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Macroeconomics 5th edition Stephen d. Williamson - Solutions

In the course of producing its output, a firm causes pollution. The government passes a law that requires the firm to stop polluting, and the firm discovers that it can prevent the pollution by hiring x workers for every worker that is producing output. That is, if the firm hires N workers, then xN
Suppose a firm has a production function given by Y = zK0.3 N0.7.(a) If z = 1 and K = 1, graph the production function. Is the marginal product of labor positive and diminishing?(b) Now, graph the production function when z = 1 and K = 1. Explain how the production function changed from part
Suppose that the production function zF(K, N) exhibits increasing returns to scale, to the extent that the marginal product of labor increases when the quantity of labor input increases.(a) Given this production function, what will be the representative firm's demand for labor?(b) What problems do
Supposing that a single consumer works for a firm, the quantity of labor input for the firm, N, is identical to the quantity of hours worked by the consumer, h – l. Graph the relationship between output produced, Y on the vertical axis and leisure hours of the consumer, l, on the horizontal axis,
Many negative externalities exist in cities. For example, a high concentration of automobile traffic in cities generates pollution and causes congestion, and both pollution and congestion are negative externalities. When a particular person decides to drive a car in a city on a given day, he or she
Suppose that the government decides to reduce taxes. In the model used in this chapter, determine the effects this has on aggregate output, consumption, employment, and the real wage, and explain your results.
Suppose that there is a natural disaster that destroys part of the nation’s capital stock.(a) Determine the effects on aggregate output, consumption, employment, and the real wage, with reference to income and substitution effects, and explain your results.(b) Do you think that changes in the
Suppose that total factor productivity, z, affects the productivity of government production just as it affects private production. That is, suppose that when the government collects taxes, it acquires goods that are then turned into government-produced goods according to G = zT so that z units of
Suppose that the representative consumer's preferences change, in that his or her marginal rate of substitution of leisure for consumption increases for any quantities of consumption and leisure.(a) Explain what this change in preferences means in more intuitive language.(b) What effects does this
Suppose that government spending makes private firms more productive; for example, government spending on roads and bridges lowers the cost of transportation. This means that there are now two effects of government spending, the first being the effects discussed in this chapter of an increase in G
In the one-period model, education can be represented as time spent by the representative consumer that is neither leisure time nor time applied to producing output. What the economy gains in the future is that the representative consumer then has more time available, as measured in terms of
In the simplified model with proportional taxation there can be two equilibria, one with a high tax rate and one with a low tax rate. Now, suppose that government spending increases. Determine the effects of an increase in G on consumption, leisure, labor supply, real output, and the tax rate in a
Suppose that the substitution effect of an increase in the real wage is always larger than the income effect for the representative consumer. Also assume that the economy is always in the low-tax-rate equilibrium on the good side of the Laffer curve. Determine the effects of an increase in total
Consider the model of public goods in the last section of this chapter.(a) Suppose that preferences over private consumption C and public goods G are such that these two goods are perfect substitutes, that is, the marginal rate of substitution of public goods for private goods is a constant b >
Extend the model of public goods, in the last section of this chapter, as follows. Suppose that output is produced, as in the simplified model with proportional taxation, only with labor, and that z = 1. Here, however, there is lump-sum taxation, and the PPF is given by Y = h – l – G. Now the
What does the DMP model predict would be the effects of labor-saving devices in the home, for example dishwashers, washing machines, and vacuum cleaners? Use diagrams to show the effects on the unemployment rate, the vacancy rate, the labor force, the number of firms, aggregate output, and labor
Suppose the government's goal is to reduce the unemployment rate. Some legislators propose that the government should give a subsidy s to any firm that hires a worker. Some other legislators argue that it would be more effective to simply pay consumers to stay home rather than searching for work,
Suppose that there is technological change that reduces the cost of recruiting for firms. Using the DMP model, determine the effects on the unemployment rate, the vacancy rate, the labor force, the number of firms, aggregate output, and labor market tightness. Use diagrams, and explain your results.
