the normalized form (ˆp∗i, 1). (Hint: Equilibriumprices correspond to the solutions of z(p) = 0. Once again, it is easier to first look for thesolutions of Z(θ) = 0. Then for each such solution θ∗i, express the corresponding price p∗i.)(f) Recall the function φ(ˆp) → ∆o. Explain what the sign (-1, 1, or 0) ofddˆpθ(φ(ˆp))is.(g) Write out ˆz(ˆp) and calculateindex(ˆp∗i) = −sgnddˆpˆz(ˆp)ˆp=ˆp∗ifor all iIs the economy regular? Notice that the Index Theorem is indeed true.(h) Suppose we change f (θ) slightly so that f is still smooth but now f > 0 except at asingle point x ∈ (0,π2) where f (x) = 0 - rename this altered function fx(·). Consider thenew excess demand function zx(p) where the old f is replaced with the new fx(convinceyourself zx(p) still satisfies the conditions of D-G-K-N). The corresponding economy is nolonger regular. The summation in the Index Theorem can now take the values 0, 1, and 2depending on the value x. Which values of x correspond to which values of the