San Francisco State University ECON 302 Business Cycles: The Classical Approach Introduction Michael Bar Recall from the introduction that the output per capita in the U.S. is groing steady, but there are uctuations about the trend. These uctuations are called business cycles. Figure shos the ln of real GNP per capita in the U.S. in the last century, together ith a linear trend. The linear trend ts the data pretty ell, hich means that the original variable, GNP per capita, as groing at constant rate. Figure : n of real GNP per capita in U.S. 4 ln(real GNP per capita) 3.5 3 2.5 2.5 900 920 940 960 980 2000 2020 Year The study of the long run groth trend belongs to the eld of economic groth. In these notes e focus on the uctuations of the output around the trend. Subtracting the groth trend from the time series in gure, results in a series of deviations from trend, displayed in gure 2. The series of deviations from trend is called detrended real output, or the cyclical part of the real output. The questions that e ant to ask in these notes are:. What causes business cycles? 2. Can the government smooth out the business cycles? 3. Should the government smooth out the business cycles? In order to anser these questions, economists use models. We ill see that di erent models give di erent ansers to those questions.
Figure 2: Cyclical part of GNP per capita in U.S. 0.3 Deviations from trend 0.2 0. 0 900 920 940 960 980 2000 2020 0. 0.2 0.3 0.4 0.5 Year 2 The Classical Model 2. The description of the model The model consists of a representative consumer, representative rm, and a government. The consumer receives income from supplying his labor and from dividends from the rm he ons. The consumer chooses his consumption and time allocation beteen labor and leisure. The rm is oned by the consumer, it ons a xed amount of capital, and it chooses the optimal amount of labor to maximize pro ts. The government consumption is exogenous to the model. The government balances its budget by collecting taxes at the amount of expenditures. The formal description of the model economy:. Consumer: max ln C + ( C;l s:t: ) ln l C = [(h l) + ] ( t) here C is consumption, l is leisure, is real age, h is time endoment (say 00 hours per eek), is the pro ts or dividends from the rm, t is the at tax rate. Thus, the time spent orking (labor supply) is S = h l 2. Firm: max = AK D D D 2
here A is productivity parameter (called Total Factor Productivity, TFP), K is the capital stock, and D is labor employed by the rm. The productivity parameter re- ects the idea that ith technological improvement (A ") more output can be produced ith the same inputs. The total output in the economy is thus Y = AK. 3. Government: collects taxes on all income at the rate of t, and spends them on government consumption. The government budget is G = t ( + ) 4. De nition: Competitive equilibrium consists of (; C; G; ; l; ; Y ) such that (a) Given ;, the values of (C; l) solve the consumer s problem, (b) Given, the value of solves the rm s problem, (c) Markets are cleared: i. D = S = (labor market), ii. C + G = Y ( nal goods market). 2.2 Important remarks about models in general This section is a philosophical discussion of our approach in general. It is essential to read it in order to understand the material of this entire course, and many other courses that you are taking. You should come back and read this again after you have practiced orking ith the classical model.. The competitive equilibrium is the model s prediction about the endogenous variables. Endogenous variables are determined inside the model, i.e. the variables hich the model is trying to explain. The exogenous variables are those that are determined outside of the model. For example, in the model of a market the endogenous variables are price and quantity traded, and exogenous variables are those that determine the location of supply and demand curves (such as income, prices of related goods, etc.). In the classical model the exogenous variables are: (A; t; K), and the endogenous variables are: (; C; G; ; l; ; Y ). 2. Causality: hat causes hat? In any model, the exogenous variables are causing the endogenous variables. For example, e can change A (the technology level) and observe the changes in real age, employment, consumption, output, etc. All the endogenous variables are caused by the exogenous variables. If e don t change any of the exogenous variables, no change in the endogenous variables can occur. Thus, in this model e cannot say that output causes employment, since both output and employment are endogenous variables, and cannot change unless e change some of the exogenous. Be very careful about making statements of causality in the real orld. 3. Why models? 3
(a) Models can explain some features of the real orld. Models don t tell us hat the orld looks like. Instead, they tell us hat e can expect to happen in the orld if the orld as like the model. For example, the supply and demand diagram doesn t look anything like the markets in the real orld. The diagram does not sho the identities of the buyers and sellers, their feelings and emotions, their physical appearance. The supply and demand diagram only captures to features of real markets: () buyers typically ant to buy less hen the price goes up, and (2) sellers ant to produce more hen the price goes up. It turns out the supply and demand diagram is very useful in explaining hy prices di er across goods and hy there are changes in prices. After testing the predictions of the model ith the data e conclude that indeed the to features of buyers and sellers that e included in the model ere important. (b) Models can be used to perform controlled experiments. In the real orld many things change at the same time; the technology changes, government policies change, etc. In the model e can perform controlled experiments of changing one thing at a time. This is impossible to do ith actual economies. 