Adapt the DMP model to include government activity as follows. Suppose that the government can operate firms, subject to the same constraints as private firms. In particular, the government must incur a cost k to post a vacancy. Supposing that the government operates G firms, then the number of
Show that, in the Keynesian DMP model, if the wage is judged to be inefficiently high, so that unemployment is inefficiently high, the government can pay a subsidy to firms that corrects the problem. Explain your results. Does it matter whether the government subsidizes firms that post vacancies or
Suppose that all social programs simultaneously become more generous. In particular suppose that there is an increase in UI benefits, and also an increase in welfare benefits, which are represented in the DMP model as payments to everyone who is not in the labor force. What will be the effects on
In the Malthusian model, suppose that the quantity of land increases. Using diagrams determine what effects this has in the long-run steady state and explain your results.
In the Malthusian model, suppose that there is a technological advance that reduces death rates. Using diagrams determine the effects of this in the long-run steady state and explain your results.
In the Solow growth model, suppose that the marginal product of capital increases for each quantity of the capital input, given the labor input.(a) Show the effects of this on the aggregate production function.(b) Using a diagram, determine the effects on the quantity of capital per worker and on
Suppose that the economy is initially in a steady state and that some of the nation’s capital stock is destroyed because of a natural disaster or a war.(a) Determine the long-run effects of this on the quantity of capital per worker and on output per worker.(b) In the short run, does aggregate
Modify the Solow growth model by including government spending as follows. The government purchases G units of consumption goods in the current period, where G = gN and g is a positive constant. The government finances its purchases through lump-sum taxes on consumers, where T denotes total taxes,
Consider a numerical example using the Solow growth model. Suppose that F(K, N) = K0.5 N0.5, with d = 0.1, s = 0.2, n = 0.01, and z = 1, and take a period to be a year.(a) Determine capital per worker, income per capita, and consumption per capita in the steady state. (b) Now, suppose that the
Suppose that we modify the Solow growth model by allowing long-run technological progress. That is, suppose that z = 1 for convenience, and that there is labor-augmenting technological progress, with a production functionY = F(K, bN),where b denotes the number of units of “human capital” per
Alter the Solow growth model so that the production technology is given by Y = zK, where Y is output, K is capital, and z is total factor productivity. Thus, output is produced only with capital.(a) Show that it is possible for income per person to grow indefinitely.(b) Also show that an increase
Consider a numerical example. In the Solow model, assume that n = 0, s = 0.2, d = 0.1, and F(K,N) = K0.3 N0.7. Suppose that initially, in period t = 0, z = 1 and the economy is in a steady state.(a) Determine consumption, investment, savings, and aggregate output in the initial steady state.(b)
Consider the following data:(a) Calculate the Solow residual for each year from 1995 to 2007.(b) Calculate percentage rates of growth in output, capital, employment, and total factor productivity for the years 1996 to 2007. In each year, what contributes the most to growth in aggregate output? What
Suppose that the depreciation rate increases. In the Solow growth model, determine the effects of this on the quantity of capital per worker and on output per worker in the steady state. Explain the economic intuition behind your results.
If total factor productivity decreases, determine using diagrams how this affects the golden rule quantity of capital per worker and the golden rule savings rate. Explain your results.
Determine the effects of a decrease in the population growth rate on the golden rule quantity of capital per worker and on the golden rule savings rate. Explain your results.
Could differences across countries in population growth account for the persistence in income disparity across countries? Use the Solow growth model to address this question and discuss.
In the Solow growth model, suppose that the per-worker production function is given by y = zk.3, with s = 0.25, d = 0.1, and n = 0.02.(a) Suppose that in country A, z = 1. Calculate per capita income and capital per worker.(b) Suppose that in country B, z = 2. Calculate per capita income and
Suppose that there are two countries with different levels of total factor productivity, and that these differences exist because of barriers to technology adoption in the low-productivity country. Also suppose that these two countries do not trade with each other. Now, suppose that residents of
Suppose, in the Solow growth model, that learning by doing is captured as a cost of installing new capital. In particular, suppose that for each unit of investment, r units of goods are used up as a cost to firms.(a) Determine how r affects the steady state quantity of capita per worker, and per
Introduce government activity in the endogenous growth model as follows. In addition to working u units of time in producing goods, the representative consumer works v units of time for the government and produces gvH goods for government use in the current period, where g > 0. The consumer now
Reinterpret the endogenous growth model in this chapter as follows. Suppose that there are two groups of people in a country, the low-skilled workers and the high-skilled workers. The low-skilled workers have less human capital per person initially than do the high-skilled workers. In the economy
Suppose there are two countries. In the rich country, the representative consumer has Hr units of human capital, and total factor productivity is zr. In the poor country, the representative consumer has Hp units of human capital, and total factor productivity is zp. Assume that b and u are the same
In the endogenous growth model, suppose that there are three possible uses of time. Let u denote the fraction of time spent working, s the fraction of time spent neither working nor accumulating human capital (call this unemployment), and 1 – u – s the fraction of time spent accumulating human
Suppose that z, the marginal product of efficiency units of labor, increases in the endogenous growth model. What effects does this have on the rates of growth and the levels of human capital, consumption, and output? Explain your results.
