ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE

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1 ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs. Neither will this synopsis necessarily suffice for exam preparation. Contents 1. Introduction 1 2. Model Primitives The representative household The representative firm One key simplifying assumption 3 3. Characterizing the Equilibrium The household s problem inter- and intra-temporal conditions Equilibrium conditions wage and rental rates The balanced growth path 5 4. Deviations from the Balanced Growth Path 6 5. Discussions and Conclusions 8 References 8 1. Introduction In this lecture, we will explore the basic model that could potentially describes the fluctuations we see in macroeconomic quantities and prices such as (un)employment and output or wage and rental rates. We focus on the movements of these real economic variables (as opposed to nominal or monetary ones such as money demand) in order to try to understand the sources of movements between peaks and troughs in the economy. We tend to take a stand point such that without any uncertainty (or shocks ) the economy should grow at a constant rate with various variables showing balanced growth between each other. That being said, we are inclined to understand the fluctuations in macroeconomic quantities and prices as irregularities as results from exogenous and, granted, hard to explain shocks. The word cycle in this lecture is indeed misleading in the sense that we are not trying to establish the observed fluctuations as some repeated and systematic pattern. Instead, we simply use the word to represent the highs and lows and the economy s almost persistent commute between them. This synopsis is mildly different from Chapter 4 in Romer (2006) in how certain results are obtained. 1

2 2 YUAN TIAN 2. Model Primitives 2.1. The representative household. Consider an economy in discrete time with infinite time horizon where there is one representative household. The number of members of the household (the population) in period t is N t and the population grows at approximately rate n. More accurately, let ln N t = N + nt where N is the natural-logarithm of the population in period 0. The household derives utility from consumption and leisure. Let c t and l t denote the consumption and labor supplied per capita by the household, the household s total utility in period t is given by N t (θ ln c t + (1 θ) ln(1 l t )). The household discounts the future at rate ρ, which implies that the discounted lifetime utility of the household with consumption and labor per capita stream {c t, l t } t=0 is given by (1) U ({c t, l t } t=0 ) = [exp( ρt) N t (θ ln c t + (1 θ) ln(1 l t ))]. t=0 In any period, the household has two sources of income: the labor income from supplying labor and the rental income from renting the capital it owns to the firm. Let w t and R t denote the wage and rental rates in period t. Prices of the consumption and investment goods are both normalized to 1 in any period. Let C t, L t, and K t denote the total consumption, labor supply, and capital of the household. Naturally, C t = c t N t, and similarly for labor supply and capital. The household has two outlets for its income in any period: consumption and investment into capital. Capital depreciates at rate δ. The law of motion for capital owned by the household is thus given by (2) K t+1 = (1 + R t δ)k t + w t L t C t. The household s objective is to maximize its discounted lifetime utility given in (1) subject to the law of motion for capital given in (2) and its initial capital stock and population, also taking wage and rental rates in any period as given The representative firm. There exists one representative firm in the economy which simply maximizes its profit in all periods. It hires labor and rents capital from the household at the competitive wage and rental rates and produces a single output that can be used as consumption and investment goods. The production function of the firm, with K t and L t being capital and labor employed, is given by Y t F (K t, A t L t ) = Kt α (A t L t ) 1 α where A t represents technology and is assumed, without loss of generality thanks to the Uzawa s Theorem, to be labor-augmenting. The path of the technology is not deterministic in the sense that it is subject to random shocks to the economy. Specifically, assume that the technology grows approximately at rate a without any shock. Let à t denote the shock to the technology that happens in period t. We assume the technology in period t is determined by ln A t = Ā + at + Ãt. Moreover, we assume some correlation between the shocks within two adjacent time periods for an extreme instance, the effect of war on technology is fairly persistent and it is reasonable to assume it affects more

