Question 3 · Introducció a la Macroeconomia UPF 2021

Question 3 (1.5 points). Solow Growth Model with Technological Progress

Consider the canonical Solow growth model with labor-augmenting technological progress and population growth. The aggregate production function is Cobb-Douglas:

$$Y = K^{\bar{a}} (AN)^{1 - \bar{a}}$$

where $K$ is physical capital, $N$ is labor input, $A$ is technology, and $\bar{a}$ (with $0 < \bar{a} < 1$) is an exogenous constant. Technology and population grow at constant exogenous rates $g_A$ and $g_n$, respectively:

$$\frac{\Delta A}{A} = g_A, \quad \frac{\Delta N}{N} = g_n$$

A constant fraction $\delta$ (with $0 < \delta < 1$) of physical capital depreciates each period. The aggregate saving rate is constant at $s$.

a.

Define capital per effective worker as $k \equiv \frac{K}{AN}$. Derive the steady-state level of capital per effective worker, $k^*$, as a function of the model parameters. [0.3 points]

b.

Express the steady-state level of output per effective worker, $y^* \equiv \frac{Y}{AN}$, as a function of the parameters. [0.2 points]

c.

Derive the steady-state level of consumption per effective worker, $c^*$, as a function of the parameters. [0.3 points]

Now assume the economy has reached its steady state. Due to the long-term disruptions of the pandemic, investment in basic research and R&D falls permanently, causing a permanent reduction in the rate of technological progress from $g_A$ to $g_A' < g_A$.

d.

Using the Solow steady-state diagram, explain how the growth rate of income per effective worker responds in the initial transition years following this shock. [0.3 points]

e.

Is the long-run growth rate of output per capita (income per worker) affected by this shock? Justify your answer. [0.2 points]

f.

What happens to the steady-state level of capital per effective worker ($k^*$)? [0.1 points]

g.

Will long-run living standards (measured by the level and trajectory of output per worker over time) increase or decrease as a result of this permanent drop in R&D? [0.1 points]