Lecture notes on Real Business Cycle Modeling

For 18.380, March 2001

Real Business Cycle (RBC) models all have these key features:

- all markets clear in the short run through adjustment of prices and wages (and interest rates and exchange rates, except that the models are rarely elaborated enough to have financial and forex markets). There is NO sluggishness in adjustment of price levels.

- purely nominal shocks to aggregate demand (caused by changes in the nominal money supply) are therefore totally crowded out in the short run by matching changes in the price level; output fluctuations must be accounted for by changes in supply of inputs (i.e. of capital and labour supply)

- all macroeconomic outcomes are derived explicitly from optimizing choices of rational agents, reacting to changes in the feasible sets of outcomes available to them. As a result, all macroeconomic outcomes represent what microeconomic agents want (that is, a Pareto-optimum), so it is not at all obvious how government stabilization policy would improve matters.

In these notes (a) I specify a relatively simple RBC model, (b) I report its 'solution', (c) I discuss what it implies for macroeconomic behaviour, and (d) I report what RBC theorists do to 'test' their theory against reality. The reading on reserve, from Mankiw and Scarth, does a good job of laying out the drawbacks of this approach, so these notes are only intended to explain what the approach involves.

(a) A Simple RBC model:

The economy consists of a large number of identical households and firms, plus a government whose behaviour is entirely exogenous. All firms are owned by households. All households live forever, and form all expectations rationally (that is, they all know exactly how the economy works). All markets clear by adjustment of wages and prices, so this is a Walrasian general equilibrium model.

Firms

Firms choose output and input levels (of labour and capital) to maximize profits p, defined as output less costs of hiring labour and capital in competitive markets. For the ith firm (and suppressing time subscripts),

p_i = Y_i - w A L_i - r K_i

Where L_i = labour employed by firm i, K_i = capital stock, r = yield promised on capital, w = wage rate for unskilled labour, and A is the technological skill level of the actual workers hired.

Each firm is subject to the constraint of a Cobb-Douglas production function, whose aggregate form for year t is

Y_t = F(K_t, A_tL_t) = K_t^a (A_t L_t)^1-a

Where the product of A_t and L_t is the number of 'effective' (unskilled) workers. The technology index A_t changes over time with a trend component g and a stochastic component u that is first-order autoregressive (i.e. persistent):

A_t = A_o e^gte^u_t; u_t = r_a u_t-1 + e_at, where e_at is 'white noise'.

Firms hire inputs in, and sell output into competitive markets, so they are all price-takers.

Government

The government uses resources to produce public goods G, according to a process that we need not model, and finances it by a set of lump-sum taxes of equal amount. Total annual output Y is allocated among three possible uses

Y = C + I + G,

Where C is aggregate household consumption, I is gross investment in new capital, and G is government spending. Increases in G therefore reduce the resources available for the private sector to allocate among C and I in any period, and that is the only role of government in the model. The level of government spending is exogenous but stochastic: it grows at a constant trend growth rate, but with a stochastic deviation from that trend that is first-order autoregressive (that is, deviations persist for one period). The trend is the sum of population growth n and technological growth trend g, which keeps the level of G in line with the number of effective workers in the economy.

G_t = G_o e^(g+n)te^v_t; v_t = r_g v_t-1 + e_gt, where e_gt is 'white noise'.

Households

There are H households in total, where H is fixed. Each household consists of N_t/H members, where N_t is the population. N_t grows at a constant population growth rate n

N_t = N_o e^nt

Households supply to factor markets their labour and all capital assets they have accumulated. The inputs are either hired by firms or withdrawn by households (where the wage rate is below the household's reservation wage). Since capital assets have no other use, their reservation wage is zero and they are all employed by firms. Capital is accumulated by households according to the law of motion (stated here for the aggregate capital stock):

K_t+1 = K_t (1-d) + I_t

= K_t (1-d) + Y_t - C_t - G_t.

Each household makes decisions as a household on how much to consume or save, and on how to allocate time between work and leisure in each and every period. This set of decisions is made to maximize a lifetime private utility function subject to the household's lifetime budget constraint. The household also gets satisfaction from the supply of government public works and services, but that utility is assumed separable from private utility (that is, has no impact on any private marginal utilities of any period), so we ignore it as irrelevant to household decisions. The lifetime private utility function is specified per member of the household and then multiplied by household size (remember, all members are identical)

U = S^¥_t=0 [1/(1+l)^t u(c_t, 1-l_t ] N_t/H

Where c is consumption per member, and l is labour supply so that 1-l is leisure (total time available per year is normalized or re-scaled to equal 1, so l is the fraction of the year spent working), u(.) is utility per member, and l is the subjective rate of time discount that converts future utilities into present utility equivalents (so they can be added up).

