# Intermediate Macroeconomics: The Real Business Cycle Model

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3 therefore a measure of how series co-move. Positive correlations mean that one series being above trend typically coincides with the other series being above trend, and vice-versa. Since we will not be doing any serious quantitative analysis, we will focus on correlations and not volatilities. Below is a table with the correlations of the HP filtered series with HP filtered GDP: Series Correlation with GDP GDP 1.00 Consumption 0.78 Investment 0.85 Hours 0.87 Real Wage 0.14 Real Interest Rate Price Level We can see that GDP, its components, and total hours worked are very strongly correlated with one another the pairwise correlations between these series are all greater than Movements in GDP are positively correlated with movements in the real wage, though this correlation is much weaker. We say that these series are all procylical. A series is said to be procyclical if it is positively correlated with GDP in words, when output is above trend, the series is above trend. The price level and the real interest rate are countercyclical in the sense that they are negatively correlated with GDP, though the correlation between the real interest rate and GDP is pretty close to zero. 3 Can Our Model Explain These Facts? In the previous section we saw that output, hours, consumption, and investment all tend to go up and down together in the data. The real wage is also procyclical, but less so than these other series. The real interest rate and the price level are countercyclical. In this section we want to ask whether or not our equilibrium model of the business cycle can account for these co-movements. In addressing this question, it is important to go back to a point we made in the previous set of notes. That point was the following: in our equilibrium model under standard assumptions on preferences, the only way that consumption and hours worked can move in the same direction is if A t changes. Changes in the three other exogenous variables A t+1, G t, and G t+1 generate negative co-movement between consumption and hours, and hence negative co-movement between consumption and investment and consumption and output. We can see this by combining the first order condition for labor supply from the household side with the first order condition for labor demand from the firm: v (1 N t ) = u (C t )A t F N (K t, N t ) 3

4 Suppose that A t and K t do not change, and suppose that N t goes up. This makes F N (K t, N t ) smaller and v (1 N t ) bigger. The only way for the condition to hold is if u (C t ) gets bigger, which means that C t must fall. Put differently, if neither A t nor K t change, then C t and N t must move in opposite directions. A corollary of this is that, if neither A t nor K t change, then C t and Y t must move in opposite directions too, which is flatly at odds with the data. In the table above we see that consumption and output are positively correlated. Consumption is also positively correlated with hours worked (correlation of 0.65) and total investment (correlation of 0.72). This means that changes in A t+1, G t, or G t+1 cannot be the main driving force behind the business cycle, at least in the context of this model. The only exogenous variable in our model that could possibly account for these correlations is A t. Suppose that there is an exogenous increase in A t. We can work through the implications of this change as we have done before. Starting in the labor market, we see that the labor demand curve shifts out. This, in conjunction with the production function itself shifting, leads to a rightward shift in the Y s curve. There is no shift in the Y d curve. This means that Y t increase and r t falls. r t falling with Y t rising means that I t will rise and C t will rise. The rise in C t works to shift the labor supply curve back in such that it intersects the labor demand curve at a level of employment consistent with the new equilibrium level of output. This means that the real wage is definitely higher, while the effect on employment is ambiguous. I have drawn the picture where total hours worked goes up. Moving down to the money market, the real interest rate (and hence the nominal rate) being lower and consumption being higher means the money demand curve pivots to the right. Along a stable money supply curve, this means that the price level must fall. 4

5 Based on the diagram and discussion above, in response to an increase/decrease in A t, the model produces movements in the endogenous variables that are broadly consistent with what we observe in the data. Output, consumption, investment, and the real wage all move in the same direction, while the real interest rate and the price level move in the opposite direction of output. The movement in hours is technically ambiguous but for plausible specifications of preferences hours should rise if A t increases, so that hours will be positively correlated with output. Hence, qualitatively at least, all of these movements are consistent with what we observe in the data. At least as a first pass, we can see that the model does a reasonable job of fitting the data. 4 Productivity Shocks in the Data We see that our real business cycle model can do a pretty good job of producing co-movements among variables that are qualitatively in line with what we observe in the data if it is driven by 5

