Project for managerial economics for decision making

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Demandcostanalysisfunction.pdf

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PRODUCTION, COST, AND SUPPLY FUNCTIONS

Production function

• Firm: transform inputs into outputs

• Production funciton:

q = f (x1, ..., xn)

• 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

1 2 3

...... ...... ...... ...... ...... ............ ...... ...... ...... ...... .....

...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......

# planes

# pilots

q = 1 q = 2

1 2 3

Nebraska beef (tons)

Texas beef (tons)

q = 3

Isoquants and cost minimization

...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......

...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......

...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......1

1.333

1 2 3 4

q = 1 q = 2 q = 3

•

• •

K∗=1.6

L∗=2.5

.................................

.... ..... ..−w/r

...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... .....

•

q = 2

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: Ld

• How does Ld 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

qi = ωi K α i L

β i

• Labor productivity: qi/Li • Total factor productivity (TFP): ωi

Estimating TFP

• Estimate coefficients from (e.g.) Cobb-Douglas production function:

α̂ = r Ki qi

β̂ = w Li qi

where r, w is cost of capital, labor

• Take logarithms and solve production function w.r.t. ωi :

ln ω̂i = ln qi − α̂ ln Ki − β̂ ln Li

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

0 40 48 56

...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......

......

Cost functions: more general case

p◦

p′

q◦ q′

•

T-shirt factory output choice

• Scenario A: BenettonTM, sole buyer of T-shirts, offers price p = $1.8 per T-shirt (for any number of T-shirts)

• Should factory increase output beyond 40 T-shirts/week, thus operating on Saturdays?

• p = 1.8, AC = 1.5, MC = 2.

• Although factory is making money at q=40 (because p > AC), profits would be lower if it produced more (because p < MC); it would lose money at the margin. (Verify this: compute profit at q=40, 41.)

T-shirt factory output choice

• Scenario B: BenettonTM, sole buyer of T-shirts, offers price p = $1.3 per T-shirt (for any number of T-shirts)

• No matter how much factory produces, price is below per-unit cost; i.e., no matter how much factory produces, it will lose money:

p < AC implies q × p < q × AC implies Revenue < Cost

• Optimal decision is not to produce at all

Use marginal cost to decide how much to produce. Use average cost to decide whether to produce.

T-shirt factory supply function

0 40 48 56

MC SLR

T-shirt factory supply function

• Suppose fixed cost has already been paid for the week; then it’s a sunk cost

• Define Average Variable Cost (AVC) as average cost excluding fixed cost

• Short-run supply switches to zero at min AVC

T-shirt factory supply function

0 40 48 56

AVC

MC SSR

Supply by price taking firm

•

•min AVC

min AC

q◦ SR

q◦ LR

AVC

MCSLRSSR ..................................................................................................................................... .... ... .... .... ...

.......................................................... ...... .....

Takeaways

• Use marginal cost when deciding how much to produce, average cost when deciding whether to produce. In other words, marginal costs for marginal decisions.