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Production and Costs in the Long Run

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Production and Costs in the Long Run

- The long run is the time frame longer or just as long as it takes to alter the plant.
- Thus the long run is that time period in which all inputs are variable.

An isocost line includes all possible combinations of labor and capital that can be purchased for a given total cost.

In equation form the total cost is

TC = PLL + PKK,

where TC = Total cost,

PL= the wage rate,

L = the amount of labor taken,

PK = the rental price of capital, and

K = the amount of capital taken. This equation can be re-expressed as

K = TC/ PK - (PL/ PK) L.

As an example say labor is $6 per unit and capital is $10 per unit. Then if we look at a total cost of $100 we see various combinations of inputs:

L = 10 and K = 4 or

L = 0 and K = 10 or

L = 16.67 and K = 0, amoung others.

On the next screen we can view the isocost line in a graph.

K

This is the isocost line at $100. If we wanted to see higher costs we would shift the line out in a parallel shift and a lower cost we have a shift in.

L

cost and output

K

On this slide I want to concentrate on one level of output, as summarized by the isoquant. Input combination L1, K1 could be used and have cost summarized by 4th highest isocost shown. L2, K2 would be cheaper, and L*, K*

K1

K2

K*

L

L1 L2 L*

is the lowest cost combination of inputs to produce the given level of output. Here the cheapest cost of the output occurs

at a tangency point between an isocost and isoquant.

cost and output

K

On this slide I want to concentrate on one level of cost, as summarized by the isocost line. Input combination L1, K1 could be used and have this cost but more output would be obtained if L*, K* were used.

K1

K*

L

L1 L*

Here, the most output for a given cost occurs at a tangency point.

On the last two screens we have seen the tangency of an isoquant and isocost line shows either

1) the cheapest way to produce a certain level of output, or

2) the most output that can be obtained for a given amount of cost.

These two things are different sides of the same coin and profit maximizing firms would be expected to reach the tangency positions in the long run.

In the long run when a firm is able to change all inputs we see the firm will go to a point where the slope of an isocost line is tangent to a isoquant. This means the slopes are equal.

Thus, slope of isocost = (PL/ PK) = MRTS = slope of isoquant.

Remember we said MRTS can be shown to be the ratio of marginal products of labor to capital. Thus

PL/PK = MPL/MPK means MPK/PK = MPL/PL. This is a statement that the “bang for the buck” should be the same for both inputs. In other words the additional output for each input per dollar spent should be equal across inputs

when all is said and done.

If you happen to be at a point where the ratios are not equal take more of the input that has the higher ratio because its marginal product will diminish with a greater amount taken. You will probably have to take less of the other input.

Expansion path

K

Once we have a unit cost of capital and labor we can draw many isocosts, each one that is farther out has a higher cost. We can see the tangency of each isocost with an isoquant (output level).

L

In the long run the firm will be at one of the points of tangency. When connect all those points we have the expansion path. In the long run the firm will be on the expansion path.

The Short Run and the Long Run and seeing the connection between the two.

The exception to reaching the tangency in the long run would be the short run when the amount of some input can not be changed to reach the tangency. In the long run all inputs can be changed in amount and thus the tangency point could be reached.

short run

K

Here the cheapest way to produce the output level as depicted in the isoquant would be to hire L*, K*. But the firm has committed to having K1 units of capital. Thus the cost of this output is indicated by the fourth highest isocost line.

K1

K2

K*

L

L1 L2 L*

We could follow K1 out and see costs of other levels of output(by putting in more isoquants).

As you follow along K1 maybe one output level will occur where that short run point is exactly the same as the long run point. In that one case the cost level is the same in both the short run and the long run.

Remember the output level shown on the previous screen in the short run with K1 has a higher cost to produce that output than would occur in the long run.

So, in the long run, you make the cost of a certain level of output the lowest by not only adjusting labor to the right amount but capital to the right amount. But, if in the short run you are not at the right amount of capital then you will produce the output at a higher cost because in the short run you are stuck at a certain level of capital.

On the next slide I have two short run average cost curves. Each one represents the average cost with different amounts of the fixed input capital. SO, maybe ATC1 could have 1 unit of capital and ATC2 could have two units of capital. There really should be lots more of these curves but I show two to get to the next point.

If output will be less than Q in the long run, then in the short run costs might be too high if we have two units of capital. But, in the long run capital would be switched to 1 unit. Similarly, output above Q has lowest cost when made with two units of capital.

ATC

ATC2

ATC1

Q

Q

- When we switch from one unit of capital to two units, we have the long run because all inputs are then variable.
- But with the two units we would have short run curves for that level of capital.
- Now we have two sets of cost curves, one for one unit of capital and one for two units of capital.

- If output is going to be less than Q1 in the long run then only one unit of capital would be wanted because those units would be produced cheapest with one unit of capital.
- Greater than Q1 would be produced cheapest with two units of capital.

- The long run curve is parts of the short run curves. For each range of output the long run curve is the segment of the short run curve that is the lowest, representing the cheapest way to produce that range of output in the long run. The final long run curve is smooth. Let’s see.

ATC

Q

Each point on the long run curve is really just a point off the lowest short run curve at a level of output.

- The long run cost curve is said to be u - shaped, just as in the short run, but for a different reason. In the short run we had diminishing returns. In the long run we have economies of scale.

- The basic idea of economies of scale is that at least for a while when the plant size is increased the average cost curve is pushed down, implying average costs are lowest in a bigger plant. It may be that further increases in plant size push the average cost curve back up. This would technically be called diseconomies of scale.

- Another way to view the long run is to think about different short run situations and put them together. Think of a short run with one capital unit. Think of one with two capital units, and so on.
- We would have a similar table of numbers and graphs as we did in the short run example when only one unit of capital was available.