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.
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.
To derive the steady-state growth rate of output and capital efficiency unit of labor, we will use the Solow Growth Model. In this model, output is represented by Y, capital is represented by K, and labor is represented by L.
Step 1: Define the production function.
Let Y = A * (K^α) * (L^(1-α)), where A is the level of technology, α (0<α<1) is the capital's share in output, and (1-α) is the labor's share in output.
Step 2: Define the saving and investment function.
Let s be the savings rate and δ be the depreciation rate. The investment function can be expressed as I = s * Y, and the depreciation rate as ΔK = δ * K.
Step 3: Define the capital accumulation equation.
The change in capital over time is given by ΔK = I - ΔK. Substituting the values from Step 2, we have:
ΔK = s * Y - δ * K.
Step 4: Rewrite the equation in terms of capital per unit of effective labor.
To normalize the equation, divide both sides by L. We get:
Δ(k) = s * (y) - (n + δ) * (k), where k = K/L and y = Y/L.
Step 5: Find the steady-state condition.
In the steady state, the capital per unit of effective labor remains constant. Thus, Δ(k) = 0. Setting Δ(k) = 0 in the equation, we have:
s * (y) = (n + δ) * (k).
Step 6: Find the steady-state growth rate of output and capital efficiency unit of labor.
Substituting the value of y from Step 1, we have:
s * [A * (k^α) * (L^(1-α))] = (n + δ) * (k).
Simplifying, we get:
s * A * (k^α) * (L^(1-α)) = (n + δ) * (k).
The steady-state growth rate of output (g_y) is given by g_y = (n + g), where g is the growth rate of technology.
The capital efficiency unit of labor (k*) can be found by solving the equation for k.
To illustrate graphically, we can plot the production function and the capital accumulation equation on a graph of k against y. The intersection point of the two equations represents the steady state growth rate of output and capital efficiency unit of labor.
Economic institutions play a crucial role in promoting technological progress and facilitating positive population growth. These institutions can involve policies that promote innovation, entrepreneurship, education, and access to resources like capital. By fostering an environment conducive to technological progress and population growth, economic institutions can contribute to long-term economic development and increased standards of living.