Ejercicio 5 - Section C: The Solow growth model · Introducció a la Macroeconomia UPF 2024

a. Straight ray from the origin

Solución: depreciation per worker ($\delta \frac{K}{N}$)

a. Upper concave curve

Solución: production function per worker ($\frac{Y}{N} = f(\frac{K}{N})$)

a. Lower concave curve

Solución: investment (saving) per worker ($s f(\frac{K}{N})$)

b. Should we expect the capital stock per worker to increase or decrease over time?

Solución: increase

b. Explain in terms of the curves.

Solución: At $K_0/N < (K/N)^*$, the investment curve lies above the depreciation line ($s f(k) > \delta k$), generating positive net investment $\Delta k > 0$ until steady state is reached.

c. Compute the exact steady-state capital stock per worker, $(K/N)^*$.

Solución: 8

d. Steady-state capital stock per worker $(K/N)^{*\prime}$ after increase in saving rate to $s' = 0.3$

Solución: 27

d. Describe and illustrate how this change affects the curves and the steady-state capital stock per worker.

Solución: An increase in the saving rate to $s' = 0.3$ shifts the investment curve $s f(k)$ upward. The depreciation ray $\delta k$ and production curve $f(k)$ remain unchanged. The new intersection occurs at a higher capital-labor ratio ($k^{*\prime} = 27$).

e. Does consumption per worker increase or decrease in the long run as a result of this increase in the saving rate?

Solución: increase

e. Provide a rigorous economic and mathematical justification based on the Golden Rule.

Solución: Steady-state consumption per worker is $c^* = f(k^*) - \delta k^*$. Maximizing consumption yields the Golden Rule condition $f'(k_G) = \delta$: $$\frac{2}{3} k_G^{-1/3} = \delta \implies s_G = \frac{\delta k_G}{f(k_G)} = \frac{f'(k_G) k_G}{f(k_G)} = \frac{2}{3} \approx 0.67$$ Since $s = 0.2 < s' = 0.3 < s_G = 2/3$, the economy is below the Golden Rule level of capital. An increase in saving raises steady-state output by more than required depreciation ($f'(k^*) > \delta$), leading to an unambiguous increase in long-run consumption per worker.