Macro-Economic Modelling 2
Short Term Dynamics: Fluctuations and the Business Cycle
The previous chapter has focused on long-term dynamics and the process of capital accumulation. One is however often interested in the short-term dynamics of business cycles, that is fluctuations in output and employment around a long-term trend.Some stylised facts characterising the expansions and contractions of the business cycle are:
- Fluctuations are recurrent but with no trivial cyclical pattern, and hence hard to predict
- Persistence of both expansions and contractions through time
- Employment and consumptions are procyclical, that is they co-move with output
- Real wages exhibit little fluctuation with and are only weakly procyclical if at all
By and large there exist two main schools of thought ...
The idea that business cycles are driven by technological changes dates back to Schumpeter (1939) [2]
The Real Business Cycle model
The so called Real busiless cycle model (RBC) was introduced in 1982 by Kydland and Prescott [3] building on a discrete-time version of the neoclassical growth model framework. Crucially however, uncertainty is introduced in the form of exhogenous productivity shocks, so that log-productivity \(\log(A_t)\) follows the \(AR(1)\) process \begin{equation} \label{eq:productivity_process} \log(A_{t+1}) = \varphi \log(A_{t}) +\varepsilon_{t+1}. \end{equation} Moreover, a choice between the fraction of time devoted to labour \(l_t\) and leisure \(h_t\) is also introduced with the natural constraint \(l_t+h_t = 1\). We consider a Cobb-Douglas per-capita production function \begin{equation} f(k_t, l_t, A_t) = A_t k_t^\alpha l_t^{1-\alpha}, \end{equation} with as usual \(\alpha\) being the output elasticity of capital. Because of the stochasticity of the productivity process \eqref{eq:productivity_process} consumers will aim at maximising expected utility, so that the optimisation problem reads \begin{equation} \label{eq:rbc_opt_problem} \boxed{ \begin{split} &\max_{c_t, h_t}\mathbb{E}\left[\sum_t\beta^t\left((1-\eta)u(c_t)+\eta\nu(h_t)\right)\right]\\ &\phantom{xxx}\text{subject to}\\ &k_{t+1} = l_t w_t + (1+r_t)k_t - c_t \end{split} } \end{equation} with \(\beta\) the discount factor. Notice that the household has two control variables: consumption \(c_t\) and leisure \(h_t\). Here \(0 < \eta < 1\) is a mixing parameter determining the relative relevance of consumption utility versus leisure utility in the total utility function ...Problem \eqref{eq:rbc_opt_problem} has Bellman equation \begin{equation} V(k_t) = \max_{}\Bigl\{(1-\eta)u(c_t)+\eta\nu(h_t)+\beta\mathbb{E}\left[V(k_{t+1})\right]\Bigr\}. \end{equation} For convenience, denote the right-hand side argument of \eqref{eq:opt_rbc_bellman} by \(\overline{V}_t = (1-\eta)u(c_t)+\eta\nu(h_t)+\beta\mathbb{E}\left[V(k_{t+1})\right]\). Notice that \begin{equation} \frac{d \overline{V}_t}{d c_t} = (1-\eta)u'(c_t)-\beta\mathbb{E}[V'(k_{t+1})], \end{equation} so that \(\overline{V}_t\) is maximised with respect to consumption \(c_t\) for \begin{equation} \label{eq:opt_rbc_bellman} \beta \mathbb{E}[V'(k_{t+1})] = (1-\eta)u'(c_t). \end{equation} Then, observe that \begin{equation} \label{eq:V_prime} \frac{d \overline{V}_t}{d k_t} = \beta\mathbb{E}[V'(k_{t+1})](1+r_t). \end{equation} Thus, \(V'(k_{t+1}) = (1-\eta)u'(c_{t+1})(1+r_{t+1})\), and from \eqref{eq:opt_rbc_bellman} one has \begin{equation} \begin{split} u'(c_t) &= \beta\frac{\mathbb{E}[V'(k_{t+1})]}{1-\eta}\\ &=\beta\frac{1}{1-\eta}\mathbb{E}[(1-\eta)u'(c_{t+1})(1+r_{t+1})]\\ &=\beta(1+r_{t+1})\mathbb{E}[u'(c_{t+1})] \end{split}\quad . \end{equation} Similarly, for labour-leisure choice one has \begin{equation} \label{eq:opt_rbc_bellman2} \frac{d V(k_t)}{d h_t} = \eta\nu'(h_t)-\beta\mathbb{E}[V'(k_{t+1})]w_t = 0, \end{equation} meaning that \(\beta\mathbb{E}[V'(k_{t+1})] = \eta\nu'(h_t) / w_t\). Therefore, putting this result together with \eqref{eq:V_prime}: \begin{equation} V'(k_{t+1}) = \frac{\eta\nu'(h_{t+1})}{w_{t+1}} (1+r_{t+1}) \end{equation} and \begin{equation} \begin{split} \nu'(h_t) &= \frac{w_t}{\eta}\beta\mathbb{E}[V'(k_{t+1})]\\ &= (1+r_{t+1})\beta\mathbb{E}[\frac{w_t}{w_{t+1}}\nu'(h_{t+1})] \end{split}\quad. \end{equation} That is to say, the Euler equations are: \begin{equation} \label{eq:rbc_euler_equations} \begin{split} u'(c_t) &= (1+r_{t+1})\beta\mathbb{E}[u'(c_{t+1})],\\ \nu'(h_t) &= (1+r_{t+1})\beta\mathbb{E}[\frac{w_t}{w_{t+1}}\nu'(h_{t+1})]. \end{split} \end{equation} Furthermore, notice that \eqref{eq:opt_rbc_bellman} and \eqref{eq:opt_rbc_bellman2} taken together imply \begin{equation} \label{eq:rbc_labour_supply0} (1-\eta)u'(c_t)w_t = \eta\nu'(h_t) = \eta\nu'(1-l_t), \end{equation} which provides an equation for the supply of labour \(l_t\).
In order to clarify the meaning of this result it is useful to look at the deterministic case with log-utility: \begin{equation} U(c_t, h_t) = (1-\eta)u(c_t) + \eta \nu(h_t) = (1-\eta)\log(c_t) + \eta \log(h_t). \end{equation} In this case, equations \eqref{eq:rbc_euler_equations} become \begin{equation} \begin{split} \frac{c_{t+1}}{c_{t}} &= (1+r_{t+1})\beta,\\ \frac{h_{t+1}}{h_{t}} &= (1+r_{t+1})\beta\frac{w_t}{w_{t+1}}. \end{split} \end{equation} Therefore, higher wages increase the supply of labour proportionally to the level of interest rates. The labout supply \eqref{eq:rbc_labour_supply0} then reads \begin{equation} \label{eq:rbc_labour_supply} l_t = 1-\frac{\eta}{1-\eta} \frac{c_t}{w_t}. \end{equation} Notice that the labour-leisure choice is therefore static, being determined in each specific period from only concurrent states.
Summing up, the RBC model is therefore specified by the dynamical system: \begin{equation} \label{eq:RBC_system_of_dyn_eqs} \begin{cases} y_t = A_t k_t^\alpha l_t^{1-\alpha}, & \text{Production function}\\ r_t = \alpha A k_t^{\alpha-1}l^{1-\alpha} - \delta, & \text{Real Rate of Interest}\\ w_t = (1-\alpha)y_t, & \text{Wage}\\ c_{t+1} = \beta(1+r_{t+1}) c_t, & \text{Euler Equation for Consumption}\\ k_{t+1} = (1 - r_t)k_t + l_tw_t - c_t, & \text{Equation of Motion for Capital}\\ l_t = 1-\frac{\eta}{1-\eta} \frac{c_t}{w_t}, & \text{Labour Supply}\\ \log(A_{t+1}) = \varphi \log(A_{t}) +\varepsilon_{t+1}, & \text{Log-Productivity AR(1) process} \end{cases} \end{equation} Systems like \eqref{eq:RBC_system_of_dyn_eqs} do not admit closed form solutions, so that the standard approach is to employ perturbation methods around the non-stochastic steady state, and by log-linearisation. The core idea, for a given state \(x_t\), is to consider the steady state \(\overline{x}\) and the log deviation from it \(\hat{x}_t\equiv\log\left(\frac{x_t}{\overline{x}}\right)\), so that one can write \(x_t = \overline{x}e^{\hat{x}_t} = \overline{x}(1+\hat{x}_t) + \mathcal{O}(\hat{x}_t^2)\), where the power series expansion has been truncated to first order, but can in principle be taken at arbitrarily higher orders. For more details, the interested reader can refer to xxx.
