高级宏观经济学Chapter4.doc

Chapter 4 REAL-BUSINESS-CYCLE THEORYl Introduction: Some Facts and Theories of Fluctuationsl A Baseline Real-Business-Cycle Modell Empirical Applicationsl Extension and LimitationsIntroduction: Some Facts and Theories of Fluctuationsa. Some Facts about Economic Fluctuations b. Theories of FluctuationsA Baseline Real-Business-Cycle ModelThe model is a discrete-time variation of the Ramsey model of Chapter 2.The economy consists of a large number of identical, price-taking firms and a large number of identical, price-taking households. Households are infinitely lived. the production function Labor and capital are paid their marginal products.The real wage: The real interest rate: the utility function, , the behavior of the two driving variables, technology and government purchasesthe effects of the shocks: the random component of follows AR(1)., where the s are white-noise disturbances. If is positive, this means that the effects of a shock to technology disappear gradually over time., where the s are white-noise disturbances that are uncorrelated with the .Household BehaviorThe two most important differences between this model and the Ramsey model are the inclusion of leisure in the utility function and the introduction of randomness in technology and government purchases. Intertemporal Substitution in Labor Supply(Consider the case where the household lives for one period and has no initial wealth, only one member.The Lagrangian is (4.15)Labor supply is independent of the wage. The income and substitution effects of a change in the wage offset each other.Consider the case where the households horizon is more than one period, e.g. for two periods. In addition, assume that there is no uncertainty about the interest rate or the second-period wage.The households lifetime budget constraint is The Lagrangian isThe first-order conditions for l1 and l2 are needed to show the effect of the relative wage in the two periods on relative labor supply.(4.21) These responses of labor supply to the relative wage and the interest rate are known as intertemporal substitution in labor supply.If w1 rises relative to w2, the household increases first-period labor supply relative to second-period supply. A rise in r raises first-period labor supply relative to second-period supply. A rise in r increases the attractiveness of working today and saving relative to working tomorrow. This effect of the interest rate on labor supply is crucial to employment fluctuations in real-business-cycle models. Household Optimization under UncertaintyHousehold faces uncertainty about rates of return and future wages. Because of this uncertainty, the household does not choose deterministic paths for consumption and labor supply. Instead, its choices of c and l at any date potentially depend on all the shocks to technology and government purchases up to that date.IF the household is behaving optimally, a marginal change of a small amount in period t must leave expected utility unchanged. Equating the utility cost and expected benefits impliesThis simplifies to (4.23) The Tradeoff between Consumption and Labor Supply(4.26) A Special Case of the ModelSimplifying AssumptionsWe eliminate government, and we assume 100 percent depreciation each period.(4.27) (4.28) The solution to the model focuses on two variables, labor supply per person, l, and the fraction of output that is saved, s. We will focus on the tow conditions for household optimization, (4.23) and (4.26).We will find that s and labor supply is independent of technology and the capital stock. Consider (4.23) first. (4.31) Technology (A) and Capital (K) do not enter (4.31). Thus there is a constant value of s that satisfies this condition. (4.32) Consider (4.26). (4.36) Thus labor supply is constant. An improvement in technology raises current wages relative to expected future wages, and thus acts to raise labor supply. But, by raising the amount saved, it also lowers the expected interest rate, which acts to reduce labor supply.DiscussionThis model provides an example of an economy where real shocks drive output movements. Because the economy is Walrasian, the movements are the optimal responses to the shocks. Observed aggregate output movements represent the time-varying Pareto optimum. The specific form of the output fluctuations is determined by the dynamics of technology and the behavior of the capital stock. (4.40) where is the difference between and the value it would take if equaled each period.(departures of log output from its normal path)can be written as a linear combination of its two previous values plus a white-noise disturbance. AR(2) (4.42) (4.42) can cause output to have “hump-shaped” response to disturbances.This special case of the model does not do a good job of matching major features of fluctuations. l The saving rate is constant and labor input does not vary? Investment varies much more than consumption, and employment and hours are strongly procyclical.l The real wage is highly procyclical, ? In actual fluctuations, in contrast, the real wage appears to be only moderately procyclical.Introducing depreciation of less than 100 percent and shocks to government purchases improves the models predictions concerning movements in employment, saving, and the real wage.Solving the Model in the General CaseOnce model is presented, parameter values are chosen, and the models quantitative implications fro the variances and correlations of various macroeconomic variables are discussed.Log-Linearizing the Model around the Balanced Growth Path(Campbell, 1994) If we log-linearize the model around the nonstochastic balanced growth path, the rules for consumption and employment (that are endogenous) must take the form(4.43) (4.44) Consumption and employment are equal to their balanced-growth-path value when K, A, and G are all equal to theirs.For a set of as to be a solution to the model, they must imply that households are satisfying optimization conditions, (4.23) and (4.26).The intratemporal First-Order ConditionConsider (4.26).(4.45) Log-linearizing (4.45) around the balanced growth path yieldswhere l* is the value of l on the balanced growth path. Thus the as must satisfy(4.48) l Consider the extreme case of no depreciation and no growth, so that investment is zero in the absence of shocks. A positive technology shock, by raising the marginal product of capital in the next period, makes it optimal for households to undertake some investment.l The saving rate rises. The fact that saving is temporarily high means that expected consumption growth is higher than it would be with a constant saving rate; from consumers intertemporal optimization condition, (4.23), this requires the expected interest rate to be higher.l A higher interest rate increases current labor supply. Thus if labor supply and consumption respond to changes in capital, households must increase either labor supply or consumption (or both).(4.49) l An improvement in technology raises the wage for a given level of labor supply.l If neither labor supply nor consumption responds(aggregate), households can raise their utility by working more and increasing their current consumption. Thus if labor supply and consumption respond to changes in technology, households must increase either labor supply or consumption (or both).(4.50) l Since an increase in government purchases increases households lifetime tax liability, it reduces their lifetime wealth.l This causes them to consume less leisure-that is, to work more.l When labor supply rises without any change in technology, the real wage falls, and the marginal disutility of working rises.l They will do this only if the