business quiz

Exercises 7 for half our current forests to be cleared? (Use annual compounding and solve for the fewest number of whole years.) Exercise 1.8 (Moderate) World population was about 679 million in the year 1700 and about 954 million in 1800. 1. What was the annual growth rate of population between 1700 and 1800? (Use con- tinuous compounding.) 2. Suppose that the human race began with Adam and Eve and that the annual growth rate between 1700 and 1800 prevailed in all years prior to 1700. About when must it have been that Adam and Eve were evicted from the Garden of Eden? (Hint: What was the population in that year?) Exercise 1.9 (Moderate) According to figures compiled by the World Bank, per capita real income in the U.S. was $15,400 in 1984, while the corresponding figure for Japan was $10,600. Between 1965 and 1984, per capita real income in the U.S. grew at an annual rate of 1.7 percent (using annual compounding), while the corresponding figure for Japan was 4.7 percent 1. If these two growth rates remain constant at their 1965-84 levels, in what year will per capita real income be the same in these two countries? (Again, use annual com- pounding, and use hundredths of a year.) 2. What will be the common per capita real income of these two countries at that date?Exercises Exercise 3.1 (Hard) Consider the two-period model from Section 3.2, and suppose the period utility is: "(c) = . Exercises 31 Variable Definition Overall utility t Time Consumption at period t (.) Period utility B Household's discount factor Household's income in period t, in units of con- sumption P Cost of a unit of consumption R Nominal interest rate Number of dollars of bonds bought at period t Lagrange multiplier in period N Number of households Table 3.1: Notation for Chapter 3 1. Determine the Euler equation in this case. 2. Determine the representative household's optimal choices: ci, c, and &. 3. Determine the equilibrium interest rate R*. 4. Determine the effect on the equilibrium interest rate R* of a permanent negative shock to the income of the representative household. (Le., both y, and y2 go down by an equal amount.) How does this relate to the case in which u(c,) = In(c, )? Exercise 3.2 (Easy) Refer to equation (3.12), which gives the equilibrium interest rate R* in the two-period model. 1. Suppose the representative household becomes more impatient. Determine the di- rection of the change in the equilibrium interest rate. (Patience is measured by B. You should use calculus.) 2. Suppose the representative household gets a temporary negative shock to its period-1 income y1. Determine the direction of the change in the equilibrium interest rate. Again, use calculus.) Exercise 3.3 (Moderate) Maxine lives for two periods. Each period, she receives an endowment of consumption goods: en in the first, ez in the second. She doesn't have to work for this output. Her pref- erences for consumption in the two periods are given by: u(q, (2) = In(c) + # In(c2), whereExercises Exercise 3.1 (Hard) Consider the two-period model from Section 3.2, and suppose the period utility is: "(c) = . Exercises 31 Variable Definition Overall utility t Time Consumption at period t (.) Period utility B Household's discount factor Household's income in period t, in units of con- sumption P Cost of a unit of consumption R Nominal interest rate Number of dollars of bonds bought at period t Lagrange multiplier in period N Number of households Table 3.1: Notation for Chapter 3 1. Determine the Euler equation in this case. 2. Determine the representative household's optimal choices: ci, c, and &. 3. Determine the equilibrium interest rate R*. 4. Determine the effect on the equilibrium interest rate R* of a permanent negative shock to the income of the representative household. (Le., both y, and y2 go down by an equal amount.) How does this relate to the case in which u(c,) = In(c, )? Exercise 3.2 (Easy) Refer to equation (3.12), which gives the equilibrium interest rate R* in the two-period model. 1. Suppose the representative household becomes more impatient. Determine the di- rection of the change in the equilibrium interest rate. (Patience is measured by B. You should use calculus.) 2. Suppose the representative household gets a temporary negative shock to its period-1 income y1. Determine the direction of the change in the equilibrium interest rate. Again, use calculus.) Exercise 3.3 (Moderate) Maxine lives for two periods. Each period, she receives an endowment of consumption goods: en in the first, ez in the second. She doesn't have to work for this output. Her pref- erences for consumption in the two periods are given by: u(q, (2) = In(c) + # In(c2), where32 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.Exercises Exercise 6.1 (Hard) This economy contains 1,100 households. Of these, 400 own type-a farms, and the other 700 own type-b farms. We use superscripts to denote which type of farm. A household of type j E {a, b} demands (i.e., it hires) /', units of labor, measured in hours. (The "d'" is for Exercises 53 Variable Definition Overall utility