Steady state

We close the model imposing market clearing. This affects the budget constraint:

$$\begin{eqnarray} k_{t+1} = w_{t} + r_{t+1}k_{t} + (1-\delta)k_{t} - c_{t} - \color{red}{T_{t}}, & \\\
w_{t} = f(k_{t}) - f^{\prime}(k_{t})k_{t},& \\\
r_{t} = f^{\prime}(k_{t}). & \end{eqnarray}$$

Hence the dynamics for capital are now affected by public expenditures:

$$k_{t+1} = f(k_{t}) + (1-\delta)k_{t} - c_{t} - \color{red}{T_{t}}.$$

The inclusion of the term $T_{t}$ will impact the level of consumption —and only consumption— at the steady state. Indeed, we can compute it as we did before. From the Euler equation we have:

$$u^{\prime} (c_{t}) = \beta (f^{\prime}(k_{t+1}) + 1 - \delta)u^{\prime}(c_{t+1}).$$ The budget constraint above determines the dynamics of capital. Setting $c_{t} = c_{t+1} = \bar{c}^\mathcal{Gov}$ and $k_{t} = k_{t+1}= \bar{k}^{\mathcal{Gov}}$ reveals the steady-state levels of capital and consumption:

$$1 = \beta( f^{\prime}(\bar{k}^{\mathcal{Gov}}) + 1 - \delta) \implies f^{\prime}(\bar{k}^\mathcal{Gov}) = \frac{1-\beta(1-\delta)}{\beta},$$ $$\bar{c}^\mathcal{Gov} = f \left( \bar{k}^{\mathcal{Gov}} \right) - \delta \bar{k}^{\mathcal{Gov}} - \color{red}{G_{t}},$$ where we have used $G_{t} = T_{t}.$

Two main comments are important here:

  • The $c$-locus is not affected, hence the steady-state level of capital remains the same,
  • The $k$-locus moves closer to the horizontal axis, because we subtract $G_{t} > 0$ from it. This implies that the steady-state level of consumption is reduced.

phase_diagram

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