**Intermediate Microeconomic Theory** Technology

**Inputs** • In order to produce output, firms must employ inputs (or factors of production) • Sometimes divided up into categories: • Labor • Capital • Land

**The Production Function** • To produce any given amount of a good a firm can only use certain combinations of inputs. • Production Function – a function that characterizes how output depends on how many of each input are used. q = f(x1, x2, …, xn) units of output units of input 1 units of input 2…units of input n

**Examples of Production Functions** • What might be candidate production functions for the following? • Vodka Distillary – can be made from either potatoes or corn. • Axe Factory – each axe requires exactly one blade & one handle. • Car Wash – requires both Labor and “Capital”, though not necessarily in fixed proportions. • So what are Production functions analogous to? How are they different?

**Production Functions vs. Utility Functions** • Unlike in utility theory, the output that gets produced has cardinal properties, not just ordinal properties. • For example, consider the following two production functions: • f(x1,x2) = x10.5x20.5 • f(x1,x2) = x12x22

**Isoquants** • Isoquant – set of all possible input bundles that are sufficient to produce a given amount of output. • Isoquants for Vodka? • Isoquants for Axes? • Isoquants for Cars Washed? • So what are Isoquants somewhat analogous to? How do they differ?

**Isoquants** • Again, like with demand theory, we are most interested in understanding trade-offs. • What aspect of Isoquants tells us about trade-offs in the production process?

**Marginal Product of an Input** • Consider how much output changes due to a small change in one input (holding all other inputs constant), or • Now consider the change in output associated with a “very small” change in the input. • Marginal Product (of an input) – the rate-of-change in output associated with increasing one input (holding all other inputs constant), or

**Marginal Product of an Input** • Example: • Suppose you run a car wash business governed by the production function q = f(L, K) = L0.5K0.5 • (q = cars washed, L = Labor hrs, K = machine hrs.) • What will Isoquants look like? • What will be the Marginal Product of Labor at the input bundle {L=4, K= 9}? • What will be the Marginal Product of Labor at the input bundle {L=9, K= 9}?

**Substitution between Inputs** • Marginal Product is interesting on its own (unlike marginal utility) • MP also helpful for considering how to evaluate trade-offs in the production process. • Consider again the following thought exercise: • Suppose firm produces using some input combination (x1’,x2’). • If it used a little bit more x1, how much less of x2 would it have to use to keep output constant? x2 Δx1 x2’ x2” Δx2 f(x1”,x2’) f(x1’,x2’) x1’ x1” x1

**Technical Rate of Substitution (TRS)** • Technical Rate of Substitution (TRS): • TRS = Slope of Isoquant • Also referred to as Marginal Rate of Technical Substitution (MRTS) or Marginal Rate of Transformation (MRT) • So what would be the expression for the TRS for a generalized Cobb-Douglas Production function F(x1,x2) = x1ax2b?

**Substitution between Inputs (cont.)** • We are often interested in production technologies that exhibit: • Diminishing Marginal Product (MP) in each input. • Diminishing Technical Rate of Substitution (TRS).

**Diminishing MP** machine hrs (K) 9 9.5 9 6.7 cars 6 cars 4 5 9 10 worker hrs (L)

**Diminishing TRS** machine hrs (K) 16 4 4 cars 1 4 worker hrs (L)