Here however, we will solve the model through the convenient Julia library Macromodelling.jl [5]. We write the dynamical equations \eqref{eq:RBC_system_of_dyn_eqs} adding for completeness the (redundant) implicit equation for the saving rate \(s_t\). The choice of the parameters is taken from the empirical literature [6].
using MacroModelling
import StatsPlots
@model RBC begin
y[0] = A[0] * k[0]^α * ℓ[0]^(1-α) # Production Function
r[0] = α * A[0] * k[0]^(α-1) * ℓ[0]^(1-α) - δ # Real Rate of Interest
w[0] = (1-α) * y[0] # Wage: Marginal Product of Labour
c[1] / c[0] = β * (1+r[1]) # Euler Equation for Consumption
k[0] = (1 - r[-1]) * k[-1] + ℓ[-1]*w[-1] - c[-1] # EoM
ℓ[0] = 1-η/(1-η) * c[0] / w[0] # Labout Supply
log(A[0]) = ϕ * log(A[-1]) + σᶻ * ε[x] # AR(1)
# -----------------------
c[0] = (1-s[0]) * y[0] # Saving rate
end
@parameters RBC begin
σᶻ= 0.00763
ϕ = 0.9
δ = 0.025
α = 0.36
β = 0.99
η = 0.497
end
| Variable | Std [%] | \(\rho_{\cdot, y}\) | \(\rho_{\cdot, A }\) | \(\rho_{\cdot, c }\) | \(\rho_{\cdot, k }\) |
|---|---|---|---|---|---|
| \(y\) | 1.8 | 1 | 0.51 | 0.83 | x |
| \(A\) | 0.03 | 1 | 0.28 | x | |
| \(c\) | 1.7 | 1 | 1.91 | ||
| \(k\) | 25 | 1 |
Money neutrality - xxx
The New Keynesian model
If the RBC framework is based on the notion that business cycles are optimal adjustment of the economy to external shocks, the New Kensian view is instead characterised at its core by a prominent role of imperfect competition.For simplicity we start by considering a non-stochastic continuous time model: this is helpful to fix ideas, and to highlight the core dynamics. Then, as done before, we will consider a discrete-time stochastic version which lends itself more to practical applications.