3 ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE 3 than one periods. In particular, suppose the technology shock follows an autoregressive process of order 1: Ã t+1 = φãt + ε t+1. ε t+1 is the disturbance to the technology shock that happens in period t + 1. We introduce this disturbance term so that future technology shocks cannot be perfectly predicted based on previous shocks (which is the sheer meaning of the existence of shocks per se). Suppose the disturbance in any period is independently and identically distributed has zero mean and finite variance. Also, assume φ < α < 1. The firm operates competitively in both the input and output markets, from which profit-maximization implies the wage and rental rates being w t = (1 α)a t Kt α (A t L t ) α ; (3) R t = αkt α 1 (A t L t ) 1 α. Combining the household and the firm, the inter-temporal resource constraint of the economy in period t is (4) Y t = C t + K t+1 (1 δ)k t One key simplifying assumption. To better focus on explaining the deviations of the economy from the balanced growth path, we make the following key simplifying assumption. Assumption 2.1. Assume capital depreciates at 100% in this economy, i.e. δ = Characterizing the Equilibrium In this section, we will characterize the equilibrium in this economy. We will take a not-so-standard approach, mixing procedures of optimization and market-clearing along the way and hopefully be able to draw definitive conclusions from this simplified model regarding real business cycles The household s problem inter- and intra-temporal conditions. The household s problem in this model is a bit more complicated than the one we faced in the neoclassical growth model in the sense that the household does not know for sure the future wage and rental rates. Recall from (3) that these two rates are determined by the marginal products, which in turn depends on the stochastic process of the production technology {A t } t=0. Therefore, we approach the household s problem without explicitly setting up the utility maximization problem instead simply working with key conditions that the optimum must satisfy. We first recognize that we should be able to represent any solution to the household s problem by describing how much to consume and how much labor to supply in period t as functions of the wage and rental rates and capital in period t and expectations of the wage and rental rates in period t + 1. Essentially, this argument is assuming that by solving the household s problem in this period-by-period way, we will be able to obtain any solution as we solve the same problem from the perspective of the household s infinite lifetime. We break down the household s problem into two layers: inter-temporal and intra-temporal. We first examine the inter-temporal consumption choice of the household and see what this implies about the household s investment decisions. Then we look at the intra-temporal labor supply decision to obtain a relationship between the household s choices on how much to consume and how much to work. Note that with these two conditions, we can even characterize the substitution the household faces between how much to work now versus how much to work in the future.

4 4 YUAN TIAN c t For the inter-temporal consumption choice, consider the following calculus of variations argument. Let be the household s optimal consumption per capita in period t. Suppose now the household decides to change its c t by an infinitesimal c. Moreover, assume that other than the periods t and t+1, consumptions in all other periods stay the same as in the optimum. This implies that such an adjustment of c must be paid back in period t + 1, although must be in a different amount. For c t and c t+1 to be optimal, such a change must result in a zero change in its discounted utility. Without loss of generality, suppose the household decreases its consumption in t by c. The loss in utility by such a change can be approximated by the marginal utility at c t multiplied by c, which is (5) θ exp ( ρt) N t c c, t where recall that the marginal period utility at c t is θ/c t due to the natural-logarithm utility in consumption. Now, how much does this decrease in consumption in period t imply about the increase in consumption in period t + 1? Recall that prices of consumption and investment are both normalized to 1 in any period, which means that a decrease in consumption by c in period t is an increase in investment in period t by c. However, the extra amount of consumption in t + 1 is not simply c. An extra c units of capital is worth [(1 + R t+1 δ) c] units of consumption in period t + 1. Furthermore, the household does not know the rental rate of capital in period t + 1 thus can only form an expectation in period t about this rental rate. Hence, the expected utility gain from the adjustment in consumption is given by [ (6) θe t exp( ρ(t + 1)) N t+1 (1 + R ] [ ] t+1 δ) c (1 + Rt+1 δ) c = θ exp( ρ(t + 1)) N t+1 E t t+1 c c t+1 where the symbol E t [ ] denotes the expectations formed in period t and, in this case, of the rental rate in period t + 1. Some of the expectations are know, such as the discount rate exp( ρ(t + 1)) and N t+1 while others contain uncertainty. In particular, notice that the expectation of c t+1 is not known since it could potentially be a function of the wage and rental rates in period t + 1. Optimality of c t and c t+1 requires that (5) must be equal to (6), which results in [ ] exp(ρ n) (1 + Rt+1 δ) (7) c = E t t c. t+1 Note that without uncertainty, (7) is simply the Euler equation as we saw from previous inter-temporal choice problems where we can interpret (1 + R t+1 δ) as the real interest rate in this case. For the intra-temporal part, focus on period t and let c t and l t be the optimal consumption and labor supply in this period. Now, take any pair of the amount of capital in periods t and t + 1. This implies that the difference between wlt and c t is a constant given this pair of capital in period t and t + 1. Hence, the household s problem in period t is now transformed into max {θ ln c + (1 θ) ln(1 l)} subject to wl c = B {c,l} for some constant B. The quotient of the first-order conditions here can be easily confirmed to be (8) θ 1 θ 1 l t c t = 1 c t = θw t(1 lt ). w t 1 θ Equations (7) and (8) are the major tools we will use in this model to characterize the equilibrium Equilibrium conditions wage and rental rates. We will now put the representative firm and household together and characterize the equilibrium conditions. Notice that the Euler equation in (7) is