4. Models are not realistic and are not supposed to be. When the object of study is very complicated, e need models that ill highlight some important features of the object and leave out many other features. For example, hen e study the economy of an entire country ith millions of people, thousands of markets and rms, it is di cult for us to understand the behavior of the economy by just looking at it. Moreover, if e don t have any models to ork ith, e don t even kno hat data should be collected about the object of our study. For example, the model of supply and demand tells us that e don t need to collect data of all the names of buyers and sellers of the market in order to understand ho it orks. 2.3 Working ith the classical model The de nition of competitive equilibrium is instructive about ho the model should be solved. The de nition suggests the folloing steps: () solve the consumer s problem to get the labor supply, (2) solve the rm s problem to get the labor demand, and (3) use the market clearing conditions to nd the real age, the equilibrium employment, and the rest of the endogenous variables. 2.3. Mathematical solution Step : solving the consumer problem The consumer s problem can be ritten as max ln C + ( C;l s:t: ) ln l C + ( t) l = (h + ) ( t) 4
This is a standard consumer choice problem ith to goods: C and l, the prices of the goods are and ( t) respectively, and the consumer s income is (h + ) ( t). We kno already ho to solve a consumer choice problem ith Cobb-Douglas preferences. Thus, the demand is C = (h + ) ( t) (h + ) ( t) l = ( ) ( t) = ( ) h + and the labor supply is S = h ( ) h + () Observe that consumption is increasing in and, and decreases in t. The labor supply is increasing in, decreasing in and does not depend on taxes. The intuition hy the labor supply is decreasing in goes as follos. The dividend income is non labor income, so hen it goes up the consumer does not need to ork as much. Figure 3 shos the graph of the labor supply curve. i.e. ho much labor the consumer ants to supply at any given age, holding everything else xed. This means that changes in are re ected by movements along the curve, hile changes in ill shift the entire curve. Figure 3: abor supply curve Real age () 0 9 8 7 6 5 4 3 2 0 abor Supply Curve 0 0 20 30 40 50 60 70 abor (N) s Step 2: Solving the rm s problem The rm s problem is max = AK D D The rst order condition D @ = ( ) AK D = 0 @ D ( ) AK D = (2) 5
hich tells us that the rm maximizes pro t hen it equates the marginal product of labor the the real age. Equation (2) thus gives us the labor demand of the rm. We can solve for D explicitly from equation (2) to get ( D = ) AK = Observe that this curve is decreasing in. Figure 4 shos the labor demand curve, i.e. ho much labor the rm ants to employ at any given age, holding everything else constant. Thus, changes in are re ected by movements along the curve hile changes in A ill shift the entire curve. The pro t is therefore given by Figure 4: abor demand curve Real age () 0 9 8 7 6 5 4 3 2 0 abor Deamand curve 0 0 20 30 40 50 60 70 abor (N) d = AK D ( ) AK D D = AK D (3) Step 3: equilibrium in the labor market etting S = D = and substituting equations (2) and (3) into equation () gives = h ( ) h + AK ( ) AK Solving for equilibrium : = h ( ) h + ( ) ( ) = h ( ) h ( ) 6
+ ( ) ( ) = h ( ) + = h + ( ) = h = h Equilibrium employment: = ( ) h Once e found the equilibrium employment, all the other endogenous variables can be found in terms of. Equilibrium leisure: l = h To solve for equilibrium age, use equation (2): = ( ) AK Equilibrium output: Y = AK Equilibrium pro t, using equation (3): = AK To nd equilibrium consumption e use the budget constraint: Equilibrium government expenditures: C = [ + ] ( t) C = ( ) AK + AK D ( t) C = ( t) Y G = Y C = ty Summary of equilibrium: ( ) h = l = h = ( ) AK Y = AK = AK C = Y ( t) G = ty 7
As you can see, an increase in productivity A, causes an increase in equilibrium output, equilibrium real age, equilibrium consumption, equilibrium government consumption, and equilibrium pro t. Equilibrium employment does not depend on the level of technology, even though the real age ent up. The e ect of higher K is similar because A and K alays appear together in the equations. An increase in the tax rate a ects only the distribution of the total output beteen the private sector and the government sector. If t = 30% for example, then the government consumes 30% of the total output, hile the private consumers get to consume the rest 70%. 8
2.3.2 Graphical analysis The classical model can be analyzed graphically ith only to diagrams, the labor market and the production function, as shon in gure 5. Figure 5: Classical model: graphical illustration Y Production function Y * * S abor market * D * These graphs correspond to the folloing equations: Production function : Y = AK abor supply curve : S = h ( ) h +, here = AK ( abor demand curve : D = ) AK = It is important to repeat here that labor supply curve is increasing in. On the other hand, if increases, this leads to a shift of the entire supply curve to the left. The labor demand curve is decreasing in. If A or K increase, the entire labor demand curve ill shift to the right. No e use this graphical frameork in order to perform 3 experiments ith the model:. An increase in productivity (A "). 9