Consider an economy with no capital, where households provide labour \(L\) in exchange of wage \(W\), can invest in a bond \(B\) paying a nominal interest rate \(i\), and maximise utility with respect to both consumption \(C\) and labour. The households' maximisation problem is \begin{equation} \label{eq:nk_opt_problem} \boxed{ \begin{split} &\max_{C, L}\int_0^\infty e^{-\rho t}u(C(t), L(t)) dt\\ &\phantom{xxx}\text{subject to}\\ &P(t)C(t) + \dot{B}(t) = i(t) B(t) + W(t)L(t) \end{split} } \end{equation} The present-value Hamiltonian is \begin{equation} H = u(C(t), L(t))e^{-\rho t}+\lambda(t)\left[iB+WL-PC\right], \end{equation} with associated co-state equation \begin{equation} \label{eq:costate_nk0} \dot{\lambda} = -\frac{\partial H}{\partial B} =-\lambda(t) i(t). \end{equation} On the other hand, Pontryagin's maximum principle requires \begin{equation} \frac{\partial H}{\partial C} = e^{-\rho t} \partial_C u - P(t)\lambda(t)= 0 \end{equation} that is \begin{equation} \label{eq:pmp_nk0} \lambda(t) = \frac{\partial_C u }{P(t)}e^{-\rho t} . \end{equation} Taking the first order time derivative and accounting for \eqref{eq:pmp_nk0}, one finds \begin{equation} \frac{\dot{\lambda}}{\lambda} = - \frac{\dot{P}}{P} + \frac{\partial_C^2 u}{\partial_C u}\dot{C}-\rho. \end{equation} The inflation rate \(\pi(t)\) is defined by \begin{equation} \label{eq:inflation_rate_definition} \pi = \frac{\dot{P}}{P}. \end{equation} Therefore, pluggin in \eqref{eq:costate_nk0} and \eqref{eq:inflation_rate_definition}, and recalling the identification \(\partial_Cu/\partial^2_Cu = -\sigma C \) with \(\sigma\) the elasticity of intertemporal substitution in consumption, one finally obtains the Euler equation for consumption: \begin{equation} \frac{\dot{C}}{C} = \sigma\left(i-\pi-\rho\right). \end{equation} Clearing in the goods markets requires that all goods produced are consumed, that is \(Y(t)=C(t), \quad \forall t\).
\begin{equation} \dot{Y} = \sigma(i_t - \pi_t-\rho) \end{equation} Assume there exists a natural level of output \(\overline{Y}\) so that one can deine a notion of output gap \begin{equation} X_t \equiv \frac{Y_t}{\overline{Y}} \end{equation} with dynamics described by \begin{equation} \frac{\dot{X}_t}{X_t} = \frac{\dot{Y}_t}{Y_t}-g \end{equation} with \(g\) the percentage growth rate of the natura level of output, depending on preferences and productivity growth. Finally, defining the so called Wicksellian interest rate also known as natural interest rate \(r^n\equiv \rho+\frac{g}{\sigma}\) \begin{equation} \boxed{ \dot{x}_t = \sigma (i_t - \pi_t - r^n) } \end{equation} On the supply side we consider a continuum of monopolistically-competitive firms indexed by \(j \in [0,1]\), producing different goods using the same technology according to the production function \begin{equation} Y_j(t) = A L_j(t)^{1-\alpha}, \end{equation} and setting their price \(P_j(t)\). Aggregate output is thus \(Y(t) = \int_0^1Y_j(t) dj\).
Following Calvo [8], each firm may adjust its price at any point in time with probability \(\theta\) and independently from previous price adjustments and other firms; that is to say, price adjustments follow independent Poisson processes. Denote the log-price of firm \(j\) at time \(t\) by \(p_j(t)=\log{P_j(t)}\), and the log-price set by firm \(j\) when it adjusts as \(p^*(s)\), independent of which firm is adjusting. Further, define the indicator \(I_j(t)\) to unity if firm \(j\) adjusts its price at time \(t\), zero otherwise. Then one can write the current price of firm \(j\) as a sums of contributions of all past price settings weighted by the probability that they remain unchanged until the current time \(t\): \begin{equation} p_j(t) = \int_{-\infty}^t p^*(s)I_j(s)e^{-\theta(t-s)} ds. \end{equation} The average price set by adjusting firms at time \(s\) is \(\int_{0}^1 p^*(s)I_j(s) dj = p^*(s)\theta\). Therefore, the (log) aggregate price level is \begin{equation} \label{eq:log_price_level_calvo} \begin{split} p(t) &\equiv \int_0^1 p_j(t) dj\\ &= \int_0^1\int_{-\infty}^t