Assume a positive population growth (n>0) and technological progress (g=0), derive analytically the steady state growth rate of output and capital efficiency unit of labor. Illustrate your answer graphically and briefly discuss the economic institution.

Answer 1

To derive the steady state growth rate of output and capital efficiency unit of labor, we can use the Solow-Swan model of economic growth. In this model, the steady state occurs when the rate of growth of output per worker (g_y) is equal to the rate of depreciation of capital per worker (δ) and the capital efficiency unit of labor (E) is constant.

The production function in this model is:

Y = K^α * (AL)^(1-α)

where Y is output per worker, K is capital per worker, L is labor, A is total factor productivity, and α is the share of capital in output.

The capital per worker is given by the equation:

K' = sY - (n + δ)K

where s is the savings rate, n is the population growth rate, and δ is the depreciation rate.

To find the steady state growth rate of output per worker, we set g_y=0:

0 = αK^(α-1) * (AL)^(1-α) * (dK/dt) + (1-α)K^α * (AL)^(-α) * (dA/dt) + (1-α)K^α * A * (dL/dt)

Since g=0, dA/dt = 0 and dL/dt = nL (population growth rate times labor), we can simplify the equation:

0 = αK^(α-1) * (AL)^(1-α) * (dK/dt) + (1-α)K^α * A * nL

From this equation, we can solve for dK/dt, which represents the steady state growth rate of capital per worker:

dK/dt = (αK^α * AL^(-1) * (1-α)AnL) / αK^(α-1) * AL^(1-α)

dK/dt = (1-α)AnL

The steady state growth rate of capital per worker is (1-α)AnL.

To find the steady state growth rate of output per worker (g_y), we set dK/dt = 0, and simplify the equation:

0 = (1-α)AnL

From this equation, we can solve for g_y, which represents the steady state growth rate of output per worker:

g_y = 0

In other words, in the steady state, output per worker doesn't grow (g_y=0).

Now, to discuss the economic institution, the steady state growth rate of output and capital efficiency unit of labor (E) being constant implies that there is no technological progress (g=0). This means that the level of technology remains constant over time and there is no change in productivity.

In this scenario, economic institutions play a crucial role in promoting economic growth and capital accumulation. Institutions that promote savings and investment, such as well-functioning financial markets, property rights protection, and stable political environment, are essential for maintaining a positive level of capital per worker and output per worker. These institutions provide incentives for individuals and firms to save, invest, and innovate, leading to sustained economic growth.

Graphically, the steady state growth rate of output per worker (g_y=0) is represented by a flat line in the production function graph, indicating that output per worker remains constant over time. The capital efficiency unit of labor (E) is also constant, represented by a horizontal line at a specific level.

Overall, economic institutions play a fundamental role in ensuring steady state growth of output and maintaining a constant level of capital efficiency unit of labor.