To be specific and make the model tractable mathematically, we impose a specific (logarithmic) functional form on the utility function:

u(c, 1- l) = ln c + b ln (1- l)

Where b is a taste parameter (formally, the elasticity of indifference curves in c, 1- l space). (This utility function has the neat property that du/dc = 1/c, and du/d(1- l ) = b/(1- l))

Households have a lifetime budget constraint that the present value of their expected lifetime consumption choices c_o, c₁, c₂, ... c_¥ not exceed the present value of their lifetime after-tax income from labour supply A_ow_ol _o, A₁w₁l ₁, .... A_¥w_¥l _¥, all present value calculations being made with the expected yields from current and future capital r₀, r₁, r₂, ... r_¥. Let the yield r_t refer to the yield on saving in year t, which is the marginal product of extra capital in year t+1.

(b) Solutions

Government determines G according to the stochastic rule for G, and sets taxes to match.

Firms meet the government's demand for output, and with the capacity that is left they produce private output so as to satisfy their first-order conditions: they employ labour up to the point where the wage rate w equals the marginal product of (unskilled) labour:

w_t = F_Lt = (1-a) [K_t/A_tL_t]^a A_t

Firms employ capital up to the point where its rental rate r + d equals its gross marginal product:

r_t + d = F_kt = a [A_tL_t/K_t]^1-a

Firms' aggregate output uses up all the labour and capital supplied at those wages and rental rates, so there is full employment at all times.

Households' behaviour is more complex, because they make so many more decisions. They choose lifetime time paths for consumption (and therefore for saving and the accumulated capital stock), and for leisure and therefore labour supply. Their decisions are best described by looking at three first-order conditions or points of tangency of their feasible set with their indifference curve map.

Within any period households trade off more consumption for less leisure (that is, for more labour supply to earn the income for more consumption) to the point where the marginal utility of consumption per member equals the wage rate times the marginal utility of leisure, which is equivalent to (with logarithmic utility)

c_t / (1-l_t) = A_t w_t / b

This choice is illustrated (in microeconomics) on the indifference curve diagram used to talk about labour supply, Figure 1. The frontier of available combinations of c and leisure 1-l is defined by the wage rate A_tw_t which translates less leisure into greater income to finance extra consumption. The outcome is at the point of tangency of the frontier with the highest accessible indifference curve. That solution point is described by the ratio c_t/ (1-l_t). Since this ratio is just vertical distance over horizontal distance of the optimal point B from the origin, it can be thought of as the slope of a line from the origin to the optimal point B in Figure 1.

Between periods households trade off labour or leisure now for labour or leisure later (holding lifetime consumption choices constant), and consumption or saving now for consumption or saving later (holding lifetime labour supply constant). The first choice can be illustrated quite simply if we consider only two periods (now (t) and next year (t+1)) and if we treat the future as uncertain. Households' first-order condition for the choice between holidays now versus holidays later becomes

(1-l_t+1)/(1-l_t) = [(1+r_t) A_tw_t] / [(1+l) A_t+1w_t+1]

where the ratio on the left is of future to current leisure at the optimal point, r_t is the return on saving in period t and the yield on extra capital in period t+1, and l is the subjective rate of discount of future utility.

This choice is illustrated in Figure 2. The frontier gives combinations of current and future labour supply that will leave total lifetime income unaffected (to hold lifetime consumption unchanged), so its slope is (1+r_t) A_t W_t / A_t+1 w_t+1, reflecting technology levels and the yield r_t. The indifference curves have slope (1-l_t+1)(1+l)/(1-l_t), reflecting logarithmic utility and the subjective rate of time discount l. The optimal solution is at the point of tangency B. The expression on the left is a ratio of optimal distances vertically and horizontally from the origin to point B, which can be interpreted as the slope of a line from the origin through point B. The expression on the right depends on technology and the marginal product of capital in period t+1.

Households' choice of consumption now versus consumption later can similarly be illustrated simply if we take only two periods and again ignore the uncertainty of future stochastic shocks to productivity and government demands. The first-order condition becomes

c_t+1 / c_t = (1+r_t)/(1+l)

This choice can be illustrated by another figure that would look almost identical to Figure 2, with the axes relabeled with c instead of 1-l for each period. The optimal choice of current consumption and therefore of current saving and the future capital stock depends on the yield on capital r_t.

