Production and Costs in the Long Run. 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. Isocost lines.
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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.
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.
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*
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.
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.
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.
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).
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 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.
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.
L1 L2 L*
We could follow K1 out and see costs of other levels of output(by putting in more isoquants).
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.
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.
Each point on the long run curve is really just a point off the lowest short run curve at a level of output.