6 exogenous changes in A t. Before declaring the model a success, we d like to know if there is any external validity to this story. Are there frequent changes in A t in the data, and do they correspond with what we see happening to output? While we cannot get a direct measure of A t, we can get an indirect measure if we are willing to make some assumptions about the economy s production function. Suppose that the production function is Cobb-Douglas: Take logs: Y t = A t K α t N 1 α t Solve for ln A t : ln Y t = ln A t + α ln K t + (1 α) ln N t ln A t = ln Y t α ln K t (1 α) ln N t In principle, Y t, N t, and K t are objects that we can measure. α is an other object that can be backed out of the data, since optimization by firms implies that (1 α) = wtnt Y t, or labor s share of total income, which is about two-thirds. Hence, α is about 1/3. This means that we can construct a measure of ln A t by subtracting α ln K t and (1 α) ln N t from measured output. Put differently, we can measure ln A t as output less share-weighted inputs, where α and (1 α) are the shares. Sometimes we call this a residual, specifically sometimes the Solow residual as this kind of exercise is often attributed to Robert Solow (the same Solow of growth model fame). Alternatively we call this empirical measure total factor productivity. Below is a time series plot of detrended total factor productivity and detrended real GDP:.04 TFP and Output Percent Deviation from Trend Filtered GDP Filtered TFP We can see that these two series look very similar. All post-war US recessions are associated with 6

7 drops in measured A t that line up nicely with drops in Y t. In other words, empirically there is some evidence that there are in fact large and frequent changes in A t that line up well with movements in Y t. Indeed, the correlation between de-trended TFP and de-trended output is In other words, not only is the model capable of qualitatively matching the data when driven by changes in A t, there is some relatively model-free evidence that there are changes in A t that line up well with observed changes in Y t. 5 The Planner s Problem We have thus far established that (i) our equilibrium real business cycle model is capable of qualitatively matching actual business cycle data when driven by changes in A t, and (ii) there is some evidence consistent with there actually being changes in A t that line up well with the actual business cycle. This means that there is some empirical evidence to suggest that the model is a pretty good model. Given that we think the model fits the data fairly well, we want to ask what kinds of policy implications the model has. We want to analyze the problem of a fictitious social planner who acts to benevolently maximize household utility subject to the resource constraints that the economy as a whole faces. There will be no explicit prices (e.g. the real interest rate and the real wage) in the planner s problem, since prices serve to coordinate actions in a decentralized equilibrium. Since the presence of money does not affect the real allocations we abstract from money altogether here. Think about the planner coordinating actions by choosing quantities. Our objective is to find the solution to the social planner s problem and then to compare it to the competitive equilibrium outcome. To the extent to which the two solutions do not coincide there may be welfare gains to be had from government intervention of one sort or the other. The planner faces the resource constraint that consumption plus investment plus government spending (which is taken as an exogenous value) be equal to total production. There is one resource constraint for each period: C t + K t+1 (1 δ)k t + G t = A t F (K t, N t ) C t+1 (1 δ)k t+1 + G t+1 = A t+1 F (K t+1, N t+1 ) In writing these constraints I have eliminated investment using the capital accumulation equation, so that I t = K t+1 (1 δ)k t, and have imposed the previously discussed terminal condition that I t+1 = (1 δ)k t+1. The planner s objective is to maximize household lifetime utility subject to these two constraints: max U = u(c t ) + v(1 N t ) + βu(c t+1 ) + βv(1 N t+1 ) C t,c t+1,n t,n t+1,k t+1 s.t. 7

