questionsare given belows

Problem 1 (Consumption Based Capital Asset Pricing Model): We will now use the Euler Equation to derive an asset pricing equation. Recall the Euler Equation: We will again linearize Euler Equation, but this time we will allow for many different types of assets, R, R, .., R. ..., R with stochastic returns. Ri = exp(r) + o'e; - (a') /2), where a; is stochastic and has unit variance and all other terms on the right- hand-side are fixed. We denote the risk-free return: R . Specifically, we will assume that at time t - 1, R, is known. We will now solve for the equilibrium relationship between asset returns and consumption innovations. You can think of this as an economy in which asset prices are endogenous and the consumption process is fixed. However, what we will derive below applies to an economy with either (or both) endogenous asset prices or endogenous consumption. Assume u is in the class of isoelastic (i.e., constant relative risk aversion) utility functions, u(c) = 1 - 7 1. Rearrange the Euler Equation to find (hint: see lecture notes from lecture 2), 1 = Enexp -p+ riti + o'citi - (o')"/2-ylnati] where - In 6 = p. 2. Assume that ely, and A Inc1 are conditionally normally distributed. Show that 0=r+1 - (0') /2-p- YEAInc+ + =V(o'eh - 7Inc+1). (1) where the expectation operator E and the variance operator V are conditional on information at date t. Set r+1 = r+1, to show that 0=r41 - p- YEAInc+ + = V(yAlnc+1). Difference Eq. 1 for assets i and j, to find 14-14 = =[(@ )2 - (@1)? - V(o'et - yAlnc+1) + V(jet - YAinc+1). 3. Prove the following statistical lemma: If A and B are random variables, then, V(A + B) = V(A) + V(B) + 2Cov(A, B) 4. Using the statistical lemma, show that 171 = 70 ic - 70 je where n' = p - r, and Gic = Cov(o's', A Inc). 5. Now consider the special case where i =equities and j =risk free asset. Show that Trend ity. =70equity,c. Interpret this equation.free bond. In particular, we will assume that the amount is sufficiently small that it does not effect the properties of the consumption process. Hence, we can use the stochastic dynamics of the consumption process, which were derived above, to price the bond. 1. Note that the Euler Equation must hold for all assets in the consumer's portfolio. Explain intuitively why this is the case. 2. If the risk free bond has interest rate R (with In Rf = rf), show that the Euler Equation for the risk free asset will be, u'(a) = E.R exp(-p)u'(a+1). 3. Manipulate the Euler Equation to show that, (R) = Enexp {-p-Alnati}. 4. Show that A In q+1 is distributed normally with mean } (r - p) +$0'- ' and variance o'. Note that we usually just assume that A Incit1 is distributed normally. For this problem, we can show it exactly. You can use the intermediate results derived on the first problem set (which are summarized above). 5. Use the result of questions 3 and 4 to derive the equilibrium interest rate of the risk free bond. Now use the Consumption Capital Asset Pricing Equation (previous problem) to derive the equilibrium interest rate of the risk free bond. Your results should be the same, since both derivations are based on the Euler Equation. 6. Defend the assumption that the amount of risk free bond is in zero net supply (i.e., that the net amount available is zero) 7. In this model economy Gequity,e = equity. Is this true in the real world? Why or why not?Problem Set 3 A. Demand Elasticities 1. This is your chance to calculate demand elasticities for health care. Suppose you are collecting data from a country (like Japan) where the government sets the price of health care. Each prefecture in Japan has a different set of prices (for example, Tokyo has higher prices than rural Hokkaido). Suppose you observe the following data for 1999: # doctor visits per month Price per doctor visit Tokyo 1.0 visits 20 Yen Hokkaido 1.5 visits 10 Yen What is the arc price elasticity of demand for health care consumers in Japan (using only these data)? Using your estimated elasticity, what would the demand for health care be if the price in Tokyo were raised to 30 Yen per visit? What would the demand in Hokkaido be if the price were lowered to 5 Yen per visit? 2. You continue you observations of the Japanese health care system into the year 2000. For inscrutable reasons having to do with internal Japanese politics, the government changed the price in both Tokyo and Hokkaido that year, and you observe the following demand: # doctor visits per month Price per doctor visit Tokyo 0.9 visits 30 Yen Hokkaido 1.4 visits 15 Yen Calculate the price elasticity of demand for health care in Japan using only data from the year 2000. 3. Use data from both years to calculate the elasticity of demand for health care for Tokyo and Hokkaido separately. Using your estimated elasticities, what would the demand for health care in each prefecture be if the price were raised to 60 Yen per visit next year (for both prefectures)? Suppose the population of Tokyo and Hokkaido are given in the following table: Population Tokyo 10 million Hokkaido 2 million 4. Combine the Tokyo and Hokkaido estimates from question 3 to get a single estimate of the health care demand elasticity for all of Japan. 5. Which set of estimates in questions 1-4 come closest to a "natural experiment" - the estimates that rely on differences in price between Tokyo and Hokkaido, the estimates that rely on difference in price in 1999 and 2000, or the combined estimates? What are strengths and weaknesses of each of the estimates? (Hint: What could introduce error in the estimates? What makes each situation different from a randomized trial?). Would a similar study in the U.S. (where the government does not set the price of health care) be as convincing as your study in Japan? 