Question

Consider a Solow growth model without technological progress. The production function is \[ Y_t = K_t^{\alpha} N_t^{1-\alpha}, \] where $Y_t$, $K_t$, and $N_t$ are aggregate output, capital, and population at time $t$, respectively. The population grows at a constant rate $g_N>0$, savings rate is constant at $s \in (0,1)$, and capital depreciates at a constant rate $\delta \ge 0$. Denote per capita capital as \[ k_t = \frac{K_t}{N_t}, \] and define the steady state as a situation where $k_{t+1} = k_t = k^*$, where $k^*$ is a positive constant. Suppose the population growth rate exogenously increases to $g'_N$. At the new steady state, the aggregate output will grow at a rate

• A. $g_N$
• B. $g'_N$
• C. $(1 - \alpha)g_N$
• D. $(1 - \alpha)g'_N$

Hint:

In the Solow model without technological progress, steady-state output per worker remains constant. Aggregate output grows only at the rate of population growth.

Answer 1

The Correct Option is B

Step 1: Relation between total and per capita output.
In the Solow model without technological progress, \[ Y_t = N_t y_t, \quad \text{where } y_t = k_t^{\alpha}. \] In steady state, $k_t = k^*$ is constant, so $y_t$ is constant. Hence, output per worker does not grow.
Step 2: Aggregate output growth.
Since $Y_t = N_t y_t$ and $y_t$ is constant, \[ \frac{\dot{Y}}{Y} = \frac{\dot{N}}{N} = g_N. \] When population growth increases to $g'_N$, the new steady-state aggregate output growth will equal the new population growth rate $g'_N$.
Step 3: Conclusion.
Hence, the aggregate output at the new steady state grows at rate $g'_N$.