Exercise 4.13

An SOE-RBC Model with Cobb-Douglas Preferences

Problem

Modify the period utility function of the EDEIR model as follows:

\[ U(c,h) = \frac{\left[ c^{1-\omega}(1-h)^{\omega} \right]^{1-\sigma}-1} {1-\sigma}. \]

All other features of the model are unchanged.

Table 4.1: Calibration of the EDEIR Small Open RBC Economy

Parameter	$\sigma$	$\delta$	$r^*$	$\alpha$	$\bar{d}$	$\omega$	$\phi$	$\psi_1$	$\rho$	$\tilde{\eta}$
Value	2	0.1	0.04	0.32	0.7442	1.455	0.028	0.000742	0.42	0.0129

Derive analytically the steady state of the model.
Set all parameters of the model as in Table 4.1, except for $\omega$. Calibrate $\omega$ to ensure that in the deterministic steady state, hours equal 1/3 (i.e., ensure that in the steady state, households spend one-third of their time working). Calculate the implied value of $\omega$.
Produce a table of predicted second moments similar to Table 4.2. When performing this step, you might find it convenient to use as a starting point the Matlab programs for the EDEIR SOE-RBC model posted online.
Compare the predictions of the present model with those of its GHH-preference counterpart.

Answer

1. Steady State with Cobb-Douglas Preferences

\[ -\frac{U_h}{U_c} = \frac{\omega}{1-\omega}\frac{c}{1-h} \]

Then (4.11) is:

\[ \frac{\omega}{1-\omega}\frac{c_t}{1-h_t} = (1-\alpha)A_t(k_t/h_t)^{\alpha} \tag{*} \]

Steady state: Given

\[h = 1/3\]

Calibrated parameters are: $\alpha$, $\bar{d}$, $r^*$, $A = 1$, and $\delta$. From page 82:

\[ \beta^{-1} = 1 + r^* \]

\[ \kappa = \frac{k}{h} = \left(\frac{\alpha}{\beta^{-1} - 1 + \delta}\right)^{\frac{1}{1-\alpha}} \]

\[ k = \kappa h \]

By (4.16):

\[ c = \kappa^{\alpha}h - r^*\bar{d} - \delta k \]

By (*):

\[ \frac{\omega}{1-\omega}\frac{c}{1-h} = (1-\alpha)\kappa^{\alpha} \]

Solve for $\omega$:

\[ \omega = \frac{Z}{1+Z}; \quad \text{where } Z \equiv \frac{(1-\alpha)\kappa^{\alpha}(1-h)}{c} \]

2.

\[ \omega = 0.6567 \]

3.

Table 4.2: Cobb Douglas

Variable	Std. Dev.	Serial Corr.	Corr. with $y$
$y$	2.7853	0.6546	1.0000
$c$	0.8461	0.8775	0.2520
$i$	11.2857	-0.0815	0.5053
$h$	1.7995	0.7523	0.9529
$tb/y$	2.5948	0.4772	0.4477
$ca/y$	2.3534	0.4129	0.3622

In This table: Original

Variable	Std. Dev.	Serial Corr.	Corr. with $y$
$y$	3.0826	0.6170	1.0000
$c$	2.7064	0.7822	0.8441
$i$	9.0391	0.0686	0.6688
$h$	2.1186	0.6170	1.0000
$tb/y$	1.7782	0.5085	-0.0435
$ca/y$	1.4529	0.3220	0.0503

4. Comparison

With income-elastic labor supply (Cobb-Douglas), hours are, as expected, somewhat less volatile than under income-inelastic labor supply (GHH). The largest difference in second moments, however, appears in consumption. Consumption is much less volatile (0.85 percent under Cobb-Douglas versus 2.71 percent under GHH) and less correlated with output (0.25 versus 0.84). The different cyclicality of consumption also affects the trade balance, which becomes procyclical, with a correlation of 0.45 with output.

The impact response of the trade balance (not shown) remains negative because investment rises sharply on impact under both Cobb-Douglas and GHH preferences. Since investment exhibits near-zero serial correlation in both cases, the post-impact dynamics of the trade balance are driven largely by consumption. Because consumption increases only modestly during the output boom triggered by a positive technology shock, the trade balance turns positive after the impact period and is therefore procyclical overall.

