(a) By how much does output eventually rise relative to what it would have been without the deficit reduction?

(b) By how much does consumption rise relative to what it would have been without the deficit reduction?

(c) What is the inmiediate effect of the deficit reduction on consumption? About how long does it take for consumption to return to what it would have been without the deficit reduction?

id) Compare your results with the results of Problem 1.6.

3.15. Consider the following variant of our model with physical and human capital: y(f) = [(l-ajf)lf(rr[(l-afl)H(r)]-", 0<a<l, 0< <1, 0 < < L kit) = sY(t) - 8 (1),

Hit) = BlaKK(t)V\aHH{t)nA{t)L{t)V--4 - 8 , > 0, > , + <1, Ut) = ), A(t) = gA{t),

where and are the fractions of the stocks of physical and human capital used in the education sector.

This model assumes that human capital is produced in its own sector with its own production function. Bodies (L) are useful only as something to be educated, not as an input into the production of final goods. Similarly, knowledge (A) is useful only as something that can be conveyed to students, not as a direct input into goods production.

(a) Define = KIAL and h = H/AL. Derive equafions for and h.

{b) Find an equation describing the set of combinations of h and such that = 0. Sketch in {h, k) space. Do the same for h = 0.

(c) Does this economy have a balanced growth path? If so, is it unique? Is it stable? What are the growth rates of output per person, physical capital per person, and human capital per person on the balanced growth path?

(d) Suppose the economy is initially on a balanced growth path, and that there is a permanent increase in s. How does this change affect the path of output per person over time?

3.16. Use the production fimcfion, (3.43), and equations (3.55) and (3.56) to derive the expressions in equafions (3.61) and (3.62) for the marginal products of physical and human capital on the balanced growth path of the model of Part of this chapter.

3.17. Constant returns to physical and human capital together. Suppose the production fimction is Y{t) = K{t)"H{t)-" (0 < a < 1), and that and H evolve according to k{t) = Sk Y{t), H{t) = si, Y{t).

(a) Show that regardless of the initial levels of and H (as long as both are positive), the ratio / converges to some balanced-growth-path level, (K/H)*.

ib) Once K/H has converged to (K/H)*, what are the growth rates of K, H, and Y7

(c) How, if at all, does the growth rate of Y on the balanced growth path depend on Sk and sh?

id) Suppose / starts off at a level that is smaller than (K/H)*.Is the Initial growth rate of Y greater than, less than, or equal to its growth rate on the balanced growth path?

3.18. Increasing returns in a model with human capital. (This follows Lucas, 1988.) Suppose that Y(t) = K(tr[{l - ) {1)] , H{t) = ), and K{t) = sY{t), and assume « + /3 > 1 .

(a) What is the growth rate of ?

(b) Does the economy converge to a balanced growth path? If so, what are the growth rates of and Y on the balanced growth path?

3.19. The speed of convergence in the model of -t of this chapter.

(a) Use the production function, (3.48), and the equations of motion for and h, (3.49) and (3.50), to find an expression for d [Iny(t)]/dt.

(b) Take a first-order Taylor approximation of the expression in part (a) in In and Inh around Ink = Ink*, Inh = Inh*.

(c) Usmg the expressions for Ink* and Inh* in equations (3.55) and (3.56), show that the expression you obtained in part (b) can be simplified to yield (3.65) in the text.

Lucass model differs from this formulation by letting Qh and s be endogenous and po--entially time-varying, and by assuming that the social and private returns to human capital differ.

Chapter 4

REAL-BUSINESS-CYCLE THEORY

4.1 Introduction: Some Facts about Economic Fluctuations

Modern economies undergo significant short-run variations in aggregate output and employment. At some times, output and employment are falling and unemployment is rising; at others, output and employment are rising rapidly and unemployment is falling. Consider, for example, the United States in the early 1980s. Between the third quarter of 1981 and the third quarter of 1982, real GDP fell by 2.8%, the fraction of the adult population employed fell by 1.3 percentage points, and the unemployment rate rose from 7.3% to 9.9%. Then over the next two years, real GDP grew by 11.0%, the fraction of the adult population employed rose by 1.9 percentage points, and the unemployment rate fell back to 7.3%.

Understanding the causes of aggregate fluctuations is a central goal of macroeconomics. This chapter and the two that follow present the leading theories concerning the sources and nature of macroeconomic fluctuations. Before turning to the theories, this section presents a brief overview of some major facts about short-run fluctuations. For concreteness, and because of the central role of the U.S. experience in shaping macroeconomic thought, the focus is on the United States.

A first important fact about fluctuations is that they do not exhibit any simple regular or cyclical pattem. Figure 4.1 plots seasonally adjusted real GDP quarterly since 1947, and Table 4.1 summarizes the behavior of real GDP in the nine postwar recessions. The figure and table show that output declines vary considerably in size and spacing. The falls in real GDP range from 0.8% in 1960 to 4.1% in 1973-1975; the times between the end of one recession and the beginning of the next range from

The formal datmg of recessions for the tJnited States is not based solely on the behavior of real GDP. Instead, recessions are identified judgmentaUy by the National Bureau of Economic Research (NBER) on the basis of various indicators. For that reason, the dates of the official NBER peaks and troughs differ slightly from the dates shown in Table 4.1. Burns and Mitchell (1944) and Moore and Zarnowltz (1986) describe the modern NBER methodology. C. Romer (1994) describes the NBER methodology for the pre-World War II era.