Assuming a Cobb-Douglas production function with constant returns to scale, then, as L rises with K and A constant, it will be the case that:Group of answer choicesThe marginal product of labour will rise and the marginal product of capital will fallBoth the marginal product of labour and the marginal product of capital will fallBoth the marginal product of labour and the marginal product of capital will riseThe marginal product of labour will fall and the marginal product of capital will rise
Question
Assuming a Cobb-Douglas production function with constant returns to scale, then, as L rises with K and A constant, it will be the case that:Group of answer choicesThe marginal product of labour will rise and the marginal product of capital will fallBoth the marginal product of labour and the marginal product of capital will fallBoth the marginal product of labour and the marginal product of capital will riseThe marginal product of labour will fall and the marginal product of capital will rise
Solution
The Cobb-Douglas production function is given by Y = A * L^alpha * K^beta, where Y is the total production, A is the total factor productivity, L is the labor input, K is the capital input, and alpha and beta are the output elasticities of labor and capital, respectively.
Assuming constant returns to scale means that alpha + beta = 1.
The marginal product of labor (MPL) is the additional output produced by adding one more unit of labor, holding all other inputs constant. It is given by the derivative of the production function with respect to L, which is alpha * A * L^(alpha-1) * K^beta.
The marginal product of capital (MPK) is the additional output produced by adding one more unit of capital, holding all other inputs constant. It is given by the derivative of the production function with respect to K, which is beta * A * L^alpha * K^(beta-1).
If L rises with K and A constant, then the MPL will fall and the MPK will rise. This is because as more labor is added, the additional output produced by each additional unit of labor (MPL) decreases due to the diminishing marginal returns. On the other hand, the additional output produced by each additional unit of capital (MPK) increases because the same amount of capital is now being used with more labor.
So, the correct answer is: The marginal product of labour will fall and the marginal product of capital will rise.
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