5 ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE 5 given in consumptions per capita, hence we start by representing the rental rate in t + 1 in per-capita terms. Recall from (3) that (9) R t+1 = αk α 1 t+1 (A t+1l t+1 ) 1 α = αy t+1 K t+1 = αy t+1 k t+1 Now, define s t as the savings rate in period t, i.e. (10) s t 1 C t Y t = 1 c t y t c t = (1 s t )y t. Imposing Assumption 2.1 on (4), we have (11) K t+1 = s t Y t k t+1 = s ty t N t N t+1 = exp( n)s t y t. Substituting (9) through (11) into (7), we know that for a time path {c t } t=0 to be optimal for the household, [ ] [ ] [ ] exp(ρ n) αyt+1 exp(ρ n) αy t+1 α = E t = E t = E t c t k t+1 c t+1 (1 s t )y t k t+1 (1 s t+1 )y t+1 s t y t (1 s t+1 ) The expectation in period t of any variable in period t is simply the value of the variable in period t. Thus, [ ] exp(ρ n)s t 1 (12) = E t. α(1 s t ) 1 s t+1 One solution to the stochastic difference equation (12) is that the savings rate is constant over time. Note that in this statement, we are not asserting that this is the unique solution to the household s problem but merely one working conjecture although it does turn out to the unique solution, which can be verified by another independent argument beyond the scope of this lecture. Imposing this conjecture on the difference equation (12) and let the constant savings rate be s 1, exp(ρ n)s [ ] 1 (13) α(1 s = E t ) 1 s = 1 1 s s = α exp(n ρ). Moreover, observe that w t = (1 α) Y t L t = (1 α)y t. Recall the intra-temporal condition and substitute in the constant savings rate and the wage rate, (14) (1 s )y t = θ 1 θ (1 α)y t(1 lt ) lt = 1 1 θ 1 s θ 1 α which is also constant over time. Therefore, (13) and (14) together offer the equilibrium path in this model: the investment of the household in capital will be a constant share of the output in any period while the household will also supply a constant share of the total population to the firm over time The balanced growth path. With the equilibrium conditions characterized, we will examine the effect of the technology shocks on the aggregate variables of the economy in the next section. To set up a baseline for discussion, we first focus on the path of these variables without any shock Ãt = 0, t. Not surprisingly, the economy exhibits a balanced growth path in this scenario. We will now calculate the constant growth rates of these variables and verify the balanced growth path. Let g Y be the growth rate of the aggregate output and similarly for the growth rates of other variables. We start with some implications of the constant labor supply and savings rate. First of all, constant labor supply implies that g L L t+1 1 = N t+1 1 = exp(n) 1, t. L t N t

