Convergence Towards the Steady State

We examine how the capital stock changes over time. In the current time period, the 1.00 units of capital per worker $k$ produce 1.00 units of output per worker $f(k)$.

Because 55 percent of output is saved and invested and 45 percent is consumed, investment per worker $i$ is 0.55 and consumption per worker $c$ is 0.45.

Because 25 percent of the capital stock depreciates, $\delta k$ is 0.250. With investment of 0.550 and depreciation of 0.250, the change in the capital stock $\Delta k$ is 0.300.

Thus, capital per worker $k$ will be 1.30 in the next period. Because investment exceeds depreciation, new capital is added and output grows.

Over many periods, the economy converges to a steady state $k^*$ with 4.80 units of capital per worker. In this steady state, investment of 0.75 exactly offsets depreciation of 1.20, so $\Delta k = 0$ (i.e. the capital stock and hence output are constant).

Changing the savings rate will lead the economy to converge to a different steady state: