Exercise 9.13

Productivity Shocks in the Nontraded Sector

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Problem

Consider an economy like the one developed in Section 9.1, in which the nontraded good is produced with the technology \(y^N_t= e^{z_t} h_t^{\alpha}\), where \(z_t\) denotes an exogenous and stochastic productivity shock. Assume that \(z_t\) evolves according to the law of motion \(z_t = \rho z_{t-1} + \mu_t\), where \(\rho\in [0,1)\) is a parameter and \(\mu_t\) is an i.i.d. disturbance with mean zero and standard deviation \(\sigma_\mu\). Suppose that the endowment of tradables is constant and equal to \(y^T>0\) and that the interest rate is constant and equal to \(r\). Assume that \(r\) satisfies \(\beta(1+r)=1\). Assume that the period utility function and the aggregator function are given by (9.28) and (9.29), respectively, with \(\xi = 1/\sigma<1\). Suppose that \(d_0=0\) and that the economy was operating at full employment in \(t=-1\).

  1. Find the equilibrium process of consumption of tradables, \(c^T_t\).

  2. Derive the optimal devaluation rate, \(\epsilon_t\), as a function of present and past values of the productivity shock.

  3. Provide a graphical analysis of the effect of an increase in productivity (\(z_0>z_{-1}\)) under the optimal exchange-rate policy and under a currency peg. Provide intuition.

  4. Suppose that the monetary authority follows the optimal exchange-rate policy that makes the domestic currency as strong as possible relative to the foreign currency at all times. Find the unconditional correlation between the net devaluation rate, \(\ln \epsilon_t\) and the growth rate of productivity, \(z_t/z_{t-1}\). Provide intuition.

  5. How would the sign of the correlation obtained in question 4 and the intuition behind it change in the case \(\xi = 1/\sigma>1\).

Answer

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