6 6 YUAN TIAN Constant savings rate implies that g Y = g C = g K. Take any two adjacent periods, g Y Y ( ) α ( ) 1 α t+1 Kt+1 At+1 1 = Lt+1 1 = (1 + g K ) α (exp(a + n)) 1 α 1, Y t K t A t L t which, in turn, implies (15) 1 + g Y = (1 + g Y ) α exp[(a + n)(1 α)] g Y = exp(a + n) 1 = g K = g C. It is easy to confirm that the growth rate of output, capital, and consumption per capita is simply g y = g k = g c = exp(a) 1 = g A. This conclusion echoes the results we obtained in the Solow growth model where we imposed the constant savings rate condition (and ignored the leisure in utility) in the first place, which partially justifies the assumption of constant savings rate in hindsight. 4. Deviations from the Balanced Growth Path As mention in the Introduction, we tend to capture the fluctuations of key variables in the economy as deviations from their normal states without any shocks. As we have just shown, the normal states are merely the corresponding balanced growth path where all variables are growing at constant rates. To understand the deviations from the balanced growth path driven by the technology shocks, we focus on how overall consumption and capital deviate from their paths where they grow at the same constant rate. Starting from now on, we will pay special attention to the natural-logarithm of the levels of the variables, mostly for the simplicity in notation and calculation. Let C t be the deviation of the natural-logarithm aggregate consumption in period t from the balanced growth path of consumption, and similarly for K t and Ỹt. Note that this notation is consistent with the way we define the process of the production technology. Similarly, let C + (a + n)t be the natural-logarithm of the aggregate consumption on the balanced growth path and similarly for Ȳ + (a + n)t and K + (a + n)t. With the results on constant savings rate and constant labor supply, we are able to characterize the exact paths of these deviations. Naturally, Observe that C + (a + n)t = ln(1 + s ) + Ȳ + (a + n)t. C t = (1 s )Y t ln C t = ln(1 s ) + ln Y t C t ln C t C (a + n)t = ln(1 + s ) + ln Y t ln(1 + s ) Ȳ (a + n)t = Ỹt ln Y t Ȳ (a + n)t. Similarly, (16) Kt+1 = Ỹt. Therefore, to quantify the deviations of the economy from the balanced growth path, we can simply focus on the process of Ỹt. Recall the production function of the firm and take the natural-logarithm, Y t = K α t (A t L t ) 1 α ln Y t = α ln K t + (1 α)(ln A t + ln L t ). Converted into deviations from the balanced growth path, (17) Ỹ t = α K t + (1 α)ãt.

7 ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE 7 From (16) and (17), we can obtain a difference equation on the deviations in the aggregate output as a function of the technology shocks. Push (17) one period forward and substitute in (16), (18) Ỹ t+1 = αỹt + (1 α)ãt+1, which shows that the deviation of the output in period t + 1 is a combination of the deviation in output in the previous period and the technology shock in the same period. We are now ready to look at the effect of a technology shock, say, at time 0 (assuming that the economy follows the balanced growth path before 0) on the paths of deviations in aggregate output, capital, and consumption from the balanced growth path. Note that by converting all the variables into natural-logarithms, we have actually quantified these deviations as percentage deviations from the balanced growth path. The obvious convenience of such a transformation is the possibility of comparisons across these variables since now they are all in percentages and without units. This type of exercise is what we usually refer to as obtaining the impulse response functions, i.e. how endogenous variables respond a change or shock in the exogenous ones. Without loss of generality, let Ã0 = 1, which implies that the deviations in the aggregate output is now interpreted as the percentage deviation from the balanced growth path due to a one percentage deviation in technology from its path without any shocks. By the autoregressive process of the technology shock, the deviation in technology in any period is now solely determined by the disturbances. For the impulse response functions, our ultimate goal is to obtain and expression of Ỹt+1 in terms of the history of the disturbances up to period t + 1. To that end, we will first express Ỹt+1 as a function but of the technology shocks then express the technology shocks as a function of the disturbances. Push (18) one period backwards, (19) Ỹ t = αỹt 1 + (1 α)ãt. Substitute (19) into (18), (20) Ỹ t+1 = α 2 Ỹ t 1 + (1 α) ( ) αãt + Ãt+1. We can conjecture from (20) feel free to repeat the recursive substitution for a couple of more periods if you feel more comfortable doing so that (21) Ỹ t+1 = α t+1 Ỹ 0 + (1 α) t i=0 (α i à t+1 i ). Note that the upper bound in the summation in (21) is t instead of t + 1. Since we have assumed that the economy was on the balanced growth path before 0, Ỹ 1 = 0, which implies K 0 = 0 by (16), which in turn implies that Ỹ0 = (1 α)ã0. Substitute this into (21) for the Ỹ0. (22) Ỹ t+1 = α t+1 (1 α)ã0 + (1 α) t ) t+1 ) (α i à t+1 i = (1 α) (α i à t+1 i. i=0 Note that now the upper bound of the summation is t + 1. What s left is simply to express the technology shocks in terms of the disturbances up to period t + 1. From the process of the technology shocks and by recursive substitution, we can get t i t i (23) à t+1 i = φ t+1 i à 0 + φ j ε t+1 i j = φ t+1 i + φ j ε t+1 i j. j=0 j=0 i=0