(c) Implications for macroeconomic behaviour

So far we have just been deriving individual behaviour. However, since all individuals are identical, it is easy to aggregate to economy-wide behaviour: just multiply all individual decisions by the population N_t. (That is the attraction of working with identical-agent models!). Let aggregate labour supply be L, and aggregate consumption C.

How do these macroeconomic outcomes behave? Let's think just about total income Y. It is always in general equilibrium, because prices and wages always adjust (quickly) to ensure balance of demand and supply in all markets. All that can disturb this equilibrium are the trend changes and the shocks u_t and v_t to technology and government demand for resources. Shocks to government demand affect households by raising their utility (in ways that do not affect any other household choices at all, by our assumption that G enters household utility functions separately from consumption and leisure) and by raising taxes (which lowers households' lifetime income constraint by the present value of those taxes). Shocks to technology, by contrast, affect the marginal products of labour and capital, and persistently over at least two periods.

The impacts of a positive technology shock (spread over two periods, since the shocks are assumed to have that much persistence) are sketched out in the arrow diagram below.

u_t>0 ==> A ==> Fk = r_t ==> ¯ C_t, ==> K_t+1 (in a C_t, C_t+1 figure, by net of income less substitution effect)

==> Y_t+1 by marginal product of extra capital supply

C_t+1 in the same figure by income plus substitution effect

==> L_t in Figure 2, by net of substitution less income effect

==> Y_t by marginal product of extra labour supply

==> ¯ L_t+1 in Figure 2, by income plus substitution effects

==> ¯ Y_t+1 by marginal product of drop in labour supply

==> A_t w_t ==> C_t (in Figure 1, by both income and substitution effect)

==> L_t, ¯ L_t+1 via income and substitution effects in Figure 2,

==> Y_t, ¯Y_t+1 by marginal products of changes in labour supply

u_t+1 (from persistence) ==> same effects in t+1, including effect on K_t+2 and therefore Y_t+2.

The chain of effects is entirely through the supply of inputs L and K, and the direct effect of technology itself in the production function. Effects on consumption are relevant only as indirect evidence of changes in savings, which in turn is relevant only because it represents a change in future capital. Also, for effects on household choices of L and K to be large enough to matter, households must be quite sensitive to the slopes of frontiers facing them. For instance, households must increase their labour supply significantly when the interest rate they can earn on current income goes up, or else the effects of technology shocks on income will be quite small.

Why does this matter? Because the job of this RBC theory is to explain the actual fluctuations of real GDP, which are quite large, with technology shocks that have generally not been all that large.

An important result of the arrow diagram above is that a shock to technology that persists for only one period has effects on Y that persist for two periods. That is, an AR(1) process for technology shocks produces an AR(2) process for real income. That is important because the actual real GDP series is well described by an AR(2) process. AR(1) processes can only have time patterns of geometric decay following a shock..AR(2) processes can have time patterns of adjustment that have humps - that is, the adjustments first rise, then fall. Many actual patterns of adjustment are hump-shaped.

(d) Testing the RBC models against reality

The claims of RBC modelers are not that they can replicate time series paths that mimic the actual clockwise cycles we see in inflation and unemployment. Their models are too stylized and abstract for that (there is no institutional detail, for instance), and the modeling approach demands that the models be stylized and abstract or else the models are impossible to solve. Instead, they claim that the processes included in their models are capable of generating time series behaviour with similar properties to those of the actual historical data. Similar properties? They mean relative variability of different parts of the model (real GDP, consumption, interest rates, wage rates, productivity levels, etc.), and patterns of correlation among those endogenous variables.

To test these claims, they specify levels of taste and technology parameters (such as b, l, and a in our simple model) and then have a computer simulate the model's behaviour over some period like 20 years, for thousands of simulations. Each simulation draws a different sample of technology and government spending shocks. From the set of simulated time paths, they then calculate the pattern of variances and covariances of important variables. So the output of an RBC model is not an estimated macro model, or a set of forecasts, but a table of covariances.

The drawbacks of the RBC approach are well discussed by Mankiw and Scarth in the reading on reserve. Those drawbacks are serious enough that many macroeconomists are drifting away from it as an unpromising line of research. However, the RBC approach of deriving macroeconomic behaviour from the rational optimizing behaviour of micro agents, in fully specified microeconomic contexts, is one that has caught on with many macroeconomists who do not also adopt the approach of assuming prices to be always flexible. So there are more and more Keynesian modelers whose work looks something like that of RBC modelers. Expect to see more of this micro-style macro modeling if you go on to graduate work in economics.