14.01 Principles of Microeconomics, Fall 2007 Chia-Hui Chen October 15, Lecture 13. Cost Function

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1 Short-Run Cost Function. Principles of Microeconomics, Fall Chia-Hui Chen October, ecture Cost Functions Outline. Chap : Short-Run Cost Function. Chap : ong-run Cost Function Cost Function et w be the cost per unit of labor and r be the cost per unit of capital. With the input abor () and Capital (K), the production cost is w + r K. A cost function C(q) is a function of q, which tells us what the minimum cost is for producing q units of output. We can also split total cost into fixed cost and variable cost as follows: C(q) = FC + V C(q). Fixed cost is independent of quantity, while variable cost is dependent on quantity. Short-Run Cost Function In the short-run, firms cannot change capital, that is to say, r K = const. Recall the production function given fixed capital level K in the short run (refer to ecture ) (see Figure ). Suppose w =, the variable cost curve can be derived from Figure. Adding r K to the variable cost, we obtain the total cost curve (see Figure ). Average total cost is TC FC + V C rk w(q; K) ATC = = = +. q q q q With the definition of the average product of labor: q AP =, Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

2 Short-Run Cost Function Q 9 Figure : Short Run Production Function. TC C VC 9 q Figure : Short Run Cost Function. Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

3 Short-Run Cost Function we can rewrite ATC as in which the average variable cost is ikewise, we rewrite the marginal cost: rk w ATC = +, q AP V C w(q; K) w = =. q q AP dtc dv C d(q) w w MC = = = w = =. dq dq dq q MP In ecture, we discussed the relation between average product of labor and marginal product of labor (see Figure ). We draw the curves for AV C and 9 AP MP 9 Figure : Average Product of abor and Marginal Product of abor. MC in the same way (see Figure ). The relation between MC and AV C is: If AV C decreases; if AV C increases; MC < AV C, MC > AV C, Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

4 ong-run Cost Function MC ATC C AVC 9 Figure : Average Cost, Average Variable Cost, and Marginal Cost. if AV C is minimized. MC = AV C, Now consider the total cost. Note that the difference between ATC and AV C decreases with q as the average fixed cost term dies out (see Figure ). The relation between MC and ATC is: If AT C decreases; if AT C increases; if AT C is minimized. MC < ATC, MC > ATC, MC = ATC, Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

5 ong-run Cost Function.. k Figure : Isoquant Curve. ong-run Cost Function In the long-run, both K and are variable. The isoquant curve describes the same output level with different combination of K and (see Figure ). The slope of an isoquant curve is MP MRTS = MPK. Similarly, the isocost curve is constructed by different (K, ) with the same cost (see Figure ). The isocost curve equation is: rk + w = const, therefore, it is linear, with a slope w. r Now we want to minimize the cost rk +w subject to an output level Q(K, ) = q. This minimum cost can be obtained when the isocost curve is tangent to the isoquant curve (see Figure ). Thus the slopes of these two curves are equal: MP w MRTS = =. MP K r Now consider an increase in wage (w). The slope of the isocost curve increases (see Figure ), and the firm use more capital and less labor. The firm s choice of input moves from A to B in the figure. The expansion path shows the minimum cost combinations of labor and capital at each level of output (see Figure 9). Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

6 K ong-run Cost Function Figure : Isocost Curve. 9 K 9 Figure : Minimize the Cost Subject to a Output evel. Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

7 ong-run Cost Function 9 K B A 9 Figure : The Change of Cost Minimized Situation. 9 Expansion Path K 9 Figure 9: Expansion Path. Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

8 ong-run Cost Function Example (Calculating the Cost.). Given the production function q = K. In the short run, q C SR (q; K) = rk + w, K where K is fixed. In the long run, according to the equation MP MP K w =, r we have K w =. r Then the expansion path is w K =. r Substituting this result into the production function, we obtain r = q ( ), w w K = q ( ). r Hence, the long-run cost function is: C R (q) = w + rk = q wr. Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

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