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Consider the version of the RCK model where there is a government that runs a constant balanced budget.

  • At $t_0$, the economy is in steady state, with constant tax $T_{old} > 0$
  • Then, at $t_1$, it is announced that at some later moment in time ($t_2$), but not right now, tax will be permanently reduced to $T_{new} < T_{old}$.

Draw a graph of changes in the phase diagram that characterizes the dynamics of the economy, and heuristic graphs of the time paths of the following variables:

  • consumption per effective worker
  • capital per effective worker
  • marginal return on capital per effective worker ($r_t$)

The memo gave the following:

phase diagram

time path of consumption

time path of capital

time path of the real interest rate

Why do the time paths of capital and the real interest rate look like that?

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When the economy transitions between time $t_1$ and $t_2$ (the red line in the phase diagram), capital reduces (because of the leftwards movement). That is why the green line showing the path of capital over time goes down initially. Because capital reduces, the marginal product of capital increases (as seen by $r$ increasing in the time path between $t_1$ and $t_2$.

Then, the economy is on the saddle path, and gradually returns to the new balanced growth path, which is a point with the permanently lower tax rate (and thus permanently higher consumption spending). When this gradual change happens (shown as the black arrow in the phase diagram), capital increases (movement to the right), so in the time path, the green line goes back up to the initial level. Because capital returns to its initial level, the return on capital also decreases to its initial level.

Consumption initially jumps up because disposable income over time is discounted, according to the formula $$\int_{t=t_1}^\infty e^{-R(t)}(W(t) - T(t)) \mathrm dt $$ where $R$ is the discount factor, $W$ is the wage and $T$ is tax, so even though disposable income doesn't decrease immediately, the anticipated decrease in the future affects current consumption. Other than that, the anticipated permanent change in government spending puts consumers on the new saddle path.

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