6. Estimates of the price elasticity of demand for health care vary greatly among studies. Studies that base their estimates on natural experiments tend to find demand elasticities that are similar to the ones found in the RAND Health Insurance Experiment, while non-natural experiment studies tend to find much larger elasticities. Which set of estimates is more plausible? What are some reasons to prefer one set of estimates over another?1. Consider the following information on Alfred's demand for visits per year to his health clinic, if his health insurance does not cover clinic visits (100% coinsurance rate). P Q 5 10 9 15 20 8 25 30 35 40 (a) Alfred has been paying $30 per visit. How many visits does he make per year? Draw his demand curve. Before insurance, Alfred will consume 6 visits at P = $30. (b) What happens to his demand curve if the insurance company institutes a 40% coinsur- ance feature (Alfred pays 40% of the price of each visit)? What is his new equilibrium demand? After insurance Alfred will consume 9 visits, up to a price of $37.50; (0.4 * $37.50 = 15). Above $37.50, he will reduce quantity. For example, at P = $50, Alfred's cost is $20, so that he will make 8 visits. 2. Consider the market demand for labor LD = 1000 - 20W and the supply of labor Ls = -200 + 40W, where W is the market wage. (a) What are the equilibrium market wage and employment level? W = 20.00 and L = 600.00 (b) Calculate the equilibrium market wage and employment level if the workers negotiate a benefit worth $1 that costs the employers $2. Employers will consider the cost of the benefit in hiring decisions, so the demand curve will shift so that LD = 1000 - 20(W + 2) and the benefit will increase the real compen- sation for working so that labor supply will increase: Ls = -200 + 40(W + 1). These new curves give the new equilibrium of W = 18.67 and L = 586.67. (c) Calculate the equilibrium market wage and employment level if the workers negotiate a benefit worth $2 that costs the employers $1. The new labor demand is LD = 1000 - 20(W + 1) and the new labor supply is Ls = -200 + 40(W + 2). The new equilibrium is W = 17.33 and L = 613.33. 3. Suppose that Charlie's Pizzeria in Kalamazoo, Michigan, employs 10 workers at a wage level of $8 per person. All other costs (ovens, rent, advertising, return to capital) total $40 per hour, and the pizzeria sells 12 pizzas per hour at a cost of $10 per pizza. Suppose the state of Michigan mandates health coverage that can only be covered at a cost of $1 per hour, if it is offered at all. Charlie finds that if he offers insurance, he could maintain production 1/4 by letting one worker go and running his pizza overs a little hotter, leading to costs of $45 per hour. (a) What are Charlie's original profits? Original profits = Revenues - labor costs - non-labor costs = 12 . 10 - 8 . 10 - 40 = 0. (b) What is Charlie's elasticity of demand for labor? How is this calculated? Because we can't calculate derivatives here, we'll need to calculate an arc elasticity, which is an average across the two levels of labor inputs and wages. This arc elasticity of demand for labor is = / = Compiz = =492 =-8.5/9.5 = -0.895. (c) What will happen to Charlie's profits in the short run if he chooses to pay for mandated insurance? New profits = 120 - (8 + 1) . 9 workers - 45 = -6. (d) What will Charlies' long-run decision be? Why? Charlie's long run decision would be not to offer the insurance. If he does he cannot make enough profits to stay in business. This is assuming that wages do not adjust from an increase in the supply of labor (i.e., real compensation is higher, so more people will want to work for this firm).2. The Trade Theorems The construction of the production possibility curve shown in Figures 13.2 and 13.3 can be used to illustrate three important "theorems" in international trade theory. To get started, notice in Figure 13.2 that the efficiency line 0,,0, is bowed above the main diagonal of the Edgeworth Box. This shows that the production of good x is always "capital intensive" relative to the production of good y. That is, when production is efficient. () > no matter how much of the goods are produced. Demonstration of the trade theorems assumes that the price ratio, p = p,/p, is determined in international markets - the domestic economy must adjust to this ratio (in trade jargon the country under examination is assumed to be "a small country in a large world"). a. Factor Price Equalization Theorem: Use Figure 13.4 to show how the international price ratio, p, determines the point in the Edgeworth Box at which domestic production will take place. Show how this determines the factor price ratio, w/v. If production functions are the same throughout the world, what will this imply about relative factor prices throughout the world? b. Stolper Samuelson Theorem: An increase in p will cause the production to move clockwise along the production possibility frontier - x production will increase and y production will fall. Use the Edgeworth Box diagram to show that such a move will decrease k// in the production of both goods. Explain why this will cause w/v to fall. What are the implications of this for the opening of trade relations (which typically increases the price of the good produced intensively with a country's most abundant input). c. Rybczynski Theorem: Suppose again that p is set by external markets and does not change. Show that an increase in & will increase the output of x (the capital-intensive good) and reduce the output of y (the labor-intensive good). 3. Exercises with the General Equilibrium Simulation Here are three exercises to complete using the general equilibrium model described in the Nicholson/Westhoff paper. Exercise 1 One important phase of any analysis using a mathematical model is called "calibration". In this phase the parameters of the model are adjusted so that the model's results more closely approximate data from the actual economy. Although it would probably be impossible to calibrate the N/W model in any realistic way, it should be possible to explore how the choices of parameters in the model matter. For that purpose you are to take the model specified in illustration 1 (without taxes) and describe how you would change one of its basic parameters in an interesting way. You are then to make this change and describe your results. Exercise 2 Person 1 is the rich person in the base case simulations because he/she owns 80 percent of the capital. Suppose a government wishes to institute a tax on wages accompanied by a redistribution of the proceeds to person 2 with the end goal of equalizing the utility of these two people: 1. Is it possible to do this? Explain 2. Run a few more simulations to describe the "equity-efficiency tradeoff" in this model. Exercise 3 Use your own imagination to extend one of the first three "illustrations" in the N/W paper in some interesting way