Figure 6: An increase in productivity (A ") Y Y 2 Production function Y S abor market W 2 W D Figure 6 shos the e ects of an increase in A in the classical model. As A ", there is an increase in labor demand (shift of the labor demand curve to the right) and a decrease in labor supply (shift of the supply curve to the left) and an increase in production function. The e ect of the increase in labor demand on employment is an increase in employment, hile the e ect of a decrease in labor supply on employment is a decrease in employment. Thus, ithout solving the model ith particular functional forms e cannot tell hat is the e ect of A " on equilibrium employment. In the pervious section hoever e solved the model ith Cobb-Douglas technology and preferences and found that the equilibrium employment does not change as A ". In other ords, the e ect on employment of a decline in labor demand and of an increase in labor supply cancel each other. Both e ects hoever increase the equilibrium real age. 2. An increase in K. The e ect of an increase in K is the same as the e ect of an increase in A. Notice that A and K alays appear together as AK. 3. An increase in the tax rate (t "). 0
Neither the labor demand nor the labor supply depend on the tax rate, hence nothing ill change in the labor market. The production function does not depend on the tax rate as ell, and therefore none of the curves in gure 5 ill shift. As e have seen before, the only e ect that an increase in the tax rate has on the economy is the increase in the government share of the total output. 2.3.3 Ansering the questions No e are ready to anser the questions e posed in the beginning of these notes, ithin the frameork of the classical model.. What causes business cycles? The exogenous variables in this model are (A; t; K). As e have seen before, a positive shock to productivity (A ") increases the equilibrium output hile a negative shock to productivity (A #) decreases it. We can think of shocks to productivity as agglomeration of many factors such as innovations, shocks to oil prices, eather, political events, etc., that change the amount produced ith the same inputs. So this model suggests that business cycles might be a result of productivity shocks. As e have seen before, changes in t do not a ect the equilibrium output. Ho about K? It is possible that a hurricane, or a terrorist attack ould destroy part of the nation s capital and cause a decline in output. It is harder to think of ho the stock of capital can experience a sudden increase. In any case, hen e look at the data on capital stock, it looks very smooth and does not exhibit uctuations that can potentially be the cause of business cycles. Moreover, in applied versions of this model, K is endogenous (accumulated ithin the maodel through investment), and therefore, in such models e can t exogenously change K. 2. Can the government smooth out the business cycles? We have seen before that changes in the tax rate in this economy do not a ect the equilibrium output. Changes in the tax rate only a ect the fraction of total output that is consumed by the government. Recall that C = ( t) Y G = ty So the anser to the question is, NO, the government cannot smooth the business cycles (in this model). 3. Should the government smooth out the business cycles? It shouldn t because it can t (in this model). For a more detailed discussion about productivity shocks see the next section.
2.4 Real business cycle doctrine Real business cycle theory suggests that the main source of business cycles is shocks to productivity. The real business cycle school is led by Edard Prescott and Finn Kydland. They ere aarded a Nobel Prize in Economics in economics in 2004 "for their contributions to dynamic macroeconomics: the time consistency of economic policy and the driving forces behind business cycles". Finn Kydland and Edard Prescott developed a methodology that allos them to anser the folloing quantitative equation: "ho much of the uctuations in output around a trend can be accounted for by random shocks to productivity?". Their anser as 2/3. Kydland and Prescott used a model that is a more complex version of the classical model (their model is called "the Neoclassical Groth Model"). But the idea can be illustrated ith the classical model. Step : Choose functional forms for utility and production function. In the data, although the real age ent up in the last decades, the average orktime did not change. Notice that in our model ith Cobb-Douglas utility function, e get the same result, i.e. in equilibrium the orktime is constant and does not depend on the real age. In the data, the labor share of total output is roughly constant over time. The Cobb- Douglas production function delivers this property. Recall that the capital share is and the labor share is, and these are constant. Step 2: Choose the parameter values for the utility and production functions. In the data the labor share is about 2=3 of the total output. Thus, set = =3 so that ( = 2=3). In the real orld people have approximately 00 hours per eek that they can allocate beteen labor and leisure activity (24 hours per day, minus 8 hours of sleep and 2 hours of maintenance such as bathroom, eating, resting). In the data the average orktime is 40 hours per eek, so using our equilibrium equation for employment e can nd as follos: ( ) h = 40 = 3 00 3 0:4 = 2 3 3 0:4 0:4 3 = 2 3 :2 0:4 = 2a :2 = 2:4a = 0:5 Thus, = 0:5, = 3. Step 3: Estimate the shocks to productivity We assume that aggregate output is produced ith Y = AK (4) 2
We have data on real GDP (Y ), on capital (K) and labor employed (). This means that e can nd A from the above equation as a residual. Because of this procedure A is called the Solo Residual, since e nd it as the residual that ould equate the left hand side and the right hand side of equation (4). Step 4: Model simulation Having found the time series of A e can simulate the model and generate time series of consumption and output. We have seen that an increase in A causes an increase in output in the classical model and a decline in A ill cause a decline in output. It turns out that the time series of Y generated by the model is very similar to the data in gure 2. In fact, the variance of the output generated by the model is about 2/3 of the actual variance of the real GDP/capita in the data. This means that random shocks to productivity can explain most, but not all the variation in real GDP/capita over the business cycles. 3