26. RBC: Wedge Accounting

In section 25, we discussed a model that includes labor productivity shock. But that is not enough for the reality since we also have frictions (distortions) in consumption, labor supply, investment because of the taxes, government expenditure, net export shocks, banking system pessimism and so on. So, the wedge accounting model discussed in this section generalizes the model in section 25 by introducing four types (three additional variables to the model in section 25) of wedges to account for those frictions. Basically, wedge accounting models admit and introduce more variables to the model to make the model better mirror the reality.

26.1 The four wedges

• Labor wedge \(1-\tau_{h,t}\): \(\tau_{h,t}\) is the labor income tax or other frictions related to labor income that could be regarded as effective tax; the actual labor income of working \(h\) hours at wage \(w\) is \((1-\tau_{h,t})wh\).
• Effective wedge \(Z_t\): \(Z_t\) is the labor productivity as we have seen in section 25; when \(Z_t\) is low, it is not because people become stupider. It's just the economy channels money to less productive units.
• Investment wedge \(\dfrac{1}{1+\tau_{x,t}}\): \(\tau_{x,t}\) is the investment tax or other frictions related to investment that could be regarded as effective tax; the actual investment cost of having \(x\) more unit of capital is \((1+\tau_{x,t})x\).
• Government wedge \(G_t\): \(G_t\) is the government expenditure plus the net export.

26.2 The primitive model with the four wedges embedded

26.2.1 Social planner's problem

Similar as the model in section 25, the social planner's problem is to maximize the expected value (in perspective of period 0) of the additively time separable discounted utility of the infinitely living representative household, i.e. for exogenously given \(\{G_t,\tau_{h,t},\tau_{x,t},Z_t\}_{t=0}^{\infty}\):

$$ \max_{\{C_t(s^t),H_t(s^t),K_{t+1}(s^t)\}_{t=0}^{\infty}}\sum_{t=0}^{\infty}\sum_{s^t}\beta^t\pi_t(s^t)\big[\ln C_t(s^t)-v(H_t(s^t))\big] \tag{26.1} $$

$$ \text{s.t.}\quad C_t(s^t)+X_t(s^t)+G_t=\underbrace{\big[K_t(s^{t-1})\big]^{\alpha}\big[Z_t(s^t)H_t(s^t)\big]^{1-\alpha}}_{\equiv Y_t(s^t)}\ \text{for }\forall t,\forall s^t \tag{26.2} $$

$$ K_{t+1}(s^t)=(1-\delta)K_t(s^{t-1})+X_t(s^t) \tag{26.3} $$

$$ \underbrace{X_t(s^t)(1+\tau_{x,t})}_{\text{actual investment cost}}+C_t(s^t)=w_t(s^t)H_t(s^t)(1-\tau_{h,t}) \tag{26.4} $$

\(K_0=K_0(s^{-1})\) given, where (26.2) is the resource constraint of the whole economy, (26.3) is the law of motion of capital, and (26.4) is the budget constraint of the representative household.

26.2.2 The Lagrangian

Since we only need to pin down the relationship between \(C_t(s^t)\), \(H_t(s^t)\), and \(K_{t+1}(s^t)\) and are not interested in solving for them in this problem, we can form the Lagrangian for the problem in (26.1) without the resource constraint (26.2) (the resource constraint is for pinning down the specific values of \(C_t(s^t)\), \(H_t(s^t)\), and \(K_{t+1}(s^t)\) such that the government expenditure requirement \(G_t\) is met; here we only want to obtain the f.o.c. to the relationships between \(C_t(s^t)\), \(H_t(s^t)\), and \(K_{t+1}(s^t)\), so not including the resource constraint won't affect the f.o.c.):

$$ \begin{aligned} \mathcal{L}=&\sum_{t=0}^{\infty}\sum_{s^t}\beta^t\pi_t(s^t)\big[\ln C_t(s^t)-v(H_t(s^t))\big]\\ &+\sum_{t=0}^{\infty}\sum_{s^t}\beta^t\pi_t(s^t)\mu_t(s^t)\Big\{w_t(s^t)H_t(s^t)(1-\tau_{h,t})-\underbrace{\big[K_{t+1}(s^t)-(1-\delta)K_t(s^{t-1})\big]}_{=X_t(s^t)}(1+\tau_{x,t})-C_t(s^t)\Big\} \end{aligned} \tag{26.5} $$

where \(\mu_t(s^t)\) is the multiplier for the household budget constraint.

26.2.3 First order conditions

Assume \(v(H)=\dfrac{\gamma\varepsilon}{1+\varepsilon}H^{\frac{1+\varepsilon}{\varepsilon}}\) (\(\varepsilon>0\), \(v'(H)=\gamma H^{1/\varepsilon}\)). Taking into consideration of the equilibrium condition for wage \(w_t(s^t)=\dfrac{\partial Y_t(s^t)}{\partial H_t(s^t)}=(1-\alpha)\big[K_t(s^{t-1})\big]^{\alpha}\big[Z_t(s^t)\big]^{1-\alpha}\big[H_t(s^t)\big]^{-\alpha}\), the f.o.c. of (26.5) are:

$$ [C_t(s^t)]\quad \frac{1}{C_t(s^t)}=\mu_t(s^t) \tag{26.6} $$

$$ [H_t(s^t)]\quad \gamma H_t(s^t)^{\frac{1}{\varepsilon}}=\mu_t(s^t)\underbrace{(1-\alpha)\big[K_t(s^{t-1})\big]^{\alpha}\big[Z_t(s^t)\big]^{1-\alpha}\big[H_t(s^t)\big]^{-\alpha}}_{=w_t(s^t)}(1-\tau_{h,t}) \tag{26.7} $$

$$ [K_{t+1}(s^t)]\quad \mu_t(s^t)(1+\tau_{x,t})=\beta\sum_{s^{t+1}>s^t}\mu_{t+1}(s^{t+1})\frac{\pi_{t+1}(s^{t+1})}{\pi_t(s^t)}\Big\{(1-\alpha)\alpha\big[K_{t+1}(s^t)\big]^{\alpha-1}\big[Z_{t+1}(s^{t+1})H_{t+1}(s^{t+1})\big]^{1-\alpha}(1-\tau_{h,t+1})+(1-\delta)(1+\tau_{x,t+1})\Big\} \tag{26.8} $$