Suppose that the government makes a onetime investment in new public school buildings, which results in a one-time reduction in consumption. The new public school buildings increase the efficiency with which human capital is accumulated. Determine the effects of this on the paths of aggregate
A consumer’s income in the current period is y = 100, and income in the future period is y' = 120. He or she pays lump-sum taxes t = 20 in the current period and toe = 10 in the future period. The real interest rate is 0.1, or 10%, per period.(a) Determine the consumer’s lifetime wealth.(b)
An employer offers his or her employee the option of shifting x units of income from next year to this year. That is, the option is to reduce income next year by x units and increase income this year by x units.(a) Would the employee take this option (use a diagram)?(b) Determine, using a diagram,
Consider the following effects of an increase in taxes for a consumer.(a) The consumer’s taxes increase by ∆t in the current period. How does this affect current consumption, future consumption, and current saving?(b) The consumer’s taxes increase permanently, increasing by ∆t in the
Suppose that the government introduces a tax on interest earnings. That is, borrowers face a real interest rate of r before and after the tax is introduced, but lenders receive an interest rate of (1 – x)r on their savings, where x is the tax rate. Therefore, we are looking at the effects of
A consumer receives income y in the current period, income y' in the future period, and pays taxes of t and t' in the current and future periods, respectively. The consumer can borrow and lend at the real interest rate r. This consumer faces a constraint on how much he or she can borrow, much like
A consumer receives income y in the current period, income y' in the future period, and pays taxes of t and t' in the current and future periods, respectively. The consumer can lend at the real interest rate r. The consumer is given two options. First, he or she can borrow at the interest rate r
Suppose that all consumers are identical, and also assume that the real interest rate r is fixed. Suppose that the government wants to collect a given amount of tax revenue R, in present value terms. Assume that the government has two options: (i) a proportional tax of s per unit of savings, in
Assume a consumer who has current-period income y = 200, future-period income y' = 150, current and future taxes t = 40 and toe = 50, respectively, and faces a market real interest rate of r = 0.05, or 5% per period. The consumer would like to consume equal amounts in both periods; that is, he or
Assume an economy with 1,000 consumers. Each consumer has income in the current period of 50 units and future income of 60 units and pays a lump-sum tax of 10 units in the current period and 20 units in the future period. The market real interest rate is 8%. Of the 1,000 consumers, 500 consume 60
In the example laid out in the subsection titled “Ricardian Equivalance: An Example,” suppose thatb < Ny' – G' / Ny – G.(a) Solve for the equilibrium real interest rate, and the consumption of lenders and borrowers in the current and future periods.(b) Does Ricardian equivalence hold in
Suppose that a consumer has income y in the current period, income y' in the future period, and faces proportional taxes on consumption in the current and future periods. There are no lump-sum taxes. That is, if consumption is c in the current period and c' in the future period, the consumer pays a
Suppose in our two-period model of the economy that the government, instead of borrowing in the current period, runs a government loan program. That is, loans are made to consumers at the market real interest rate r, with the aggregate quantity of loans made in the current period denoted by L.
Suppose that there is a credit market imperfection due to asymmetric information. In the economy, a fraction b of consumers consists of lenders, who each receive an endowment of y units of the consumption good in the current period, and 0 units in the future period. A fraction (1 – b)a consumers
Suppose there is a credit market imperfection due to limited commitment. As in the setup with collateralizable wealth we examined in this chapter, each consumer has a component of wealth which has value pH in the future period, cannot be sold in the current period, and can be pledged as collateral
Suppose that there is limited commitment in the credit market, but lenders are uncertain about the value of collateral. Each consumer has a quantity of collateral H, but from the point of view of lender, there is a probability a that the collateral will be worth p in the future period, and
Suppose a credit market with a good borrowers and 1-a bad borrowers. The good borrowers are all identical, and always repay their loans. Bad borrowers never repay their loans. Banks issue deposits that pay a real interest rate r1, and make loans to borrowers. Banks cannot tell the difference
In the example (“Limited Commitment and Market Interest Rates”), suppose that t < (v – y')y/y'. Also suppose that t falls. What effect will this have on the market real interest rate, and on consumption? Explain.