p^*(s)I_j(s)e^{-\theta(t-s)} ds dj\\ &= \int_{-\infty}^t p^*(s)\left(\int_0^1I_j(s)dj \right)e^{-\theta(t-s)} ds\\ &= \theta\int_{-\infty}^t p^*(s) e^{-\theta(t-s)} ds . \end{split} \end{equation} The optimal price adustment \(\delta p_j(t) = p^*(t)-p_j(t)\) chosen by each firm at time \(t\) is computed as the sum of all future discounted price adjustments and output gaps weighted by the probability they will have to adjust in the future REF. That is, in terms of aggregate price adjustment \(\delta p(t) = p^*(t)-p(t)\): \begin{equation} \label{eq:log_price_adjustment_calvo} \delta p(t) = \theta\int_{t}^\infty e^{-\rho(s-t)}\left(\delta p(s)+x(s)\right)e^{-\theta(s-t)} ds. \end{equation} Differentiating \eqref{eq:log_price_level_calvo} and \eqref{eq:log_price_adjustment_calvo} therefore yields \begin{equation} \begin{split} \dot{p}(t) &= \theta\delta p(t)\\ \dot{\delta p}(t) &= -\theta x(t) +\rho\delta p(t) \end{split} \end{equation} which together, and recalling definition \eqref{eq:inflation_rate_definition} for the inflation rate, give \begin{equation} \dot{\delta p}(t) = -\theta x(t) +\frac{\rho}{\theta}\pi(t). \end{equation} Finally, from the second derivative of \eqref{eq:log_price_level_calvo} we obtain \begin{equation} \label{eq:nk_phillips_curve} \boxed{ \dot{\pi}(t) = \rho \pi(t) -\theta^2x(t) } \end{equation} which is known as the New Keynesian Phillips curve. Integration of \eqref{eq:nk_phillips_curve} forward in time yields the solution \begin{equation} \pi = \theta^2\int_t^\infty x(t) e^{-\rho(s-t)}ds \end{equation} revealing that inflation can be interpreted in this model as the present discounted value of future expected output gaps.
At this point it is worth noticing that the form found in \eqref{eq:nk_phillips_curve} does not depend in general on the specific choice of the pricing model. The Calvo model is an example of a time-dependent model, but the very same dynamics can be obtained under state-dependent models (such as the Rotemberg model [12] assuming quadratic costs of price adjustments) in which price changes are determined by economic state-contingent factors. In particular, it is not much the Poisson property which is important, but rather the stickiness of prices, which breaks money neutrality.
References and notes
[1] GDP: GDP; Investment (Gross Private Domestic Investment): GPDI; Wage per Hour (Average Hourly Earnings of Production and Nonsupervisory Employees, Total Private): AHETPI; Hours worked (Nonfarm Business Sector: Hours Worked for All Workers): HOANBS; UNRATE: UNRATE; TFP (Total Factor Productivity): Federal Reserve Bank of San Francisco, quarterly TFP.[2] "Business Cycles: A Theoretical, Historical, and Statistical Analysis of the Capitalist Process", Joseph Alois Schumpeter (1939).
[3] "Time to Build and Aggregate Fluctuations", Finn E. Kydland, Edward C. Prescott, Econometrica, Vol. 50, No. 6 (1982)
[4] "", .
[5] "MacroModelling.jl: A Julia package for developing and solving dynamic stochastic general equilibrium models", T. Kockerols, (2023), Journal of Open Source Software, 8(89), 5598, https://doi.org/10.21105/joss.05598.
[6] See for instance [7]. The parameter \(\eta\) can instead be calibrated so as to allow for a labour supply of \(1/3\) (i.e. 8h per day). At first approximation notice that assuming \(c_t=w_t\) in \eqref{eq:rbc_labour_supply} yields \(\eta = 0.4\).
[7] "The labor market in real business cycle theory", Gary Hansen and Randall Wright, Quarterly Review, 1992, vol. 16, issue Spr, 2-12.
[8] "Staggered prices in a utility-maximizing framework",Guillermo A. Calvo, Journal of Monetary Economics, Volume 12, Issue 3, September 1983, Pages 383-398 .
[9] By the law of large numbers.
[10] "Nominal price rigidity, money supply endogeneity, and business cycles", Tack Yun, Journal of Monetary Economics, Volume 37, Issue 2, April 1996, Pages 345-370.
[11] "", .
[12] "Monopolistic Price Adjustment and Aggregate Output", Julio Rotemberg, Review of Economic Studies 49, 517–531 (1982).
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