14. Real Business Cycle Model

Suppose we are interested in \(y_t\), the real GDP per capita. We first detrend \(y_t\) to get the business cycle part \(\{d_t\}\). Then, we establish our theoretical model for the economy with consumers and firms and some stochastic shocks to theoretically predict the properties of \(\{d_t\}\). Finally, we compare the theoretical and empirical \(\{d_t\}\) to see how well our model works.

14.1 Choose Preferences and Technology

If we impose conditions for the Second Welfare Theorem to hold, then we can focus on the social planner's problem with a representative consumer and firm whose solution is Pareto efficient and thus achievable by lump-sum transfers and competition.

Let the preference of the representative consumer be represented by \(u(c_t,l_t)\) where \(c_t\) is consumption and \(l_t\) is the proportion of time endowment used for leisure. Let's choose the representative firm's technology as $$y_t=e^{z_t}F(k_t,n_t)$$ where \(k_t\) is capital and \(n_t\) is the proportion of time endowment used to work. Note that: - \(c,l,k,n\) are all per capita basis. - \(n+l=1\). - \(u(c,l)\) should be consistent with long run growth: the long run growth has the property that after taking out the stochastic shocks \(z\), \(c\) is growing at constant rate while \(l\) remains constant. - \(F(k,n)\) is constant return to scale (CRS) so that we can only care about the per capita production. - function \(F\) and \(u\) are stationary over time. - the productivity shock \(\{e^{z_t}\}_{t=0}^\infty\) is a stochastic first-order Markov process; to match the business cycle facts empirically, \(\{e^{z_t}\}_{t=0}^\infty\) is always set as an AR(1).

14.2 Market Structure

• Without loss of generality, we let the household own all the capital stock, and they rent it out to the firms for rents.
• The household supplies \(n\) amount of labor and receive wage from the firm.
• The household owns the firm completely, but because of the CRS property of the firm's technology, profits are depleted by the payment to factors, which implies that profits are always zero under competition.

14.3 Household's Problem

The household solves $$\max_{\{c_t,k_{t+1}\}}\left\{\mathbb E\left[\sum_{t=0}^\infty\beta^t u(c_t,1-n_t)\right]\right\}$$ subject to $$c_t+(k_{t+1}-(1-\delta)k_t)=\underbrace{w_t n_t+r_t k_t}_{=e^{z_t}F(k_t,n_t)}+\underbrace{\text{firm's profit}}_{=0,\text{ CRS}}\Rightarrow c_t+(k_{t+1}-(1-\delta)k_t)=w_t n_t+r_t k_t$$ where \(w_t\) is wage, \(r_t\) is rent and \(\delta\) is the constant depreciation rate of capital.

14.4 Firm's Problem

The firm solves $$\max_{\{k_t,n_t\}}\left\{e^{z_t}F(k_t,n_t)-w_t n_t-r_t k_t\right\}$$ In equilibrium, $$r_t=e^{z_t}F_k(k_t,n_t),\qquad w_t=e^{z_t}F_n(k_t,n_t)$$

14.5.1 Social planner's Bellman equation

• State variable is \((k,z)\).
• The Bellman equation is $$V(k,z)=\max_{\{c,n,y\}}\left\{u(c,1-n)+\beta\mathbb E_{z'}[V(y,z')\mid z]\right\}\quad\text{s.t.}\quad c+(y-(1-\delta)k)=e^z F(k,n)\tag{14.1}$$ where \(\mathbb E_{z'}[V(y,z')\mid z]=\int V(y,z')q(z,z')dz'\).
• One way to solve (14.1) is to solve it piece by piece, i.e. we can first solve \(\max_{\{c,n\}}\{u(c,1-n)+\beta\mathbb E_{z'}[V(y,z')\mid z]\}\) s.t. \(c+(y-(1-\delta)k)=e^z F(k,n)\) for any given \(y\), and denote the solution by \(c^\star(y)\) and \(n^\star(y)\). Then plug \(c^\star(y)\) and \(n^\star(y)\) into (14.1) to obtain \(V(k,z)=\max_y\{u(c^\star(y),1-n^\star(y))+\beta\mathbb E_{z'}[V(y,z')\mid z]\}\) to solve for \(y^\star\).

