Short-Run Cost FunctionLong-Run Cost FunctionCite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 1 1 Short-Run Cost Function 14.01 Principles of Microeconomics, Fall 2007 Chia-Hui Chen October 15, 2007 Lecture 13 Cost Functions Outline 1. Chap 7: Short-Run Cost Function 2. Chap 7: Long-Run Cost Function Cost Function Let w be the cost per unit of labor and r be the cost per unit of capital. With the input Labor (L) and Capital (K), the production co st is w × L + 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 o utput. We can also split total cost into fixed cost and variable cost as follows: C(q) = F C + V C(q). Fixed cost is independent of quantity, while variable cost is dependent on quan-tity. 1 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 (re fer to Lecture 11) (see Figure 1). Suppose w = 1, the variable cost curve can be derived from Figure 1. Adding r × K to the variable cos t, we obtain the total cost curve (see Figure 2). Average total cost is T C F C + V C rK wL(q; K)AT C = = = + . q q q q With the definition of the average product of labor: qAPL = , LCite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 2 1 Short-Run Cost Function 40 35 30 25 20 15 10 5 0 0 1 2 3 4 5 6 7 8 9 10 L Q Figure 1: Short Run Production Function. 0 5 10 15 20 25 30 35 40 45 C VC TC 50 0 1 2 3 4 5 6 7 8 9 10 q Figure 2: Short Run Cost Function.L 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 AP MP Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 3 1 Short-Run Cost Function we can rewrite AT C as rK w AT C = + , q APL in which the average variable cost is V C wL(q; K) w = = . q q APL Likewise, we rewrite the marginal cost: dT C dV C dL(q) w w M C = = = w = = . dq dq dq ∂q M PL∂L In L e c tur e 11, we discussed the relation between average product of labor and marginal product of labor (see Figure 3). We draw the curves for AV C and Figure 3: Average Product of L abor and Marginal Product of Labor. M C in the s ame way (see Figure 4). The relation between M C and AV C is: If • M C < AV C, AV C decreases; if • M C > AV C, AV C increases;Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 4 2 Long-Run Cost Function 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 25 30 C AVC ATCMC L Figure 4: Average Cost, Average Variable Cost, and Marginal Cost. if • M C = AV C, AV C is minimized. Now consider the total cost. Note that the difference between AT C and AV C decreases with q as the average fixed cost term dies out (see Figure 4). The relation be tween MC and AT C is: If • M C < AT C, AT C decre ases; if • M C > AT C, AT C increases; if • M C = AT C, AT C is minimized.5 2 Long-Run Cost Function k 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 L Figure 5: Iso quant Curve. 2 Long-Run Cost Function In the long-run, both K and L are variable. The isoquant curve describes the same output level with different combination of K and L (see Figure 5). The slope of an is oq uant curve is M PL −M RT S = −M PK . Similarly, the isoc ost curve is constructed by different (K, L) with the same cost (see Figure 6). The isocost curve equation is: rK + wL = const, therefore, it is linear, with a slope − w . r Now we want to minimize the co st rK +wL subject to an output level Q(K, L) = q. This minimum cost can be obtained when the isocost curve is tangent to the isoquant curve (see Figure 7). Thus the slopes of these two curves are equal: M PL w M RT S = = . M PK r Now cons ider an increase in wage (w). The slope of the isocost curve increases (see Figure 8), and the firm use more c apital 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 labo r a nd capital at each level of output (see Figure 9). Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].K 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 L Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 6 2 Long-Run Cost Function Figure 6: Iso c ost Curve. 0 1 2 3 4 5 6 7 8 9 10 L Figure 7: Minimize the Cost Subject to a Output Level. 0 1 2 3 4 5 6 7 8 9 10 KCite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 7 2 Long-Run Cost Function 0 1 2 3 4 5 6 7 8 9 10 K A B 0 1 2 3 4 5 6 7 8 9 10 L Figure 8: The Change of Cost Minimized Situation. 2 3 4 5 6 7 8 9 10 11 12 K Expansion Path 2 3 4 5 6 7 8 9 10 11 12 L Figure 9: Expansion Path.Cite as: Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 8 2 Long-Run Cost Function Example (Calculating the Cost.). Given the production function 2323q = L K . In the short run, 32qCSR(q; K) = rK + w …