Solve!!!!!!!

Theory: 1) Take the function: y = f(x;a) = a' -2x - ax . a) Find the first order condition for a maximum/minimum. b) Is this a maximum or a minimum? c) Solve for x" . d) Explicitly solve for dx and da e) Now re-write the first order condition and use the implicit da function theorem to find dx da f) Use the envelope theorem to find " and explain briefly da why this theorem allows you to simplify your calculation from (d). g) What is dy ? Would your answer differ depending on whether x" was a maximum or minimum for this problem? 2) Let a consumer's utility be given by U(x, y, z) = In(x) + In(y) + In(2 + z). The consumer has a budget constraint of x + 3y+2z =18. a) What are the optimal choices for this consumer? b) What would the choices be if the budget constraint were increased from 18 to 19? c) What is the consumer's utility in each case? d) How do you interpret the Lagrange multiplier in (b)? e) Show how you could use the Lagrange multiplier (and the envelope theorem for constrained problems) to estimate the change in consumer utility caused by increasing her budget from 18 to 19 without having to resolve for x, y, and z in (c). 3) Take the problem max z = - subject to x + 4y =90. a) Set up the Lagrangian and find xty the first order conditions b) Solve explicitly for x, y, z and 1, the Lagrangian multiplier. c) Set up the dual problem of (a), i.e., minimize the primal constraint function subject to the primal objective function being equal to the value of z" obtained in (b). d) Solve explicitly for x, y, and 1" , the Lagrangian from the dual. e) What is the relationship between 1 and 1" , and can you provide an economic explanation for this?Background: Human Capital acquisition and Signaling are two competing explanations for why people with more education receive higher earnings. The Human Capital model posits that people acquire skills in school that make them more productive in their jobs, yielding higher wages. As we have discussed in class, the Signaling model posits that schooling primarily or exclusively serves to 'signal' the higher average ability of those people who are able to successfully complete more education. In the signaling model, schooling does not necessarily increase productivity; it merely provides information about productivity to employers who would otherwise have trouble discerning it. 1. You have been appointed Education Czar by President Ralph Nader (appointed by the U.S. Supreme Court in a surprise election turn) who has given you despotic power and unlimited funding to design a social experiment using 1,000 human subjects to test whether the General Educational Development (GED) certificate raises earnings by serving as a labor market signal. Imagine that you also have the following tools at your disposal: 1) the Armed Forces Qualification Test (AFQT) which is a reliable measure of individual ability/productivity; 2) the ability to assign GEDs to whomever you like; 3) the ability to deny GEDs to whomever you like; 4) the ability to observe the earnings of participants in your experiment over several years. Describe briefly a difference-in-difference experimental design you could use to evaluate the signaling value of the GED. (Note: you are conducting a genuine economic experiment, not a 'natural experiment.') Specifically, how would you make use of the four tools described above? What would constitute evidence in favor of the signaling hypothesis? What would constitute evidence against it? Why is the difference-in- difference design important to your study? (You may want to consult the opening pages of the TMW study to clarify your thoughts on this topic.) 2. Unfortunately, President Nader was impeached for the crime of assigning extra- constitutional powers to an MIT undergraduate. Hence, you will have to form your conclusions based upon the Tyler, Murname, and Willett 'natural experiment' study. At the top of page 435, TMW sketch in words the necessary conditions for a GED signaling equilibrium to hold. Write out the key equations for these conditions. If the GED were offered for free, was not unpleasant to take, and required no preparation time (perhaps because it measures "ability' rather than knowledge), could the signaling equilibrium ever hold? Why or why not? 3. The TMW study provides compelling evidence that the GED holds a pure signaling value of approximately $1,500 for people just above the passing score versus those just below the passing score. Does the TMW study therefore demonstrate that the human capital model of education as a productivity enhancing mechanism is incorrect? Please answer in a few precise sentences. 4. The Governor of Texas, who assumed the presidency after the Nader impeachment, hears of the TMW study on Fox Headline News. He proposes the following policy to boost the low earnings of U.S. high school dropouts: the Department of Education will grant each high school dropout a GED immediately. In a few years time, this should raise their earnings by about $1,500, just as TMW demonstrate. Explain whether and why this policy is likely to yield the intended result.8. There are continuing efforts to open up the Alaskan wilderness for further energy exploration. Suppose there are only 2 goods, natural gas q1, and other goods, 92. Suppose further that the current income of the typical individual in the U.S. is $25 thousand dollars, that the current price of natural gas is p, = $5 per unit and that the price of other goods is p2 = $1 per unit. If further exploration is allowed, the price of natural gas is projected to fall to $1 per unit. However, there is a group of environmentalists who will be outraged by any damage to this natural habitat, Assume that the typical, non-environmentalist American has the utility function: 2(7, 92) = 3091 93 (a) At initial prices, what is the demand for q, for 92, and what is the initial utility of the typical American