Using both (26.6) and (26.7), we can get the stacked f.o.c., which is also the intra-temporal indifference condition:

$$ \gamma H_t(s^t)^{\frac{1}{\varepsilon}}C_t(s^t)=(1-\alpha)(1-\tau_{h,t})\frac{Y_t(s^t)}{H_t(s^t)} \tag{26.9} $$

which is a distorted because \(MRS_t\) and \(MPL_t\) is different by a \((1-\tau_{h,t})\) labor income wedge.

26.2.4 Euler equation and transversality condition

The Euler equation (EE, or call it inter-temporal indifference condition) can be obtained by using (26.6) and (26.8):

$$ \frac{1+\tau_{x,t}}{C_t(s^t)}=\beta\sum_{s^{t+1}>s^t}\frac{\pi_{t+1}(s^{t+1})}{\pi_t(s^t)}\cdot\frac{(1-\alpha)\alpha\big[K_{t+1}(s^t)\big]^{\alpha-1}\big[Z_{t+1}(s^{t+1})H_{t+1}(s^{t+1})\big]^{1-\alpha}(1-\tau_{h,t+1})+(1-\delta)(1+\tau_{x,t+1})}{C_{t+1}(s^{t+1})} \tag{26.10} $$

Or equivalently, we can write the EE in expectation form, i.e.

$$ \frac{1+\tau_{x,t}}{C_t(s^t)}=\beta\mathbb{E}_{s^{t+1}}\left[\frac{(1-\alpha)\alpha\big[K_{t+1}(s^t)\big]^{\alpha-1}\big[Z_{t+1}(s^{t+1})H_{t+1}(s^{t+1})\big]^{1-\alpha}(1-\tau_{h,t+1})+(1-\delta)(1+\tau_{x,t+1})}{C_{t+1}(s^{t+1})}\,\Big|\,s^t\right] \tag{26.11} $$

26.2.5 Summary of system of equations

We have obtained the following system of four equations:

• Intra-temporal indifference condition (stacked f.o.c.):

$$ \gamma H_t(s^t)^{\frac{1}{\varepsilon}}C_t(s^t)=(1-\alpha)(1-\tau_{h,t})\frac{Y_t(s^t)}{H_t(s^t)} \tag{26.12} $$

• Inter-temporal indifference condition (EE):

$$ \frac{1+\tau_{x,t}}{C_t(s^t)}=\beta\mathbb{E}_{s^{t+1}}\left[\frac{(1-\alpha)\alpha\big[K_{t+1}(s^t)\big]^{\alpha-1}\big[Z_{t+1}(s^{t+1})H_{t+1}(s^{t+1})\big]^{1-\alpha}(1-\tau_{h,t+1})+(1-\delta)(1+\tau_{x,t+1})}{C_{t+1}(s^{t+1})}\,\Big|\,s^t\right] \tag{26.13} $$

• Resource constraint:

$$ C_t(s^t)+X_t(s^t)+G_t=\big[K_t(s^{t-1})\big]^{\alpha}\big[Z_t(s^t)H_t(s^t)\big]^{1-\alpha}\ \text{for }\forall t,\forall s^t \tag{26.14} $$

• Production function:

$$ Y_t(s^t)=\big[K_t(s^{t-1})\big]^{\alpha}\big[Z_t(s^t)H_t(s^t)\big]^{1-\alpha} \tag{26.15} $$

26.3 Wedge accounting

The basic idea is that the wedge vector series \(\{G_t,\tau_{h,t},\tau_{x,t},Z_t\}_{t=0}^{\infty}\) is theoretically determined, and is not observed (even though the tax rates and government expenditure could possibly be observed directly, some other factors that could be classified as such wedges are never clearly defined and observable; since the wedge sequence represents all the frictions in their types, the wedge sequence is not directly observable). So, we need to back out the wedge series from the observable data.

Based on the observed data of \(\{C_t(s^t),H_t(s^t),K_{t+1}(s^t)\}_{t=0}^{\infty}\) and the system of equations (26.12), (26.13), (26.14), and (26.15), we can solve for the four unknowns \(G_t,\tau_{h,t},\tau_{x,t},Z_t\) for each period.

Then, stack such wedges into a vector

$$ \phi_t=\begin{pmatrix}G_t\\ \tau_{h,t}\\ \tau_{x,t}\\ Z_t\end{pmatrix} $$

and do the first-order Vector Autoregression (VAR-1)

$$ \phi_{t+1}=\mathbf{A}\phi_t+\varepsilon_t $$

This type of research enables us to think about:

• The relationship between the wedges: what is the effect of one wedge on the other wedges.
• The explanatory power (relative importance) of each wedge:
  • we can shut down one wedge (setting it as a series of 0's);
  • and then look at the counterfactual series of any variables from \(\{C_t(s^t),H_t(s^t),K_{t+1}(s^t)\}_{t=0}^{\infty}\);
  • if the difference between the counterfactual series and the actual observed series is huge, then it means that the wedge shut down affects that real economy variable in a significant way;
  • one thing bear in mind is that even the difference is not big, it doesn't prove the wedge is not important for that real economy variable. It is possible that the wedge affects that variable through another indirect way.