Use the social security model developed in this chapter to answer this question. Suppose that the government establishes a social security program in period T, which provides a social security benefit of b (in terms of consumption goods) for each old person forever. In period T the government
Why is a good conductor of electricity also a good conductor of heat?
Use the social security model developed in this chapter to answer this question. Suppose that a government pay-as-you-go social security system has been in place for a long time, providing a social security payment to each old person of b units of consumption. Now, in period T, suppose that the
In the example (“Limited Commitment and Market Interest Rates”), suppose that v < y' and y'/y < a < b.(a) Suppose that t = t’ = 0. Determine the equilibrium real interest rate, and the equilibrium quantities of current and future consumption for lenders and borrowers. (b) Determine an
What is the effect of an increase in d, the depreciation rate, on the representative firm’s investment decision, and on its optimal investment schedule? Explain your results carefully.
Tom lives on an island and has 20 coconut trees in the current period, which currently produce 180 coconuts. Tom detests coconuts, but he can trade them with people on other neighboring islands for things that he wants. Further, Tom can borrow and lend coconuts with neighboring islands. In the
The government wishes to bring about an increase in investment expenditures, and is considering two tax policies that policymakers think could bring this about. Under the first tax policy, firms would receive a subsidy in the current period of t per unit of current output produced. Policymakers
Suppose that we modify the model of the firm's investment behavior by assuming that any capital the firm has remaining at the end of the period can be sold at the price p'K (in our model we assumed the capital could be sold at a price of one, in terms of consumption goods).(a) Determine how this
Determine how the following affects the slope of the output demand curve, and explain your results:(a) The marginal propensity to consume increases.(b) The intertemporal substitution effect of the real interest rate on current consumption increases.(c) The demand for investment goods becomes less
The government decreases current taxes, while holding government spending in the present and the future constant.(a) Using diagrams, determine the equilibrium effects on consumption, investment, the real interest rate, aggregate output, employment, and the real wage. What is the multiplier, and how
Determine how the following affect the slope of the output supply curve, and explain your results:(a) The marginal product of labor decreases at a faster rate as the quantity of labor used in production increases.(b) The intertemporal substitution effect of the real interest rate on current leisure
Suppose that there is a shift in the representative consumer’s preferences. Namely, the consumer prefers, given the market real interest rate, to consume less current leisure and more current consumption goods.(a) Determine the effects of this on current aggregate output, current employment, the
Suppose that there is a permanent increase in total factor productivity. Determine the implications of this for current macroeconomic variables, and show how the impact differs from the case where total factor productivity is expected to increase only temporarily. Explain your results.
Suppose that z' increases and that K increases at the same time. Show that it is possible for the real interest rate to remain constant as a result. What does this say about the model’s ability to explain the differences between poor and rich countries and to explain what happens as a country’s
There is a temporary increase in the relative price of energy. Determine how the response of current aggregate output to this shock depends on the marginal propensity to consume, and explain carefully why you get this result.
Suppose that a country experiences destruction of part of its capital stock. Suppose also that the capital stock plays a role as collateral in credit contracts, so that the destruction of capital increases credit market frictions.(a) Determine how the net effects on macroeconomic variables differ
A war breaks out that is widely expected to last only one year. Show how the effect of this shock on aggregate output depends on the size of the intertemporal substitution effect of the real interest rate on current leisure, and carefully explain your results.
A macroeconomist suggests that, since aggregate output and employment have decreased, the government should increase expenditures on goods and services to increase both output and employment. Suppose that output and employment fell because of a sectoral shock.(a) Determine, using diagrams, what the
In the monetary intertemporal model, show that it is possible to have an equilibrium where money is not held and only credit cards are used in transactions. Is there such a thing as a price level in this equilibrium? Does monetary policy work? If so, how? Explain your results and what they mean for
The government decides that the use of credit cards is bad, and introduces a tax on credit card balances. That is, if a consumer or firm holds a credit card balance of X (in real terms), he or she is taxed tX, where t is the tax rate. Determine the effects on the equilibrium price and quantity of
Suppose that the nominal interest rate is zero, that is, R = 0.(a) What is the equilibrium quantity of credit card balances?(b) In what sense does the economy run more efficiently with R = 0 than with R > 0?(c) Explain your results in parts (a) and (b). Discuss the realism of these predictions.
