Production Functions

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1 Short Run Production Function. Principles of Microeconomics, Fall Chia-Hui Chen October, ecture Production Functions Outline. Chap : Short Run Production Function. Chap : ong Run Production Function. Chap : Returns to Scale 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 has only one variable (see Figure ). Average Product of abor. Slope from the origin to (,q). Marginal Product of abor. q = F (K, ) Output q AP = =. abor Input Output q MP = =. abor Input Additional output produced by an additional unit of labor. Some properties about AP and MP (see Figure ). Output is maximized. AP is increasing. MP =, MP > AP, Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

2 Short Run Production Function Q Figure : Short Run Production Function. AP MP Figure : Average Product of abor and Marginal Product of abor. Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

3 ong Run Production Function AP is decreasing. AP is maximized. MP < AP, MP = AP, To prove this, maximize AP by first order condition: q() = q q = q q = MP = AP. Example (Chair Production.). Note that here AP and MP are not continuous, so the condition for maximizing AP we derived above does not apply. Number of Workers Number of Chairs Produced AP MP N/A N/A Table : Relation between Chair Production and abor. ong Run Production Function Two variable inputs in long run (see Figure ). Isoquants. Curves showing all possible combinations of inputs that yield the same output (see Figure ). Marginal Rate of Technical Substitution (MRTS). Slope of Isoquants. dk MRTS = d How many units of K can be reduced to keep Q constant when we increase by one unit. ike MRS, we also have MP MRTS =. MP k Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

4 ong Run Production Function k= Q k= k= Figure : ong Run Production Function... k Figure : K vs, Isoquant Curve. Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

5 k Returns to Scale Proof. Since K is a function of on the isoquant curve, q(k(), ) = q dk q + = d dk MP =. d MP K Perfect Substitutes (Inputs). (see Figure ) Figure : Isoquant Curve, Perfect Substitutes. Perfect Complements (Inputs). (see Figure ) Returns to Scale Marginal Product of Capital. q(k, ) MP K = K Marginal Product of abor K constant, q? Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

6 Returns to Scale k Figure : Isoquant Curve, Perfect Complements. Marginal Product of Capital constant, K q? What happens to q when both inputs are increased? K, q? Increasing Returns to Scale. A production function is said to have increasing returns to scale if Q(K, ) > Q(K, ), or Q(aK, a) = Q(K, ), a <. One big firm is more efficient than many small firms. Isoquants get closer as we move away from the origin (see Figure ). Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

7 Returns to Scale Q= k Q= Q= Figure : Isoquant Curves, Increasing Returns to Scale. Cite as: Chia-Hui Chen, course materials for. Principles of Microeconomics, Fall. MIT

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