The growth rate of productivity due to an increase in capital income equals
A)the share of capital income in aggregate income.
B)the slope of the productivity function.
C)the growth rate of capital per hour worked times the share of capital income in aggregate income.
D)the growth rate of capital per hour worked.
E)the growth rate of technology.

Question

Quizplus · Accepted Answer

According to the Cobb-Douglas production function, the contribution to output from capital can be expressed as a function of the capital stock, labor input, and a parameter representing the elasticity of output with respect to capital. Mathematically, this can be expressed as Y = AK^αL^(1-α), where Y is the level of output, K is the capital stock, L is the labor input, A is the level of technology, and α is the output elasticity of capital. The marginal productivity of capital (MPC) is given by MPK = ∂Y/∂K = αAK^(α-1)L^(1-α). Taking the derivative of the MPC with respect to time gives us the growth rate of productivity due to an increase in capital income. Therefore, d(MPK)/dt = αA(α-1)K^(α-2)L^(1-α)dK/dt, which can be rewritten as d(MPK)/dt = α(α-1)Y/K * dK/dt, where Y/K represents output per unit of capital. Multiplying both sides by the share of capital income in aggregate income (s) gives us ds/dt * MPK/Y = α(α-1)s * dK/dt/K. The left-hand side of the equation represents the growth rate of capital income, while the right-hand side represents the growth rate of capital per hour worked. Thus, the growth rate of productivity due to an increase in capital income is equal to α(α-1)s * dK/dt/K.

The Growth Rate of Productivity Due to an Increase in Capital