Thirdly, there are no large asymmetries between rises and falls in output. Output growth is distributed roughly symmetrically around its mean. The general path is relatively long periods above the mean and brief periods relatively far below it. A fourth set of facts is that it seems fluctuations have dampened over time. The collapse of the economy into the Depression and the rebound of the 1930s and WWII dwarf any fluctuations before or since. Real GDP fell by 27% between 1929-1933 then rose at an average annual rate of 10% over the next 11 years.
5.2 An Overview of Business-Cycle Research
It is natural to begin our study of aggregate fluctuations by asking whether they can be understood using a Walrasian model-- that is, a competitive model without any externalities, asymmetric information, missing markets, or other imperfections. If they can, then analysis of these fluctuations may not require any fundamental departure from microeconomic analysis.
The Ramsey model is the natural Walrasian baseline model of the aggregate economy, excluding not only imperfections but all issues of heterogeneity. To look at fluctuations, we must add a source of disturbances-- usually to the economy's technology or government purchases. We also must adjust for variations in employment. Fully specified general-equilibrium models of fluctuations are known as dynamic stochastic general-equilibrium (or DSGE) models.
5.3 A Baseline Real-Business-Cycle Model
The model is a discrete-time variation of the Ramsey model. The economy consists of a large number of identical, price-taking firms and a large number of identical, price-taking households, with infinite lifetimes. The production is Cobb-Douglas, thus output in period t is
\(K_t = K_t ^\alpha (A_t L_t) ^{1 - \alpha}, 0 < \alpha < 1\)
Output is divided among consumption, investment and government purchases. Capital depreciates each period (by the fraction delta). Thus capital stock in period t + 1 is
\(K_{t + 1} = K_t + I_t - \delta K_t\)
\(= K_t + Y_t - C_t - G_t - \delta K_t\)
The government's purchases are financed by lump-sum taxes that equal their purchases each period. Labor and capital are paid their marginal products. Thus the real wage and the interest rate in period t are
\(w_t = (1 - \alpha) K_t ^\alpha (A_t L_t)^{- \alpha} A_t\)
\(= (1 - \alpha) (\frac{K_t}{A_t L_t}) ^\alpha A_t\)
\(r_t = \alpha (\frac{A_t L_t}{K_t})^{1 - \alpha} - \delta\)
The representative household maximizes the expected value of
\(U = \sum _{t = 0} ^\infty e^{- \rho t} u(c_t, 1 - \ell _t) \frac{N_t}{H}\)
where \(\rho\) is the discount rate, \(N_t\) is the population, \(H\) is the number of households, \(c\) is consumption per member of the household, and \(\ell _t\) is the amount each member works. Since all households are the same, we can simplify by dividing by N and taking logs:
\(u_t = \ln c_t + b \ln (1 - \ell _t), b > 0\)
Finally, the two driving variables are technology and government purchases. To capture growth, assume that g is the growth rate of technological progress but that it is also subject to random disturbances:
\(\ln A_t = \bar A + g t + \tilde{A_t}\)
Where the last term reflects departures from trend. It follows a first-order autoregressive process, that is,
\(\tilde{A_t} = \rho _A \tilde{A_{t - 1}} + \epsilon_{A,t}, -1 < \rho _A < 1\)
where the last term are the white-noise disturbances -- a series of mean-zero shocks that are uncorrelated with each other. In words, the last equation states that the random component equals the fraction of the previous period's value (\(\rho _A\)) plus a random term. If the fraction \(\rho _A\) is positive, then the effects of a shock gradually disappear over time.
Government purchases function similarly. The trend growth rate of per capita government purchases equals the trend growth rate of technology (so they fit the economy). Thus,
\(\ln G_t = \bar G + (n + g)t + \tilde{G_t}\)
\(\tilde{G_t} = \rho_G \tilde{G_{t - 1}} + \epsilon _{G, t}\)
where the last term is the white-noise disturbances.
5.4 Household Behavior
The 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
First we'll consider the case where the household lives only for one period, has no initial wealth, and only one member. In this case, the household's budget constraint will be \(c = w \ell\). The Lagrangian for the household's maximization problem is
\(\mathcal{L} = \ln c + b \ln (1 - \ell) + \lambda (w \ell - c)\)
The first-order conditions for c and \(\ell\), respectively, are:
\(\frac{1}{c} - \lambda = 0\)
\(- \frac{b}{1 - \ell} + \lambda w = 0\)
Since we have the budget constraint above, we can see that \(\lambda = \frac{1}{w \ell} \). Substituting this into the last equation yields
\(- \frac{b}{1 - \ell} + \frac{1}{\ell} = 0\)
Now, wage does not enter. Intuitively, utility is logarithmic in consumption and the household has no initial wealth, thus the income and substitution effects of a change in the wage offset each other.
The household's lifetime budget constraint is now
\(c_1 + \frac{1}{1 + r} c^2 = w_1 \ell _1 + \frac{1}{1 + r} w_2 \ell _2\)
The Lagrangian is
\(\mathcal{L} = \ln c_1 + b \ln (1 - \ell _1) + e^{- \rho} [\ln c_2 + b \ln (1 - \ell _2)] + \lambda [w_1 \ell _1 + \frac{1}{1 + r} w_2 \ell _2 - c_1 + \frac{1}{1 + r} c^2 ]\)
The household's choice variables are the period consumptions and the labor times. We can show the effect of the relative wage in the two periods on relative labor supply by taking the first-order conditions for labor period 1 and 2. The implications are:
\(\frac{1 - \ell _1}{1 - \ell _2} = \frac{1}{e^{- \rho} (1 + r)} \frac{w_2}{w_1}\)
This implies that the relative labor supply in the two periods responds to the relative wage. If, for example, wage 1 rises relative to wage 2, the household decreases first-period leisure relative to second-period leisure; or, increases first-period labor relative to second-period supply. This also means that a rise in the real interest rate raises first-period labor supply relative to second-period supply since the attractiveness of working today is increased to be able to save more when the rate goes up. These responses are known as intertemporal substitution in labor supply.
Household Optimization under Certainty
Secondly, the household faces uncertainty about rates of return and future wages. Because of this, households do not choose deterministic paths for consumption and labor supply. Instead, its choices of consumption and labor potentially depend on all the shocks to technology and government purchases up to that date.
Consider the household in period t. Simplifying this condition to only expectations will go to:
\(\frac{1}{c_t} = e^{- \rho} E_t [\frac{1}{c_{t + 1}} (1 + r_{t + 1})]\)
This is similar to the equation in the Ramsey model. It represents the tradeoff between present and future consumption, depending not just on expectations of future marginal utility and the rate of return but also on their interaction.
The Tradeoff between Consumption and Labor Supply
The household gets to choose its labor as well as consumption, thus they are interrelated. If a household is behaving optimally, a marginal change between the two must leave expected utility unchanged.
We can equate cost and benefit then simplify:
\(\frac{c_t}{1 - \ell _t} = \frac{w_t}{b}\)
This relates current leisure and consumption, given the wage. Because it involves current variables, uncertainty does not enter.