PRODUCTION, COST, AND SUPPLY FUNCTIONS Production function Firm: - PowerPoint PPT Presentation

PRODUCTION, COST, AND SUPPLY FUNCTIONS

Production function • Firm: transform inputs into outputs • Production funciton : q = f ( x 1 , ..., x n ) • In what follows, two inputs: K , L

Isoquants • Contour lines that connect points with same in ( K , L ) space producing same output level. • Similarly to indifference curves, generally convex (diminishing marginal returns). • The more convex, the the more complementary the inputs; the flatter, the closer substitutes.

Production functions: two extreme cases # planes Nebraska beef (tons) q = 1 q = 2 3 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q = 3 2 . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Texas beef (tons) . . . . # pilots . . . . 1 2 3 1 2 3

Isoquants and cost minimization K K 4 3 q = 1 q = 2 q = 3 q = 2 . . . . . − w / r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . K ∗ =1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . 1.333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L . . . L . . 1 2 3 4 L ∗ =2.5

Demand for inputs • For given input prices r , w , and for a given output level q , find optimal input mix K , L . • The resulting L , for example, is demand for labor: L d • How does L d depend on w , p and especially r ? • Example 1: desktop computer and demand for labor • Example 2: Compare two industries (hydroelectric dam construction; aircraft construction) in two countries (U.S. and India)

Productivity • Cobb-Douglas production function q i = ω i K α i L β i • Labor productivity: q i / L i • Total factor productivity (TFP): ω i

Estimating TFP • Estimate coefficients from (e.g.) Cobb-Douglas production function: α = r K i β = w L i � � q i q i where r , w is cost of capital, labor • Take logarithms and solve production function w.r.t. ω i : α ln K i − � ln � ω i = ln q i − � β ln L i

Cost functions • For given input prices r , w , and for a given output level q , find optimal input mix K , L • Determine cost r K + w L • C ( q ): minimum cost required to achieve output level q

Cost concepts • Fixed cost (FC): the cost that does not depend on the output level, C (0) • Variable cost (VC): that cost which would be zero if the output level were zero, C ( q ) − C (0) • Average cost (AC) (a.k.a. “unit cost”): total cost divided by output level, C ( q ) / q • Marginal cost (MC): the unit cost of a small increase in output − Definition: derivative of cost with respect to output, d C / d q − Approximated by C ( q ) − C ( q − 1)

Examples • Bagels: modest fixed cost (space), relatively constant marginal cost (labor and materials) • Electricity generation: large fixed cost (plant), initially declining marginal cost (large plants are more efficient, and many plants have startup costs) • Music CDs: large fixed cost (recording), small marginal cost (production and distribution)

Example: the T-shirt factory

T-shirt factory example To produce T-shirts: • Lease one machine at $20/week • Machine requires one worker, produces one T-shirt per hour • Worker is paid $1/hour on weekdays (up to 40 hours), $2/hour on Saturdays (up to 8 hours), $3 on Sundays (up to 8 hours)

T-shirt factory costs Suppose output level is 40 T-shirts per week. Then, • Fixed cost: FC = $20. Variable cost: VC = 40 × $1 = $40 • Average cost: AC = ($20+$40)/40 = $1.5 • Marginal cost: MC = $2 (Note that producing an extra T-shirt would imply working on Saturday, which costs more.) Similar calculations can be made for other output levels, leading to the cost function . . .

T-shirt factory cost function p 3 MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . q . . . . . . . . . . . . . . . . . . 0 . . . . . 0 40 48 56