End of Answer

--- title: "Exercise 4.13" subtitle: "An SOE-RBC Model with Cobb-Douglas Preferences" --- <a class="return-button" href="index.qmd">⬅ Return</a> ## Problem Modify the period utility function of the EDEIR model as follows: $$ U(c,h) = \frac{\left[ c^{1-\omega}(1-h)^{\omega} \right]^{1-\sigma}-1} {1-\sigma}. $$ All other features of the model are unchanged. **Table 4.1: Calibration of the EDEIR Small Open RBC Economy** | Parameter | $\sigma$ | $\delta$ | $r^*$ | $\alpha$ | $\bar{d}$ | $\omega$ | $\phi$ | $\psi_1$ | $\rho$ | $\tilde{\eta}$ | |-----------|----------|----------|-------|----------|-----------|----------|--------|----------|--------|----------------| | Value | 2 | 0.1 | 0.04 | 0.32 | 0.7442 | 1.455 | 0.028 | 0.000742 | 0.42 | 0.0129 | 1. Derive analytically the steady state of the model. 2. Set all parameters of the model as in Table 4.1, except for $\omega$. Calibrate $\omega$ to ensure that in the deterministic steady state, hours equal 1/3 (i.e., ensure that in the steady state, households spend one-third of their time working). Calculate the implied value of $\omega$. 3. Produce a table of predicted second moments similar to Table 4.2. When performing this step, you might find it convenient to use as a starting point the Matlab programs for the EDEIR SOE-RBC model posted online. 4. Compare the predictions of the present model with those of its GHH-preference counterpart. ## Answer #### 1. Steady State with Cobb-Douglas Preferences $$ -\frac{U_h}{U_c} = \frac{\omega}{1-\omega}\frac{c}{1-h} $$ Then (4.11) is: $$ \frac{\omega}{1-\omega}\frac{c_t}{1-h_t} = (1-\alpha)A_t(k_t/h_t)^{\alpha} \tag{*} $$ **Steady state:** Given $$h = 1/3$$ Calibrated parameters are: $\alpha$, $\bar{d}$, $r^*$, $A = 1$, and $\delta$. From page 82: $$ \beta^{-1} = 1 + r^* $$ $$ \kappa = \frac{k}{h} = \left(\frac{\alpha}{\beta^{-1} - 1 + \delta}\right)^{\frac{1}{1-\alpha}} $$ $$ k = \kappa h $$ By (4.16): $$ c = \kappa^{\alpha}h - r^*\bar{d} - \delta k $$ By (*): $$ \frac{\omega}{1-\omega}\frac{c}{1-h} = (1-\alpha)\kappa^{\alpha} $$ Solve for $\omega$: $$ \omega = \frac{Z}{1+Z}; \quad \text{where } Z \equiv \frac{(1-\alpha)\kappa^{\alpha}(1-h)}{c} $$ #### 2. $$ \omega = 0.6567 $$ #### 3. **Table 4.2: Cobb Douglas** | Variable | Std. Dev. | Serial Corr. | Corr. with $y$ | |-------------|-----------|--------------|-----------------| | $y$ | 2.7853 | 0.6546 | 1.0000 | | $c$ | 0.8461 | 0.8775 | 0.2520 | | $i$ | 11.2857 | -0.0815 | 0.5053 | | $h$ | 1.7995 | 0.7523 | 0.9529 | | $tb/y$ | 2.5948 | 0.4772 | 0.4477 | | $ca/y$ | 2.3534 | 0.4129 | 0.3622 | **In This table: Original** | Variable | Std. Dev. | Serial Corr. | Corr. with $y$ | |-------------|-----------|--------------|-----------------| | $y$ | 3.0826 | 0.6170 | 1.0000 | | $c$ | 2.7064 | 0.7822 | 0.8441 | | $i$ | 9.0391 | 0.0686 | 0.6688 | | $h$ | 2.1186 | 0.6170 | 1.0000 | | $tb/y$ | 1.7782 | 0.5085 | -0.0435 | | $ca/y$ | 1.4529 | 0.3220 | 0.0503 | #### 4. Comparison With income-elastic labor supply (Cobb-Douglas), hours are, as expected, somewhat less volatile than under income-inelastic labor supply (GHH). The largest difference in second moments, however, appears in consumption. Consumption is much less volatile (0.85 percent under Cobb-Douglas versus 2.71 percent under GHH) and less correlated with output (0.25 versus 0.84). The different cyclicality of consumption also affects the trade balance, which becomes procyclical, with a correlation of 0.45 with output. The impact response of the trade balance (not shown) remains negative because investment rises sharply on impact under both Cobb-Douglas and GHH preferences. Since investment exhibits near-zero serial correlation in both cases, the post-impact dynamics of the trade balance are driven largely by consumption. Because consumption increases only modestly during the output boom triggered by a positive technology shock, the trade balance turns positive after the impact period and is therefore procyclical overall. **End of Answer**