Question 3 · Introducció a la Macroeconomia UPF 2019

a. Capital per effective worker $k^* = \frac{K}{AL}$

Solución: $$\left(\frac{s}{\delta + g_A + g_N}\right)^{\frac{1}{1-\alpha}}$$

a. Output per effective worker $y^* = \frac{Y}{AL}$

Solución: $$\left(\frac{s}{\delta + g_A + g_N}\right)^{\frac{\alpha}{1-\alpha}}$$

b. The long-run growth rate of aggregate GDP ($Y$)

Solución: $g_Y = g_A + g_N$

c. The long-run growth rate of output per worker ($Y/L$)

Solución: $g_{Y/L} = g_A$

d. The steady-state capital-output ratio ($K/Y$)

Solución: $$\frac{s}{\delta + g_A + g_N}$$

e. Are standards of living improving while the economy is in steady state? Justify briefly.

Solución: Yes, because output per capita (and consequently per-capita consumption) grows continuously at the rate of technological progress $g_A > 0$.

Part B a. How does the growth rate of per-capita income behave during the transition in the first few years?

Solución: An increase from $s_0$ to $s_1$ shifts the saving curve upward to $s_1 f(k)$. Because investment exceeds break-even investment, capital deepening occurs. During the transitional years, the growth rate of per-capita income temporarily exceeds $g_A$.

Part B b. Is the growth rate of per-capita income affected in the long run?

Solución: No. Once the economy reaches the new steady state $(K/AN)_1$, the growth rate of per-capita income returns to its original rate $g_A$.

Part B c. Will this policy lead to permanently higher steady-state consumption per worker? Explain.

Solución: Not necessarily. While per-capita output $y^*$ is permanently higher, steady-state consumption per worker is $c^* = (1-s)y^*$. Consumption increases only if the original saving rate was below the Golden Rule level ($s < s_{GR} = \alpha$); if the economy was already beyond the Golden Rule, higher saving reduces per-capita consumption.