Short Run Production FunctionLong Run Production FunctionReturns to ScaleCite 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 Production Function 14.01 Principles of Microeconomics, Fall 2007 Chia-Hui Chen October 1, 2007 Lecture 11 Production Functions Outline 1. Chap 6: Short Run Production Function 2. Chap 6: Long Run Production Function 3. Chap 6: Returns to Scale 1 Short Run Production Function In the short run, the capital input is fixed, so we only need to consider the change of labor. Therefore, the production function q = F (K, L) has only one variable L (see Figure 1). Average Product of Labor. Output qAPL = = . Labor Input L Slope from the origin to (L,q). Marginal Product of Labor. ∂Output ∂q M PL = = . ∂Labor Input ∂L Additional output produced by an additional unit of labor. Some proper ties about AP and M P (see Figure 2). • When M P = 0, Output is maximized. • When M P > AP, AP is increasing.10 9 8 7 0 1 2 3 4 5 6 7 8 9 10 L 0 1 2 3 4 5 6 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]. 2 1 Short Run Production Function 0 1 2 3 4 5 6 7 8 9 10 L 0 5 10 15 20 25 30 35 40 Q Figure 1: Short Run Production Function. Figure 2: Average Product of Labor and Marginal Product of Labor.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 2 Long Run Production Function • When M P < AP, AP is decreasing. • When M P = AP, AP is maximized. To prove this, maximize AP by first order co ndition: ∂ q(L) = 0 ∂L L =⇒ ∂q 1 q− = 0 ∂L L L2 =⇒ ∂q q = ∂L L =⇒ M P = AP. Example (Chair Production.). Note that here APL and M PL are not con-tinuous, so the condition for maximizing APL we derived ab ove does not apply. Number o f Wor kers Number of Cha irs Produced 0 0 1 2 2 8 3 9 APL M PL N/A N/A 2 2 4 6 3 1 Table 1 : Relation between Chair Production and L abor. 2 Long Run Production Function Two variable inputs in long run (see Figur e 3). Isoquants. Curves showing all possible combinations of inputs that yield the same output (see Figure 4). Marginal Rate of Technical Substitution (M RT S). Slope of Isoquants. dK M RT S = − dL How many units of K can be reduced to keep Q constant when we increase L by one unit. Like MRS, we also have M PLM RT S = . M Pk4 Q 10 9 8 7 k=3 6 5 k=2 4 k=1 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 L 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 2 Long Run Production Function Figure 3: Long Run Production Function. Figure 4: K vs L, Isoquant Curve. 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 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 L 5 3 Returns to Scale Proof. Since K is a function of L on the isoquant curve, q(K(L), L) = 0 =⇒ ∂q dK ∂q + = 0 ∂L dL ∂L =⇒ dK M PL − = . dL M PK Perfect Substitutes (Inputs). (see Figure 5) Figure 5: Isoqua nt Curve, Perfect Substitutes. Perfect Complements (Inputs). (see Figure 6) 3 Returns to Scale Marginal Product of Capital. ∂q(K, L)M PK = ∂K Marginal Product of Labor K constant , L ↑ → q? 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 k 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 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]. 3 Returns to Scale Figure 6: Isoquant Curve, Perfect Complements. Marginal Product of Capital L constant , K ↑ → q? What happens to q when both inputs are increased? K ↑ , L ↑ → q? Increasing Returns to Scale. • A production function is said to have increasing returns to scale if Q(2K, 2L) > 2Q(K, L), or Q(aK, aL) = 2Q(K, L), a < 2. • One big firm is more efficient than many small firms. • Isoquants get closer as we move away from the origin (see Figure 7).7 k 10 9 8 Q=3 7 6 5 Q=2 4 3 2 Q=1 1 0 0 2 4 6 8 10 12 14 L 3 Returns to Scale Figure 7: Isoqua nt Curves, Increasing Returns to Scale. 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