Question

Suppose a chair manufacturer's can hire labour for €30 per hour and the rental cost of capital is є15 per hour. The chair manufacturer's production function is such that a chair can be produced using 4 hours of labour or 4 hours of capital in any combination. (For example it can produce one chair with: 4 hours of labour only; 4 hours (rental) of capital only; or any combination of capital and labour that adds up to 4 hours) 3. Write down the equation for the cost function. What is the slope? Assume Capital (K) is on the Y axis and Labour (L) is on the X-axis. (a) (b) Draw the isoquant associated with producing one chair (q1) (c) If the firm is using 3 hours of labour and 1 hour of machine time to produce a chair, is it minimizing costs? If not how can it improve. Use diagrams in your answer, where possible.

Answer 1

Solution:

We are given that wage rate, w =
We are given that wage rate, w = euro 30 (per hour), rental rate r = euro 15 (per hour) denote labour by L and capital by k through the question a) Cost function, C = wL + rK C = 30L + 15K This is the equation for the cost function With K on Y-axis (vertical axis) and L on X-axis (horizontal axis), the slope of the cost function, dK/dL becomes: slope = dK/dL = partial c/partial L/partial c/partial k = 30/15 = 2 b) According to the question, the production function can be stated as following: L + K = 4 (since any combination of L and K should add up to 4) Then, dividing whole by 4, we get L/4 + K/4 = 1 = q So, for 1 (unit of) chair, this is the production function: L/4 + K/4 = 1 Plotting this on LK-plane gives us isoquant associated with producing 1 chair:
30 (per hour), rental rate, r =
We are given that wage rate, w = euro 30 (per hour), rental rate r = euro 15 (per hour) denote labour by L and capital by k through the question a) Cost function, C = wL + rK C = 30L + 15K This is the equation for the cost function With K on Y-axis (vertical axis) and L on X-axis (horizontal axis), the slope of the cost function, dK/dL becomes: slope = dK/dL = partial c/partial L/partial c/partial k = 30/15 = 2 b) According to the question, the production function can be stated as following: L + K = 4 (since any combination of L and K should add up to 4) Then, dividing whole by 4, we get L/4 + K/4 = 1 = q So, for 1 (unit of) chair, this is the production function: L/4 + K/4 = 1 Plotting this on LK-plane gives us isoquant associated with producing 1 chair:
15 (per hour). Denoting labor by L and capital by K throughout the question.

a) Cost function, C = wL + rK

C = 30L + 15K

This is the equation for the cost function.

With K on Y-axis (vertical axis) and L on X-axis (horizontal axis), the slope of the cost function, dK/dL becomes:

slope = dK/dL = = 30/15 = 2

b) According to the question, the production function can be stated as following:

L + K = 4 (since any combination of L and K should add up to 4)

Then, dividing whole by 4, we get

L/4 + K/4 = 1 = q

So, for 1 (unit of) chair, this is the production function: L/4 + K/4 = 1

Plotting this on LK-plane gives us isoquant associated with producing 1 chair:

c) Cost can be minimized at the combination of labor and capital where the lowest isocost line becomes tangent, or simply touches the isoquant. Thus, for given combination of (L, K) = (3, 1), firms is NOT minimizing it's cost to produce a chair. We see this using the figure below: by plotting 2 isocost curves (red colors), one through the combination (L,K) = (3, 1) and another through the supposedly, lowest/minimum cost of producing a chair.

Clearly, from the figure, Isocost 1 is not the lowest lying isocost curve (symbolizing, generating minimum cost) associated with producing 1 chair. In fact, there lie many many lines below isocost 1, touching the isoquant with q = 1. Among the them, the lowest one is the Isocost 2, which uses the combination (0, 4) that is 0 hours of labor, and (all) 4 hours of capital, to generate 1 chair.

Mathematically, it can be verified:

With (L, K) = (3, 1), cost, C = 30*3 + 15*1 = 90 + 15 = 105 (using part (a))

We are given that wage rate, w = euro 30 (per hour), rental rate r = euro 15 (per hour) denote labour by L and capital by k through the question a) Cost function, C = wL + rK C = 30L + 15K This is the equation for the cost function With K on Y-axis (vertical axis) and L on X-axis (horizontal axis), the slope of the cost function, dK/dL becomes: slope = dK/dL = partial c/partial L/partial c/partial k = 30/15 = 2 b) According to the question, the production function can be stated as following: L + K = 4 (since any combination of L and K should add up to 4) Then, dividing whole by 4, we get L/4 + K/4 = 1 = q So, for 1 (unit of) chair, this is the production function: L/4 + K/4 = 1 Plotting this on LK-plane gives us isoquant associated with producing 1 chair:

With (L, K) = (0, 4), cost, C = 30*0 + 15*4 = 0 + 60 = 60

And both combinations help produce 1 chair. Of course  
We are given that wage rate, w = euro 30 (per hour), rental rate r = euro 15 (per hour) denote labour by L and capital by k through the question a) Cost function, C = wL + rK C = 30L + 15K This is the equation for the cost function With K on Y-axis (vertical axis) and L on X-axis (horizontal axis), the slope of the cost function, dK/dL becomes: slope = dK/dL = partial c/partial L/partial c/partial k = 30/15 = 2 b) According to the question, the production function can be stated as following: L + K = 4 (since any combination of L and K should add up to 4) Then, dividing whole by 4, we get L/4 + K/4 = 1 = q So, for 1 (unit of) chair, this is the production function: L/4 + K/4 = 1 Plotting this on LK-plane gives us isoquant associated with producing 1 chair:
60 <
We are given that wage rate, w = euro 30 (per hour), rental rate r = euro 15 (per hour) denote labour by L and capital by k through the question a) Cost function, C = wL + rK C = 30L + 15K This is the equation for the cost function With K on Y-axis (vertical axis) and L on X-axis (horizontal axis), the slope of the cost function, dK/dL becomes: slope = dK/dL = partial c/partial L/partial c/partial k = 30/15 = 2 b) According to the question, the production function can be stated as following: L + K = 4 (since any combination of L and K should add up to 4) Then, dividing whole by 4, we get L/4 + K/4 = 1 = q So, for 1 (unit of) chair, this is the production function: L/4 + K/4 = 1 Plotting this on LK-plane gives us isoquant associated with producing 1 chair:
105

Intuitively, it makes sense, as in our example, labor and capital are acting as perfect substitutes, so if cost of hiring 1 hour of labor (i.e., wage rate) is higher than cost of hiring 1 hour of capital (i.e., rental rate), firm can achieve cost efficiency by undertaking production by completely substituting capital for labor.