ECO456

32 The Behavior of Households with Markets for Commodities and Credit " and my are her consumptions in periods 1 and 2, respectively, and # is some discount factor between zero and one. She is able to save some of her endowment in period 1 for consumption in period 2. Call the amount she saves s. Maxine's savings get invaded by rats, so if she saves a units of consumption in period 1, she will have only (1 -6)s units of consumption saved in period 2, where & is some number between zero and one. 1. Write down Maxine's maximization problem. (You should show her choice variables, her objective, and her constraints.) 2. Solve Maxine's maximization problem. (This will give you her choices for given val- ues of e1, ez, 8, and 6.) 3. How do Maxine's choices change if she finds a way reduce the damage done by the rats? (You should use calculus to do comparative statics for changes in f.) Exercise 3.4 (Moderate) An agent lives for five periods and has an edible tree. The agent comes into the world at time t = 0, at which time the tree is of size zo. Let a be the agent's consumption at time t. If the agent eats the whole tree at time t, then c, = 2, and there will be nothing left to eat in subsequent periods. If the agent does not eat the whole tree, then the remainder grows at the simple growth rate a between periods. If at time t the agent saves 100s, percent of the tree for the future, then 241 = (1+ a )$2. All the agent cares about is consumption during the five periods. Specifically, the agent's preferences are: U = _ _, #' In(c,). The tree is the only resource available to the agent. Write out the agent's optimization problem.Borjas, Problem 4-4 This question asks how the wages of [presumably high-skilled] natives should change when there is a change in the quantity ofavailable lowskilled [immigrant] labor assuming that natives and immigrants are complements. Whenever two factors of production are complementary, an increase in one increases the returns to another. So in this case, the illegal alien hiring penalties raises the cost of unskilled labor, lowering the amount of unskilled labor that is hired which reduces the returns to skilled labor, lowering their wage. Borjas, Problem 4-6 Elasticity of demand for labor is 0.5. This implies that a 1% increase in wages reduces employment by 0.5% Alternatively [just ipping the fraction], an increase in employment by 1% reduces the wage offered by rms by 2%. [a] For now, treat each economy [North and South] as completely isolated and as separate countries and economies, essentially. Immigration has no effect on the North, so wages in the North don't change. Immigration in the South increases the population [and hence employment, since supply is perfectly inelastic] by 20,000, which represents a 5% growth in employment which [based on the above discussion] reduces wages by 10%, to $13.50. [b] Since wages fall by $1.50 in the South, 1,500 natives will migrate from the South to the North. This represents 0.25% employment growth in the North [1500(000000}, which reduces wages by 0.5%, to $14.93. Southern employment falls from its immediate postmigration level of 420,000 to 413,500 which represents a fall in employment of 0.36%, increasing wages by 0.? 2%, from $13.50 to around $13.60. The ratio of wages in the North to the South is $14.93J$13.0, or around 1.10. Wages are about 10% higher in the North after one year of migration. [c] [n the long run, people would migrate year after year [similar to b}, as long as wages are higher in one place than the other. Eventually [in the long run], wages and employment would equate. You could solve for the nal wage level by recognizing that employment would eventually be 510,000 in each area, and calculating what that would do to wages. 0r, you could recognize that once we are thinking longrun, there is essentially no difference between the North and South [given that the labor demand elasticity is the same in each], so you can think of the North and South as a single economy that be'ns with an inelasticallyemployed population of 1,000,000 and experiences employment growth of 20,000 which represents growth of 2%, reducing wages by 4% from $15.00 to $14.40. Problem 1: Learning by Doing with Spillovers Consider the model of learning by doing with spillovers [Arrow 3: Homer} presented in class and me that the production function is Cobb-Douglas, that is. Y?\" =(KF")"UI:LI\"}1'\" However. assume there are diminishing returns to technological progress, hf = 111:2\Money in the Utility Function (Final, 2005) Assume that consumer's utility in period / depends on consumption and leisure: U(Cr + V(L - aN, where L is the total time available for leisure and for trips to the banks,/, is the number of trips to the bank, and a is time spent on each trip, so L - aN, is leisure. If consumers decide to spend P, C, on consumption in period , they do this at a constant rate within the period. They need money to buy goods. They get this money by going to the bank to exchange bonds for money. They can take N, such trips. If N, = 1, they go to the bank once, at the start of the period, take M, = P,C, out, and spend it over the period. Their average money balances for the period are therefore equal to M = = = P. 1. Derive average money balances as a function of spending, P,C, and the number of trips taken, N,- 2. Replace N, by its expression in terms of average real money balances and consumption in the utility function. 3. Discuss: "Putting money in the utility function is just a short cut for capturing the idea that having larger real money balances saves on trips to the bank." 4. In light of this exercise, does it make sense to assume that utility is separable in consumption and in real money balances? 5. Would money be neutral/super-neutral in this setup? Seigniorage This question asks you to extend the discussion on "Money growth, inflation, and seigniorage" in lecture to include rational expectations. Consider the basic setup presented by Olivier in class. Money demand is given by: M = exp(-ane . We are going to use the equation in discrete time, where it takes the form (in logs): m-p = -a(EPmi - P. . 1. Show that with rational expectations, the current price level is a weighted average of expected next period prices and current money supply. 2. Replace forward to show that pr is a function of expected future money supply. 3. Assume that my - mr-1 = 00. Find the evolution of the price level. A strange model of money (Waiver, 2005) Take an economy with a continuum of individuals, indexed by i, and maximizing: Cite as: Olivier Blanchard, course materials for 14.452 Macroeconomic Theory II, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. subject to: PC; + M, = P,Y + My + Ti. Individuals take Y; as given. For the economy as a whole, output is exogenous, constant, perishable, and equal to Y. The total nominal stock is constant and equal to M. 1. Derive and interpret the first order conditions for individual i. 2. Using equilibrium conditions, derive the rate of inflation in the economy. Explain in words. 3. What is the rate of money growth? Why is the rate of inflation different from the rate of money growth? 4. Would a cash in advance give rise to different results? How would you introduce it? And what would the implications on the rate of inflation?Money as a Factor of Production (Based on Dornbusch and Frenkel, 1973) The shortcut used by Dornbusch and Frenkel to introduce money in the economy is that they assume that output available is equal to a fraction of production (G(K, 1 ), where the fraction is an increasing function of real money balances: (1-1(P) ) G(K, 1 L(. satisfies these properties: L(oo = 0, L(0 = 1, L'() 0. The households maximize: Max > BUCCHI FO S.1. PHICHi + MAHI + PARKHIM = PHI(1 -L( 7,) MIL ) G(K Hi, 1 + XHi+ Mui+ PHi(1 -6 KHi where C, M. K, and X are consumption, money balances, capital holdings, and government transfers. There is no uncertainty. 1. Show that the budget constraint can be written in real terms as: Cuit (1 + MAN Hit + KHI = (1-1( MM ) )G(KAI, 1 Xhi - Miti + (1 -6 Kris PHi PHi where 1 + Multi = Phi 2. Derive the FOCs of this problem. Characterize the solution to the problem using an intertemporal and an intratemporal condition. 3. Characterize the steady state. Is money neutral? Superneutral? 4. What are the basic differences of this approach with respect to including money in the utility function or a cash in advance constraint