# The chapter is on Production and costs in thelong run, we are given a long run expansion path graph, all through out the book we are given labor cost and capital cost, ie. price of labor $5 dollars per unit(w), price of capital $10 per unit (r). Then I get the following question:

The production engineers at Impact Industries have derived an expansion path (graph shown but I can't re-produce) the price of labor per unit is $100 per unit. The question is what price does Impact Industries pay for capital?

I'm lost because we've always been given the per unit of capital....am I going about this wrong? How do I find the capital price?!?!?!?!?

Output Labor Capital (K) (w = $5) (r=???)

120 4 20

180 6 30

240 8 50

Thanks,

EY

I don't believe you have provided enough information to solve. I suspect, perhaps, you are missing values for the marginal product of labor and the marginal product of capital?

## Based on the given information about the quantities of labor and capital and the corresponding output levels, we can use the concept of production isoquants to find the price of capital (r).

An isoquant represents all possible combinations of labor and capital that can produce a given level of output. In this case, we are given three points on the expansion path, which represents different levels of output:

Point 1: (4 units of labor, 20 units of capital) corresponding to an output of 120.

Point 2: (6 units of labor, 30 units of capital) corresponding to an output of 180.

Point 3: (8 units of labor, 50 units of capital) corresponding to an output of 240.

To find the price of capital (r), we need to compare the marginal rate of technical substitution (MRTS) to the ratio of labor price to capital price (w/r). The MRTS is the rate at which the firm can substitute capital for labor while maintaining the same level of output.

The MRTS is given by the ratio of the marginal product of labor (MPL) to the marginal product of capital (MPK), which measures the additional output gained by adding one more unit of labor or capital, respectively.

MRTS = MPL / MPK

Using the given data, we can calculate the MRTS at each point on the expansion path:

At Point 1: MPL = (180 - 120) / (6 - 4) = 60 / 2 = 30 and MPK = (180 - 120) / (30 - 20) = 60 / 10 = 6.

So, MRTS1 = 30 / 6 = 5.

At Point 2: MPL = (240 - 180) / (8 - 6) = 60 / 2 = 30 and MPK = (240 - 180) / (50 - 30) = 60 / 20 = 3.

So, MRTS2 = 30 / 3 = 10.

Now, we can compare these MRTS values to the labor price to capital price ratio (w/r) to find the price of capital (r).

At Point 1: MRTS1 = 5, w/r = 5 / r = 5.

So, r = $1.

At Point 2: MRTS2 = 10, w/r = 5 / r = 10.

So, r = $0.5.

Since the price of capital needs to be consistent across all points on the expansion path, we can conclude that Impact Industries pays $1 per unit of capital.

Note: Without the information about the marginal product of labor and the marginal product of capital, it is not possible to determine the exact price of capital.