8 8 YUAN TIAN Substituting (23) into (22) gives us the desired expression. As can be easily seen from this process, the full characterization of à t+1 is quite complicated when the disturbances in the technology shocks constant happen in the time path leading to period t + 1. For simplicity and meaningful interpretation, let s focus no the case where the disturbance is zero for all periods we will get back to this issue shortly. Define κ as The disturbances leading up to t + 1 are all zero, so Substituted into (22), κ φ α κ < 1 à t+1 i = φ t+1 i = α t+1 i κ t+1 i. t+1 i (24) Ỹ t+1 = (1 α) α i α t+1 i κ t+1 i = (1 α) α t+1 1 κt+2 1 κ = 1 α α φ (α t+2 φ t+2). i=0 Differentiate (24) with respect to t dỹt+1 dt = 1 α α φ (α t+2 ln α φ t+2 ln φ ) < 1 α α φ (α t+2 φ t+2) ln α < 0. Furthermore, it can be easily confirmed that the second derivative is positive. Thus, the deviation in aggregate output from the balanced growth path due to a technology shock will decrease at a decreasing rate to zero as time goes to infinity. The assumption about all the disturbances being zero is actually not as strong as it seems. Suppose there is a non-zero disturbance in period T > 0, ε T 0. Notice that this only affects the technology shocks in and after period T. Thus, the time path of the deviation in aggregate output after T can be viewed the impulse response to a different initial technology shock. Granted, there is one additional subtle difference, namely the economy was not following the balanced growth path before T. However, the effect of such a minor modification is fairly straightforward. It only implies that the K T 0, as opposed to K 0 = 0. This will shift the entire path of Ỹt after T by a constant amount but the trend of the deviation converging to zero over time will not be affected. In summary, the effect of a disturbance in the technology shock is lifting or pressing the entire path of deviations, which then follows the convergent trend to zero in finite time in absence of additional disturbances. 5. Discussions and Conclusions In this lecture, we have introduced a simple model as an attempt to explain the observed fluctuations in variable macroeconomic models. In our model, savings rate and labor supply are constant thanks to the simplifying assumptions of 100% capital depreciation, which is actually a fairly strong assumption. shortcomings of the model are obvious, the most significant one among which is the result on constant labor supply. Thus, this model is not suitable for explaining the fluctuations in unemployment. However, as we have seen, the results on the behavior of aggregate output, consumption, and capital are very straightforward and so are the impulse responses of these variables to the technology shocks and the relevant disturbances. In that sense, this model does provide a reasonably flexible and tractable workhorse starting point for modeling aggregate economic behavior. The References Romer, David, Advanced Macroeconomics, McGraw-Hill, 3rd Edition.

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