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Tutorial 8: The Costs of Production (cont.)

 

In Tutorial 7 we saw how we could model the production process with two variable inputs by using a graphical model of isoquants. In the second part of Tutorial 8 we learned how we could model the cost of production using two variable inputs with a graphical model of isocost lines. Now it's time to put the two together to model how a firm can choose that combination of the two variable inputs that minimizes the cost of producing a given level of output.

 

Choosing Inputs  Top of page.

 

Suppose the firm's goal is to produce Q1 units/year. They could hire any combination of labor and capital on the isoquant Q1. But logically the firm would rather choose that combination that entails the minimum cost. Stated simply, the firm's objective is to minimize the cost of producing a given level of output.

See Figure 7. Suppose the firm's output must be Q = 100 units/day. It could choose any one of the numerous combinations of L and K along the isoquant Q1 that would produce the desired output. Suppose it chose (2.5, 10)? If w = $2/unit of L and r = $2/unit of K, then that combination would cost the firm 2.5 · $2 + 10 · $2 = $25 to produce 100 units/day. Is that the cheapest it can get? Well it could hire no L and no K and that would cost them $0, but the firm is constrained by its desire to produce 100 units/day. So how about these combinations:

 

Table 1

Labor hours

Capital hours Cost Comments
3.6 7.0 $21.20 Cheaper, but...
5.0 5.0 $20.00 Looking better...
7.0 3.6 $21.20 oops!

 

Figure 7


Graph showing how isoquant and isocosts model is used to find the cost-minimizing combination of labor and capital.

 

Well the combination (5, 5) was cheapest. So what are the conditions of the model at this point? Well the slopes of the isocost and the isquant are equal there. But why would that be the cheapest?

The absolute value of the slope of isoquant = MRTS. The marginal rate of technical substitution reveals how much K the firm is willing to give up when one more L is used to produce a given level of Q. The absolute value of the slope of isocost = w/r, which reveals how many units of K the firm must trade to free up enough money to buy one more unit of L. At (2.5, 10.0), MRTS > w/r meaning firm is willing to give up more K when it hires one more L than is required by the market. So it will happily hire more L and reduce the amount of K used to produce Q because it will be cheaper. This also holds true for (3.6, 7.0) -- MRTS is still > w/r but not by as much. Finally, at (5, 5) MRTS = w/r and the firm has reached the point where further additions of L (even with reductions in K) will raise the cost of producing 100 units of output to more than $20.

So, mathemagically, the cost minimizing firm will vary L* and K* until MRTS - r/w = 0 (which is the same as saying MRTS = w/r. Why write it this way? See the Excel Workbook for this Tutorial...). But MRTS = w/r at (0, 0). True, but the firm is constrained by its desire to produce a given level of output, i.e., its objective is subject to the constraint that A*La*Kb = Q1,
where MRTS = (a/L*) / (b/K*). So, graphically, the combination of inputs that satisfies both the objective and the constraint, is found at the point of tangency between the isocost line and isquant.

 

Caution: This model is easily confused with the consumer choice model. The elements appear similar but their roles differ. While there are many similarities, there is an important difference. In the consumer choice model the budget line represents the individual's constraint on choice. Indifference curves "appear" through any market basket of goods. In the production costs model, the isoquant represents the firm's constraint on production costs. Isocost lines "appear" through any combination of labor and capital the firm hires.

 

Now it's time to "do the thing".

Click on the following link to download the Minimizing Production Costs Workbook. Work through Tutorial 8 Questions 2 - 7. This will help to improve your understanding of how we combine the isoquant and the isocost lines to model the cost-minimizing choice of two variable inputs.

Return here when you have finished.

Need help downloading the Excel file?

 

 

This third part of Tutorial 8 presents a discussion of how we can combine the isocost model together with the isoquant model developed in Tutorial 7 and use it to see how a firm can minimize the cost of producing a given level of output using two variable inputs.

In part d we will use the isocost/isoquant model to discover the differences in the cost of increasing output in the short run (when at least one input is fixed) and in the long run (when no inputs are fixed).