Question

[Solow Model] Consider a closed economy where the labor force DECREASES at a rate of 1% per year. Markets are competitive. The production function satisfies constant returns to scale and diminishing marginal returns to both labor and capital. The share of capital income in total income stays constant over time. Assume the economy is in its steady state and the real wage grows at a rate of 4% per year.

(a) Find the GROWTH RATE of the real rental rate.

(b) .Find the GROWTH RATE of consumption per worker.

(c) Find the (approximate) GROWTH RATE of (aggregate) consumption.

Answer 1

(a) The growth rate of the real rental rate is approximately
`\( \underline{3\%} \)` per year.

(b) The growth rate of consumption per worker is approximately
`\( \underline{2\%} \)` per year.

(c) The approximate growth rate of aggregate consumption is
`\( \underline{1\%} \)` per year.

Step-by-step explanation:

In a steady state, the growth rate of the real rental rate (r) is equal to the growth rate of the real wage (w), which is given as 4%. Thus, (r = w = 4%).

The growth rate of consumption per worker (c) can be found using the formula (c = (1-s)y), where (s) is the savings rate. Given that the share of capital income is constant, (s) is also constant. Since the production function exhibits constant returns to scale and diminishing marginal returns, the savings rate (s) is equal to the share of capital income. Therefore, (s) is constant, and the growth rate of consumption per worker is the same as the growth rate of output per worker. This is approximately
`\(1\%\) (\(g = n + \delta\)).`

The growth rate of aggregate consumption is the sum of the growth rate of population (n) and the growth rate of consumption per worker (c). Given that the labor force is decreasing at 1% per year (n = -1%), and the growth rate of consumption per worker is approximately 1%, the growth rate of aggregate consumption is approximately (0% + 1% = 1% ) per year.

Answer 2

The growth rate of the real rental rate of capital is influenced by the scarcity of capital as labor decreases, leading to an increase. The growth rate of consumption per worker is equivalent to the growth rate of the real wage, which is 4%. The approximate growth rate of aggregate consumption is the growth of consumption per worker minus the decrease in the labor force, resulting in a 3% growth rate.

Step-by-step explanation:

To calculate the growth rate of the real rental rate of capital, we need to consider the student's description of the economy. Since the production function operates under constant returns to scale and the share of capital income in total income remains constant, and given that the labor force is decreasing at a rate of 1% per year, this implies that capital must be becoming more scarce relative to labor. As a result, the real rental rate should increase at a rate to compensate for both the reduction in labor and the increased productivity due to capital's higher relative scarcity.

For the growth rate of consumption per worker, if we're in a steady state, then the real wage is indicative of the well-being of the worker. With the real wage growing at 4% per year, and labor becoming more scarce, consumption per worker would also grow at this rate.

When determining the growth rate of aggregate consumption, we need to take into account both the growth rate of consumption per worker and the change in the number of workers. Hence, the approximate growth rate of aggregate consumption is the growth rate of consumption per worker (4%) minus the rate at which the labor force is decreasing (1%), giving an approximate growth rate of 3% for aggregate consumption.