PRODUCTION, COST, AND SUPPLY FUNCTIONS
• Firm: transform inputs into outputs • Production funciton:
q = f (x1, ..., xn)
• In what follows, two inputs: K, L
2 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.
3 Production functions: two extreme cases
# planes Nebraska beef (tons)
3 q = 1 q = 2 3
...... 2 . 2 q = 3 ...... 1 . . 1 ...... # pilots Texas beef (tons)
1 2 3 1 2 3
4 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
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: 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)
6 Productivity
• Cobb-Douglas production function
α β qi = ωi Ki Li
• Labor productivity: qi /Li
• Total factor productivity (TFP): ωi
7 Estimating TFP
• Estimate coefficients from (e.g.) Cobb-Douglas production function: r Ki w Li αb = βb = qi qi where r, w is cost of capital, labor
• Take logarithms and solve production function w.r.t. ωi :
ln ωbi = ln qi − αb ln Ki − βb ln Li
8 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
9 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)
10 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)
11 Example: the T-shirt factory
12 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)
13 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 . . .
14 T-shirt factory cost function
p
...... 3 . . MC ...... 2 ...... AC ...... 1 ...... q 0 . 0 40 48 56
15 Cost functions: more general case
p
MC
0 ...... p . . . . AC ...... ◦ ...... p •...... q
q◦ q0
16 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.)
17 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.
18 T-shirt factory supply function
p
3 MC SLR
2 AC ......
1
q 0 0 40 48 56
19 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
20 T-shirt factory supply function
p
3 MC SSR
2 AVC
1
q 0 0 40 48 56
21 Supply by price taking firm
p
MC SSR SLR ...... AC ...... •. min AC ...... AVC ...... min AVC •...... q
◦ ◦ qSR qLR
22 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.
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