14.5.2 First order conditions and envelop condition

Rewrite (14.1) as $$V(k,z)=\max_{\{c,n,y\}}\left\{u(c,1-n)+\beta\int V(y,z')q(z,z')dz'\right\}$$ where \(c=e^z F(k,n)-(y-(1-\delta)k)\) and \(y=e^z F(k,n)-c+(1-\delta)k\). The f.o.c. are $$[c]:\quad u_c(c,1-n)=\beta\int V_y(y,z')q(z,z')dz'\tag{14.2}$$ $$[n]:\quad u_l(c,1-n)=\beta\int e^z F_n(k,n)V_y(y,z')q(z,z')dz'\tag{14.3}$$ $$[y]:\quad u_c(c,1-n)=\beta\int V_y(y,z')q(z,z')dz'$$ Note that the \([c]\) f.o.c. (14.2) and the \([y]\) f.o.c. are the same because optimal \(y\) is pinned down if \(c\) and \(n\) are optimally chosen, so the \([y]\) f.o.c. is redundant. Or equivalently, (14.2) and (14.3) give $$\underbrace{\frac{u_l(c,1-n)}{u_c(c,1-n)}}_{\text{MRS}}=\underbrace{e^z F_n(k,n)}_{\text{MRT}}\tag{14.4}$$ where the LHS is the marginal rate of substitution (MRS) measuring the number of units of consumption to forego for one extra unit of leisure that makes the consumer equally happy, and the RHS is the marginal rate of transformation (MRT) measuring the number of units of consumption to forego for one extra unit of leisure that is technologically possible for this economy.

The envelop condition is $$V_k(k,z)=u_c(c,1-n)(e^z F_k(k,n)+(1-\delta))\tag{14.5}$$ Combining (14.5) and (14.2): $$u_c(c,1-n)=\beta\int V_y(y,z')q(z,z')dz'=\beta\int u_c(c',1-n')\left(e^{z'}F_k(k',n')+(1-\delta)\right)q(z,z')dz'$$ i.e. the Euler equation $$u_c(c,1-n)=\beta\mathbb E_{z'}\left[u_c(c',1-n')\left(e^{z'}F_k(k',n')+(1-\delta)\right)\mid z\right]\tag{14.6}$$

For computational purpose, we will specialize to the following convenient functional form: $$F(k,n)=Ak^\alpha n^{1-\alpha},\qquad u(c,1-n)=\ln(c)+\theta\ln(1-n)$$ To compute the model, we need to: - assume the shock process has a unique ergodic set. - calibrate the long run expected value so that in steady state we have \(z=\bar z=0\) and \(e^z=1\). - The next step is fixing \(z=0\) and finding the deterministic steady state, which is written as \((\bar k,\bar c,\bar n)\). - Then, we do the second-order Taylor expansion to (14.1) w.r.t. \(\tilde k\), to (14.4) w.r.t. \(\tilde n\), and to (14.6) w.r.t. \(\tilde c\) respectively to approximate the optimal path of convergence of \(k,n,c\) around the steady state after a small disturbance caused by \(z\). - Then we can randomly generate a series of shocks \(z\) according to the shock process to see how would \(k,n,c\) move and conclude whether their movements coincide with the empirical data. If not, we can tweak (calibrate) the coefficients of preference and technology to make them match. Then we can derive some implications from those tweaked parameters.

14.7 Recursive Competitive Equilibrium

If we have conditions such that the Second Welfare Theorem holds, then we can solve the above social planner's (representative agent's) problem to have a solution that is also a solution to the competitive equilibrium of that economy. However, if we do have - externality - money - taxes - price setting firms

and so on, then the Second Welfare Theorem fails, which means we cannot characterize a competitive equilibrium with a social planner's problem. Instead, we have to directly model the competitive equilibrium itself. To make that model tractable, we will use recursive competitive equilibrium as we did in §10.4.

14.8 Comparative Statics

For this section, we will do the analysis with the general functional form of \(F\) and \(u\). Let's first assume \(n=\bar n\), i.e. labor supply is inelastic. Then (14.2) and (14.3) imply that $$u_c(c,1-n)=\beta\mathbb E_{z'}[V_y(y,z')\mid z]\tag{14.7}$$ Figure (paraphrased): the horizontal axis is next-period capital \(y\). The RHS \(\beta\mathbb E_{z'}[V_y(y,z')\mid z]\) is decreasing in \(y\); the LHS \(u_c\) is increasing in \(y\) (\(y\uparrow\Rightarrow c\downarrow\Rightarrow u_c\uparrow\)). Their intersection is the optimal \(y^\star\). The figure also draws the shifted "LHS with larger \(z\)" and "LHS with larger \(k\)" curves, and \(y^\star\) and \(\hat y^\star\).

The intersection of RHS and LHS of (14.7) is the optimal choice of \(y^\star\). For \(\hat k>k\), the RHS doesn't move but the LHS moves down due to the concavity assumption of \(u\), so the new optimal choice \(\hat y^\star>y^\star\), which means \(g(k)=y^\star\) is increasing in \(k\). However, for a higher \(z\), the LHS moves down for the same reason but the direction of RHS movement is unclear because we don't know how \(\beta\mathbb E_{z'}[V_y(y,z')\mid z]\) depends on \(z\) and in this set-up it doesn't make sense to assume \(z_t\)'s are i.i.d..