Exercise 4.1

Dynamics of a Linear Economy

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Problem

Consider an economy whose equilibrium dynamics has the reduced form

\[ \hat{c}_{t+1} = \gamma_{11} \hat{c}_t \]

and

\[ \hat{d}_t = \gamma_{21} \hat{c}_t + \gamma_{22} \hat{d}_{t-1}, \]

where \(c_t\) denotes consumption, and \(d_t\) denotes debt acquired in period \(t\) and maturing in period \(t+1\). A hat denotes deviation from steady state. The variable \(d_{t-1}\) is an endogenous state and is predetermined in \(t\), and the variable \(c_t\) is a control variable determined in \(t\). The coefficients \(\gamma_{11}\), \(\gamma_{21}\), and \(\gamma_{22}\) are constant parameters. Suppose that \(\gamma_{22} > 0\).

  1. Find conditions on the parameters of the model under which

    (a) The economy has a unique equilibrium converging to the steady state,
    (b) debt converges monotonically to its steady-state value, and
    (c) consumption is strictly decreasing in debt (i.e., \(c_t\) is a strictly decreasing function of \(d_{t-1}\)).

  2. Find the policy functions for debt and consumption that are consistent with the three conditions imposed in the previous item.

Answer

1.

Hint: Consult Appendix 4.14.2.

2.

Policy functions:

\[ d_t = \gamma_{11} d_{t-1} \quad \text{and} \quad c_t = \frac{\gamma_{11} - \gamma_{22}}{\gamma_{21}}. \]

Need \(\gamma_{11} \in (0,1)\), \(\gamma_{22} > 1\), and \(\gamma_{21} > 0\).