Chapter 7 Technology Intermediate Microeconomics: A Tool-Building Approach Routledge, UK © 2016 Samiran Banerjee

Production with one input • A technology or production function: q = f(l) where q is output, and l is labor • Marginal productivity of labor, MPl: MPl = = f’(l) • Average productivity of labor, APl: APl = = dq dl q l f(l) l

MPl and APl graphically Slope of • MPl is decreasing • APl is decreasing • APl > MPl • MPl is increasing • APl is increasing • APl < MPl

Consumption vs. production (with two inputs) • Utility, u, as a function of goods, x and y: u = f(x, y) • Indifference curve joins all (x, y) combinations that yield the same utility • Marginal utilities: MUx and MUy • Marginal rate of substitution: MRS • Output, q, as a function of inputs, l and k: q = f(l, k) • Isoquant joins all (l, k) combinations that yield the same output • Marginal productivities: MPl and MPk • Technical rate of substitution: TRS

Types of technologies • Four basic ones – Linear: q(l, k) = al + bk – Leontief: q(l, k) = min {al, bk} – Quasilinear: q(l, k) = f(l) + k, or q(l, k) = l + f(k) – Cobb-Douglas: q(l, k) = Alakb *The function f is generally increasing and strictly concave, i.e., f ’ > 0 and f ” < 0

Returns to scale • Given a technology, let the initial output be qo = f(lo, ko) • Scale each input by a factor, t > 1 • Let the new output be qn = f(tlo, tko) • For all input combinations (lo, ko), qn = tqo implies constant returns to scale (CRS) qn > tqo implies increasing returns to scale (IRS) qn < tqo implies decreasing returns to scale (DRS)

Returns to scale and homogeneity Homogeneity of degree 1 Ex. 1 Cobb-Douglas technology: qo = lo1/2ko1/2 qn = (tlo)1/2(tko)1/2 = t1/2lo1/2t1/2ko1/2 = tlo1/2ko1/2 = tqo (CRS) Ex. 2 Linear technology: qo = (lo + ko)2 qn = (tlo + tko)2 = t2(lo + ko)2 = t2qo (IRS) If a technology is homogeneous of degree r in the inputs, then r = 1 implies CRS r > 1 implies IRS r < 1 implies DRS Homogeneity of degree 2

Production possibilities: 1 input Given two technologies and the total amount of input resource, what are the maximum combination of the two goods possible? Example: • Technology 1: q1 = l1/5 • Technology 2: q2 = (l2)1/2 • Resource constraint: l1 + l2 = 100 From technology 1, l1 = 5q1 From technology 2, l2 = (q2)2 Then 5q1 + (q2)2 = 100. Therefore, T(q1, q2) = 5q1 + (q2)2 – 100 = 0 Transformation frontier or production possibility frontier (PPF)

Marginal rate of transformation The absolute value of the slope |dq2/dq1| of the transformation frontier shows the opportunity cost of good 1 and called the marginal rate of transformation, MRT. • T(q1, q2) = 5q1 + (q2)2 – 100 = 0 • MRT = = • At A = (15, 5), the MRT = 0.5: an additional unit of good 1 can be obtained by giving up ½ a unit of good 2 ∂T/∂q1 ∂T/∂q2 5 2q2

PPF with 2 inputs: Efficient input use Given two technologies, each produced by 2 inputs, l and k, what are the maximum combination of the two goods possible? Example: • Technology 1: q1 = (l1)1/2(k1)1/2 • Technology 2: q2 = 4l2/9 + k2 • Resource constraints: l1 + l2 = 9 and k1 + k2 = 4 • Draw input Edgeworth box • Pick any isoquant for good 1 • Maximize the production for good 2 • Repeat • Join all these Lerner efficient (LE) input combinations Input contract curve Input Edgeworth box

Deriving the PPF with 2 inputs Efficient input use happens on the input contract curve when k1 = 4l1/9 and k2 = 4l2/9 • Substitute k1 = 4l1/9 into technology 1: q1 = (l1)1/2(k1)1/2 = (l1)1/2(4l1/9)1/2 = 2l1/3 • Flip to obtain l1 = 3q1/2 • Substitute k2 = 4l2/9 into technology 2: q2 = 4l2/9 + k2 = 4l2/9 + 4l2/9 = 8l2/9 • Flip to obtain l2 = 9q2/8 • Since l1 + l2 = 9, we get T(q1, q2) = 3q1/2 + 9q2/8 – 9 = 0