Consider a profit maximizing competitive firm with the following production function: y=L^α+K^β where
0 <α< 1 and 0 <β< 1. The firm can purchase labor (L) and capital (K) it wants in competitive input market at unit prices of w and r, respectively. The market price of the firm’s output is p.
a) Find the profit maximizing quantities of labor, L* and capital, K* and output, Y*.
b) Is Y* an increasing function of output price?
c) How does the profit maximizing level of output change when the rental rate of capital is decreased?
To find the profit-maximizing quantities of labor (L*) and capital (K*) and output (Y*), we need to first calculate the firm's profit function and then differentiate it with respect to L and K to find the maximum.
a) The profit function is given by:
π = pY - wL - rK
To find the profit-maximizing quantities, we need to maximize the profit function by setting the partial derivatives to zero:
∂π/∂L = p(∂Y/∂L) - w = 0
∂π/∂K = p(∂Y/∂K) - r = 0
Taking the partial derivatives of the production function with respect to L and K, we get:
∂Y/∂L = αL^(α-1) + βK^β
∂Y/∂K = βK^(β-1) + αL^α
Setting the partial derivatives to zero, we have:
p(αL^(α-1) + βK^β) - w = 0
p(βK^(β-1) + αL^α) - r = 0
Simplifying the equations, we get:
αpL^(α-1) + βpK^β - w = 0
βpK^(β-1) + αpL^α - r = 0
From the first equation, we can solve for L:
αpL^(α-1) = w - βpK^β
L^(α-1) = (w - βpK^β) / (αp)
L = [(w - βpK^β) / (αp)]^(1/(α-1))
Similarly, from the second equation, we can solve for K:
βpK^(β-1) = r - αpL^α
K^(β-1) = (r - αpL^α) / (βp)
K = [(r - αpL^α) / (βp)]^(1/(β-1))
Substituting the expressions for L and K back into the production function, we can solve for Y:
Y = L^α + K^β
Therefore, the profit-maximizing quantities of labor, capital, and output are L*, K*, and Y* respectively.
b) To determine if Y* is an increasing function of the output price, we can take the derivative of Y with respect to p and check its sign. If the derivative is positive, then Y* is an increasing function of p.
∂Y/∂p = αL^α(∂L/∂p) + βK^β(∂K/∂p)
Since L and K are determined by w, r, and p (not directly by p), their partial derivatives with respect to p are zero. Therefore, ∂L/∂p = 0 and ∂K/∂p = 0.
This implies that Y does not depend on p and, thus, Y* is not an increasing function of p.
c) To determine how the profit-maximizing level of output changes when the rental rate of capital (r) is decreased, we need to analyze the relationship between K* and r.
From the earlier equations, we found that the profit-maximizing quantity of capital is given by:
K = [(r - αpL^α) / (βp)]^(1/(β-1))
When r is decreased, the denominator of the expression decreases, which means that K* increases. As K* increases, the production function Y = L^α + K^β also increases since both L and K are positive. Therefore, the profit-maximizing level of output (Y*) increases when the rental rate of capital is decreased.