Suppose we now care about the long run decisions of a firm that has a production function of the form q = 4L^1/2 + K
Suppose w = 1 and r = 0.5
Assume that, at the beginning when w0 = 1 and r = 0.5, the firm chose to produce q0 = 20 units of output. Then, the wage increased to w1 = 2 and in consequence the firm chose to produce q1 = 10 units of output.
(a) Calculate the optimal choices of labor and capital after the wage change.
(b) Draw the old and new isocosts lines
(c) Calculate the scale effect
(d) Calculate the substitution effect
(e) Calculate the elasticity of substitution
To find the optimal choices of labor and capital after the wage change, we need to solve the firm's cost minimization problem.
(a) Calculating the optimal choices of labor and capital:
The cost function is given by C = wL + rK, where C is the total cost, w is the wage rate, L is the quantity of labor, r is the rental rate of capital, and K is the quantity of capital.
At the beginning, when w0 = 1 and r = 0.5, the firm chose to produce q0 = 20 units of output. Therefore, we have:
20 = 4L0^0.5 + K0
After the wage increase to w1 = 2, the firm chose to produce q1 = 10 units of output. Now we need to solve for the new optimal choices of labor and capital. We can set up the following equation:
10 = 4L1^0.5 + K1
To solve for L1 and K1, we can rewrite these equations:
L0^0.5 = (20 - K0) / 4
L1^0.5 = (10 - K1) / 4
Since w0 = 1 and w1 = 2, we can set up the following equality:
w0 / w1 = 1 / 2 = (L0^0.5 / L1^0.5) = ((20 - K0) / 4) / ((10 - K1) / 4)
Simplifying the equation, we get:
2 = (20 - K0) / (10 - K1)
Cross-multiplying and simplifying, we get:
20 - K0 = 2(10 - K1)
20 - K0 = 20 - 2K1
0 = -K1 + K0
Therefore, K1 = K0
Substituting K1 = K0 in the equation 10 = 4L1^0.5 + K1, we get:
10 = 4L1^0.5 + K0
To solve for L1, we can substitute K1 = K0 into the equation L1^0.5 = (10 - K1) / 4, which gives:
L1^0.5 = (10 - K0) / 4
Simplifying, we get:
L1 = ((10 - K0) / 4)^2
The optimal choices of labor and capital after the wage change are L1 = ((10 - K0) / 4)^2 and K1 = K0.
(b) Drawing the old and new isocost lines:
The isocost line represents the combinations of labor and capital that have the same total cost. It can be represented by the equation C = wL + rK, where C is the total cost, w is the wage rate, L is the quantity of labor, r is the rental rate of capital, and K is the quantity of capital.
For the old isocost line with w0 = 1 and r = 0.5, the equation becomes:
C0 = L0 + 0.5K0
For the new isocost line with w1 = 2 and r = 0.5, the equation becomes:
C1 = 2L1 + 0.5K1
You can plot these two equations on a graph with the quantity of labor (L) on the x-axis and the quantity of capital (K) on the y-axis to obtain the old and new isocost lines.
(c) Calculating the scale effect:
The scale effect measures the change in output resulting from a change in all inputs proportionally. It can be calculated by taking the ratio of the new output (q1) to the old output (q0) when all inputs are scaled up proportionally.
In this case, since both labor and capital remain the same (L1 = L0 and K1 = K0), the scale effect is calculated as:
Scale effect = q1 / q0 = 10 / 20 = 0.5
(d) Calculating the substitution effect:
The substitution effect measures the change in output resulting from a change in the relative prices of inputs (in this case, the change in the wage rate). It can be calculated as the change in output due to the change in the wage rate while keeping the cost of production constant.
In this case, the cost of production remains constant because K1 = K0. Therefore, the substitution effect is calculated as the change in output from q0 to q1:
Substitution effect = q1 - q0 = 10 - 20 = -10
(e) Calculating the elasticity of substitution:
The elasticity of substitution measures the responsiveness of the firm's choice between labor and capital to changes in their relative prices. It can be calculated using the formula:
Elasticity of substitution = (dK/dw) * (w/K)
To find (dK/dw), we take the derivative of the optimal capital with respect to the wage rate. Since K1 = K0, the derivative is zero.
The elasticity of substitution in this case is:
Elasticity of substitution = 0 * (w/K) = 0
Therefore, the elasticity of substitution is zero.