Exercise 5.3

Slow Diffusion of Technology Shocks to the Country Premium, Household Production, and Government Spending

⬅ Return

Problem

The model presented in section 5.3 assumes that permanent productivity shocks affect not only the productivity of labor and capital in producing market goods, but also the country premium, home production, and government spending. For instance, the assumption that the country interest rate depends on \(\widetilde{D}_{t+1}/X_t\), implies that a positive innovation in \(X_t\) in period \(t\), causes, all other things equal, a fall in the country premium. In this exercise, we attenuate this type of effect by reformulating the model. Let

\[ \widetilde{X}_t = \widetilde{X}_{t-1}^{\zeta}X_t^{1-\zeta}, \]

with \(\zeta\in[0,1)\). Note that the original formulation obtains when \(\zeta=0\). Replace equations (5.3), (5.5), and (5.9), respectively, with

\[ E_0\sum_{t=0}^{\infty} \nu_t\beta^t \frac{\left[C_t - \omega^{-1}\widetilde{X}_{t-1} h_t^{\omega}\right]^{1-\gamma}-1}{1-\gamma}, \]

\[ s_t = \frac{S_t}{\widetilde{X}_{t-1}}, \]

and

\[ r_t = r^* + \psi \left( e^{(\widetilde{D}_{t+1}/\widetilde{X}_t-\overline{d})/ \overline{y}} -1 \right) + e^{\mu_t-1}-1. \]

Keep all other features of the model as presented in section 5.3.

  1. Present the equilibrium conditions of the model in stationary form.

  2. Using Bayesian techniques, reestimate the model adding \(\zeta\) to the vector of estimated parameters. Assume a uniform prior distribution for \(\zeta\) with support \([0,0.99]\) and produce 1 million draws from the posterior distribution of the parameter vector. Present the estimation results in the form of a table like table 5.4. Discuss your findings.

  3. Characterize numerically the predictions of the model. In particular, produce tables similar to tables 5.5 and 5.6. Discuss your results and provide intuition.

  4. Compute the impulse responses of output, consumption, investment, the trade-balance-to-output ratio, and the country interest rate to a one-percent innovation in \(g_t\) for three values of \(\zeta\), namely, 0, its posterior median, and 0.99.

Answer

No solution has been posted yet. If you have a solution, we invite you to share it in the comment box at the bottom of this page.