8 C t + K t+1 (1 δ)k t + G t = A t F (K t, N t ) C t+1 (1 δ)k t+1 + G t+1 = A t+1 F (K t+1, N t+1 ) This is a constrained problem with two constraints. To deal with the two constraints I am going to eliminate both C t and C t+1 as choice variables by solving for them from the constraints and substituting back into the objective function. The re-written problem is: max U = u(a t F (K t, N t ) K t+1 + (1 δ)k t G t ) + v(1 N t ) +... N t,n t+1,k t+1 + βu(a t+1 F (K t+1, N t+1 ) + (1 δ)k t+1 G t+1 ) + βv(1 N t+1 ) This is now an unconstrained problem. To characterize the solution take the derivatives with respect to the choice variables and set them equal to zero: U = 0 u (A t F (K t, N t ) K t+1 + (1 δ)k t G t )A t F N (K t, N t ) v (1 N t ) = 0 N t U = 0 βu (A t+1 F (K t+1, N t+1 ) + (1 δ)k t+1 G t+1 )A t+1 F N (K t+1, N t+1 ) βv (1 N t ) = 0 N t+1 U = 0 u (A t F (K t, N t ) K t+1 + (1 δ)k t G t ) +... K t+1 + βu (A t+1 F (K t+1, N t+1 ) + (1 δ)k t+1 G t+1 )(A t+1 F K (K t+1, N t+1 ) + (1 δ)) = 0 These can be re-arranged and re-written by substituting C t and C t+1 back in to obtain: v (1 N t ) = u (C t )A t F N (K t, N t ) v (1 N t+1 ) = u (C t+1 )A t+1 F N (K t+1, N t+1 ) u (C t ) = βu (C t+1 )(A t+1 F K (K t+1, N t+1 ) + (1 δ)) These three conditions characterize the solution to the social planner s problem. Now, let s take a step back and recall what were the equilibrium conditions of the competitive equilibrium. On the household side, we had: v (1 N t ) = u (C t )w t v (1 N t+1 ) = u (C t+1 )w t+1 u (C t ) = β(1 + r t )u (C t+1 ) On the firm side, we had: 8

10 changes in A t (which the model does not seek to explain). This means that there is no justification for activist short run stabilization policies. Rather, policies should focus on raising the level of A t in the long run without regard to what happens in the short run (as in the Solow model). 6 Criticisms of Real Business Cycle Theory Real business cycle has met with substantial criticism since its inception. Some of this criticism is empirical, questioning the idea that what show up as changes in A t in the data really represent changes in A t in the model. Some other criticisms are more theoretical, arguing that the model is not realistic because it fails to capture several aspects of the real world that the criticizers feel are relevant. A non-exhaustive list of criticisms of the real business cycle model are below: 1. In the data output and hours worked are about as volatile as one another. The model does not generate large enough movements in hours worked relative to the data. Qualitatively you can see this by noting that the change in hours worked given a change in A t is technically ambiguous, whereas Y t definitely moves in the same direction as A t. This means that Y t moves more than N t in the model, which is not true in the data. 2. The model has no unemployment the labor market is always in equilibrium. Unemployment is a real world phenomenon and is costly from a welfare perspective. 3. The model has no role for active monetary policy and features monetary non-neutrality. In the real world people certainly think that the Fed matters. 4. If A t is a measure of productivity, what does it mean for productivity to decline? Did we forget how to produce? Is it plausible that that A t moves around a lot over short horizons? 5. There have been lots of people that have questioned the measurement of A t. In particular, we may not have good measures of N t and K t. In particular, there could be unobserved utilization for example, during recessions workers sit at work idle, so actual labor input is declining, but measured labor input does not decline by much, and so too much of the movement in output gets attributed to changes in A t. 6. The model does not have any heterogeneity. Real world recessions are costly not because everyone suffers income that is slightly lower, but rather because some people lose all of their income whereas others suffer no income loss. This means that we want to be careful about taking the model s policy and welfare implications too seriously. Each of these criticisms has merit. I am least sympathetic to point (4); properly understood, A t is more than just technology, and so there is no compelling reason to insist that it can t decline or that it can t move around a lot over short horizons. One thing that could potentially manifest itself as A t is the health of the financial system if the financial system collapses, then there will 10

11 be a poor allocation of resources across the economy, which may manifest itself as low A t, much like what we observed during the recent recession. My own view is that real business cycle theory is a decent first pass at understanding economic fluctuations, though it is admittedly too simple and one probably needs to be weary about taking its policy and welfare implications too seriously. Nevertheless, I think it s a good benchmark. What was (and is) revolutionary about the real business cycle theory is that it suggests that there may be no justification for activist policies because fluctuations in economic aggregates may be the efficient reaction to changing conditions like A t. This does not mean that in all circumstances there is no need for activist economic policy; what it does mean is that proponents of activist policies need to articulate the friction or market failure that moves us beyond the real business cycle world and into a world in which policy can improve outcomes by smoothing out the cycle. 11

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