Time Consumption at period t Labor at period t u(.) Period utility B Household's discount factor Household's income in period t, in units of con- sumption f(4.) Production function P Cost of a unit of consumption R Nominal interest rate Number of dollars of bonds bought at period Lagrange multiplier in period t Lagrangean N Number of households D() Helper function, to simplify notation Table 6.2: Notation for Section 6.2 demand.) The type-/ household supplies ? of labor. (The ":" is for supply.) The household need not use its own labor on its own farm. It can hire other laborers and can supply its own labor for work on other farms. The wage per hour of work in this economy is w. This is expressed in consumption units, i.e., households can eat w. Every household takes the wage was given. Preferences are: u(ed, 1!) = In(c') + In(24 -14). where of is the household's consumption. Production on type-a farms is given by: 1 = (17)0.5 and that on type-b farms is: 1/ = 2(16)0.5 We are going to solve for the wage that clears the market. In order to do that, we need to determine demand and supply of labor as a function of the wage. If an owner of a type-a farm hires /" hours of labor at wage w per hour, the farm owner will make profit: 1" = (17)05 -w/9.1. Use calculus to solve for a type-a farmer's profit-maximizing choice of labor /; to hire as a function of the wage w. Call this amount of labor /,". It will be a function of w. Calculate the profit of a type-a farmer as a function of w. Call this profit , "* 2. If an owner of a type-b farm hires /; hours of labor at wage w per hour, the farm owner will make profit: 16 = 2(15)0.5 -w/9. Repeat part 1 but for type-b farmers. Call a type-b farmer's profit-maximizing choice of labor ?" . Calculate the profit of a type-b farmer as a function of w. Call this profit 3. If a type-a farmer works/", then that farmer's income will be: "* + u". Accordingly, the budget constraint for type-a farmers is: (" = x* + w/". A type-a household chooses its labor supply by maximizing its utility subject to its budget. Determine a type-a household's optimal choice of labor to supply for any given wage w. Call this amount of labor /"*. 4. Repeat part 3 but for type-b households. Call this amount of labor ?". 5. Aggregate labor demand is just the sum of the demands of all the farm owners. Cal- culate aggregate demand by adding up the labor demands of the 400 type-a farmers and the 700 type-b farmers. This will be an expression for hours of labor / in terms of the market wage w. Call the result - 6. Aggregate labor supply is just the sum of the supplies of all the households. Calculate aggregate supply, and call it ?;. 7. Use your results from parts 5 and 6 to solve for the equilibrium wage w*. (Set the two expressions equal and solve for w.) Exercise 6.2 (Hard) Consider an economy with many identical households. Each household owns a business that employs both capital (machinery) & and labor /, to produce output y. (The "d" is for demand.) Production possibilities are represented by y = Ak"(1:)". The stock of capital that each household owns is fixed. It may employ labor at the prevailing wage w per unit of labor . Each household takes the wage as given. The profit of each household from running its business is: (6.9) T =y - wid = Aku (la) " - wid. 1. Determine the optimal amount of labor for each household to hire as a function of its capital endowment & and the prevailing wage w. Call this amount of labor /9- Exercises 55 2. Plug /; back into equation (6.9) to get the maximized profit of the household. Call this profit .*. 3. Each household has preferences over its consumption c and labor supply /s. These preferences are represented by the utility function: u(c, /) = ci(1 -1,);. Each house- hold has an endowment of labor can be used in the household's own business or rented to others at the wage w. If the household supplies labor /s, then it will earn labor income wis. Output, wages, and profit are all quoted in terms of real goods, so they can be consumed directly. Set up the household's problem for choosing its labor supply /s. Write it in the following form: max {objective} subject to: constraints choices 4. Carry out the maximization from part 3 to derive the optimal labor supply (. 5. Determine the equilibrium wage w in this economy. 6. How does the equilibrium wage w change with the amount of capital & owned by each household? 7. What does this model imply about the wage differences between the U.S. and Mex- ico? What about immigration between the two countries?Exercises Exercise 8.1 (Easy) Consider an economy where velocity V equals 5, output grows at three percent a year, and money supply grows at five percent a year. What is the annual inflation rate? Exercise 8.2 (Hard) In the quantity theory, we assumed that velocity was constant. In reality, the velocity of money varies across countries. Would you expect countries with high inflation to have higher or lower velocity than low-inflation countries? Justify your answer. (Hint: You should draw both on Chapter 4 and Chapter 8 to answer this question.)