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Chapter 6: Firms and Production • Firms’ goal is to maximize their profit. • Profit function: π= R – C = P*Q – C(Q) where R is revenue, C is cost, P is price, and Q is quantity • Production function: the relationship between the quantities of inputs used and the maximum quantity of output that can be produced. It summarizes the technology of transforming inputs into outputs. e.g.) q = f(L,K) • Fixed input vs. variable input 1 Short-Run: At least one factor of production is fixed For production function: q = f(L,K) • Average product of labor (AP) = q/L • Marginal product of labor (MP) = △q/△L • AP increases when MP exceeds AP and decreases when MP is exceeded by AP. 2 3 Diminishing Marginal Returns (or diminishing marginal product) • If a firm keeps adding one more unit of input, holding all other inputs and technology constant, the extra output it obtains will become smaller eventually. • Why? – Too many workers per machine – Increases the cost of managing labors, etc. 4 Example: A Cobb-Douglas Function • Production Function: q 10L0.5 K 0.5 • Capital (K) is fixed. Only labor (L) is variable. • The marginal product of Labor is q 10 0.5L( 0.51) K 0.5 5 L0.5 K 0.5 L • The second derivative of q w.r.t. L is 2q ( 0.51) 0.5 1.5 0.5 5 ( 0 . 5 ) L K 0 . 25 L K 0 L2 which is negative: Concave function. Assume that K is fixed at 100. Draw the production function and the marginal product of Labor. 5 Long-Run: All inputs are variable • Firms can vary input mix to achieve the most efficient production. • Isoquant: a curve that shows the efficient combinations of labor and capital that can produce a single level of output (similar to indifference curve) q q( L, K ) • Marginal Rate of Technical Substitution (MRTS): – the extra units of one input needed to replace one unit of another input that allows a firm to produce the same level of output – slope of an isoquant (i.e., dK ) dL 6 Diminishing marginal rate of technical substitution 7 Unique Isoquants 8 MRTS and Marginal Products • By definition of isoquant: q q( L, K ) • To see the small change in q, totally differentiate an isoquant: q q dq dL dK 0 L K Marginal increase in Change output from increasing L in L Total increase in output from increasing L by dL MPL dK MRTS MPK dL 9 Example: A Cobb-Douglas Function q 10L0.5 K 0.5 • Production Function: • Capital (K) is not fixed (long-run). • The marginal product of Labor is q 10 0.5L( 0.51) K 0.5 5 L0.5 K 0.5 L • The marginal product of Capital is q 10 0.5L0.5 K ( 0.51) 5 L0.5 K 0.5 K The marginal rate of technical substitution (MRTS) is dK MPL 0.5L0.5 K 0.5 K MRTS dL MPK 0.5L0.5 K 0.5 L Draw the isoquant curve. 10 Returns to Scale • How much output changes if a firm increases all its inputs proportionately. • Long-run concept • Constant Returns to Scale (CRS): t * f(x1, x2) = f(tx1, tx2) • Increasing Returns to Scale (IRS): t * f(x1, x2) < f(tx1, tx2) • Decreasing Returns to Scale (DRS): t * f(x1, x2) > f(tx1, tx2) 11 Reasons for increasing or decreasing returns to scale - Increasing Returns to Scale (IRS): - A larger plant may allow for greater specializations of inputs. - Decreasing Returns to Scale (DRS): - Management problems may arise when the production scale is increased, e.g., cheating by workers. - Large teams of workers may not function as well as small teams. 12 13 For a Cobb-Douglas production function: q AL K • If we double all inputs, A(2L) (2K ) 2 • CRS if • IRS if • DRS if AL K 2 q 1 1 1 14 Productivity and Technical Change • Technical change: – Neutral technical change • q = A*f(L,K) – Non-neutral technical change • e.g. from labor-using to labor-saving 15 Illustration of Neutral Technical Change K Isoquants q = 30 → q = 45 q = 20 → q = 30 q = 10 → q = 15 L 16 Illustration of Non-neutral or Biased Technical Change K K-using or L-saving L-using or K-saving Original isoquants 17 L