In the monetary intertemporal model, suppose that the money supply is fixed for all time. Determine the effects of a decrease in the capital stock, brought about by a war or natural disaster, on current equilibrium output, employment, the real wage, the real interest rate, the nominal interest
A new technological innovation comes on line. What are the current effects on aggregate output, consumption, investment, employment, the real wage, the real interest rate, the nominal interest rate, and the price level? Explain your results. Also explain how they fit the key business cycle facts in
Suppose, in the Friedman-Lucas money surprise model, that there are money demand shocks and shocks to total factor productivity. Neither private sector economic agents nor the central bank can observe money demand shocks directly. Private sector economic agents cannot observe productivity shocks.
Suppose that we allow for the fact that cash can be stolen, but assume that a stolen credit card cannot be used, as it is instantly cancelled, so no one steals credit cards. Determine the effects this has on the quantity and price of credit card balances, the demand for money, and the price level.
In the Friedman-Lucas money surprise model, suppose that the central bank wants to reduce the price level. Suppose the central bank has two options: (i) announce in advance that the money supply will decrease; (ii) surprise the public with a decrease in the money supply.Which option is preferable?
Assume that there are no surprises, with all economic agents and the central bank having full information about shocks that are hitting the economy. Suppose that the central bank adopts a nominal GDP target, and interpret this in the model as a goal of maintaining some constant level of nominal
In the real business cycle model, suppose that government spending increases temporarily. Determine the equilibrium effects of this. Could business cycles be explained by fluctuations in G? That is, does the model replicate the key business cycle facts from Chapter when subjected to temporary
Suppose that temporary increases in government spending lead to permanent increases in total factor productivity, perhaps because some government spending improves infrastructure and makes private firms more productive. Show that temporary shocks to government spending of this type could lead to
In the real business cycle model, suppose that firms become infected with optimism and they expect that total factor productivity will be much higher in the future.(a) Determine the equilibrium effects of this.(b) If waves of optimism and pessimism of this sort cause GDP to fluctuate, does the
Suppose that money plays the role of a sunspot variable in the coordination failure model, so that the economy is in the bad equilibrium when the money supply is low and in the good equilibrium when the money supply is high. Explain what the monetary authority could do to make consumers better off.
In the coordination failure model, suppose that consumers’ preferences shift so that they want to consume less leisure and more consumption goods. Determine the effects on aggregate variables in the good equilibrium and in the bad equilibrium, and explain your results.
Suppose that there is a natural disaster that destroys some of the nation's capital stock. The central bank's goal is to stabilize the price level. Given this goal, what should the central bank do in response to the natural disaster? Explain with the aid of diagrams.
Suppose that the central bank observes a drop in real GDP, but does not know what caused this drop.(a) How would the central bank respond if it believed that GDP dropped because of a decline in total factor productivity, and that real business cycle theory is correct?(b) How would the central bank
Suppose, in the New Monetarist model, that there is deficient financial liquidity. If the fiscal authority were to engineer a tax cut, financed by an increase in the quantity of government debt, with the quantity of outside money held constant, what happens? What does this say about Ricardian
In the New Monetarist model, suppose that the central bank conducted a "quantitative easing" program by issuing outside money and exchanging it for privately produced liquid financial assets. What would the macroeconomic effects be? Does it matter if there is a liquidity trap where excess reserves
Repeat question (8) for the liquidity trap case where interest is paid on reserves and there are excess reserves held in the financial system. Explain your results and discuss.
Suppose government spending increases temporarily in the New Keynesian model.(a) What are the effects on real output, consumption, investment, the price level, employment, and the real wage?(b) Are these effects consistent with the key business cycle facts from Chapter? What does this say about the
In the New Keynesian model, suppose that supply is initially equal to demand in the goods market and that there is a negative shock to the demand for investment goods, because firms anticipate lower total factor productivity in the future.(a) Determine the effects on real output, the real interest

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