Production and Costs

1
Chapter 5

• Production and Costs
• Productivity determines the relationship between
  inputs and outputs and the various measures of
  costs.

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Production function

• Output is produced from inputs, given a
  particular technology
• 2 inputs labor L and capital K
• Q F(K,L)
• Efficient production generates any output level
  at the least cost.
• Short-run some input is fixed.
• Long-run all inputs are variable.

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Some Definitions

• Total product (TP) is the most that can be
  produced with given inputs.
• Average product of labor (APL) is TP/L.
• Marginal product of labor (MPL) is ?TP/?L.

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Total and Marginal Product

• TP is increasing with additional L if the MPL gt
  0.
• Diminishing marginal returns occur when MPL gt 0
  but decreasing.

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Marginal Productivity

• This is a short-run concept only one input
  changes.
• As long as marginal product is positive output is
  increasing.
• If marginal product is decreasing, then output
  increases at a decreasing rate.
• In general, firms will produce where there are
  diminishing marginal returns.

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Excel Workbook

• Compare TP with AP and MP click here

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Value of Marginal Product

• The value of marginal product of labor VMPL is
  the additional revenue generated by the
  additional input.
• If P is the price of the product, then
• VMPL P?MPL
• If w is paid for L and w lt VMPL, then the firm
  should buy more L.
• If w gt VMPL, then firm should buy less of L.

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Optimal Employment

• Firms optimal purchase of L is where w VMPL
  and VMPL is decreasing.
• This implies diminishing MPL.

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Short-Run vs. Long-run

• In the SR some inputs are fixed.
• In the LR all inputs are variable.
• In the LR the firm is able to substitute some of
  one input for some of another and keep output the
  same (on same isoquant).
• This is called the marginal rate of technical
  substitution (MRTS).

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Perfect Substitutes

• Q 4K L
• If L is decreased by 1, Q decreases by 1.
• If K is increased by 1, Q increases by 4.
• Rate of substitution is ?K/?L ? 1/4.

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Perfect Complements

• Q min K/4, L
• If K/4 gt L, then can only produce L.
• Must have 1 unit of L for every 4 of K
• No substitution fixed proportions

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No Substitution

• Q min K/4, L
• If K 4 and L 1, can produce Q(4,1) 1.
• K gt 4 when L 1 does not increase Q.

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Variable Substitution

• Q K0.5 L0.5
• The ability to substitute is
• MRTSKL MPL/MPK
• MPL 0.5K0.5L0.5-1 K0.5/2L0.5
• MPL depends on level of K and L.

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Graph of Isoquant
15
Costs

• Economic costs are the opportunity cost of
  production.
• With efficient production each level of Q is
  produced at lowest costs.
• To minimize the cost of producing any Q, find
  lowest expenditure
• C rK wL
• on K and L which produces Q.

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Iso-Cost Line

• All bundles of K and L that cost the same.
• Solve iso-cost equation for K to graph it

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Optimal Inputs for Q

• Use K and L on isoquant for Q and on lowest
  isocost.

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Bang-per-buck

• At cost minimizing K and L, slopes are equal
• MRTSKL MPL/MPK w/r
• or bang-per-buck is the same.

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Cost Definitions

• TC(Q) FC VC(Q)
• MC(Q) dTC(Q)/dQ dVC(Q)/dQ
• AVC(Q) VC(Q)/Q
• AFC(Q) FC/Q
• ATC(Q) TC(Q)/Q AFC(Q) AVC(Q)

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An Example

• If TC(Q) 200 2Q2
• FC 200
• VC(Q) 2Q2
• MC(Q) 4Q
• AVC(Q) 2Q2/Q 2Q
• AFC(Q) 200/Q
• ATC(Q) 200/Q 2Q

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Graph of Example
MC(Q) 4Q AVC(Q) 2Q AFC(Q) 200/Q ATC(Q)
200/Q 2Q
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Productivity and Costs

• The shape of the TP curve determines the shape of
  VC and TC.
• If L is variable input, K is fixed input
• VC w?L
• FC r?K
• TC VC FC
• MC is slope of TC

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• Multiply L by the wage and flip the graph.
• Vertical axis measures costs.
• ? slope of TP implies ? slope of VC.

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Graph of MC

• Initial ? MPL implies initial ? MC
• Eventual ? MPL implies eventual ? MC
• MC is U-shaped

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MC and AVC

• If MC lt AVC, then AVC is falling.
• If MC gt AVC, then AVC is rising.
• If MC AVC, AVC is minimized.

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Example

• TC(Q) 100 10Q ? 18Q2 2Q3
• MC(Q) 10 ? 36Q 6Q2
• dMC/dQ ? 36 12 Q
• MC is declining for Q lt 3
• and increasing for Q gt 3.

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Excel Worksheet

• Compare TC, AC and MC click here

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ATC and AFC

• AFC FC/Q is declining.
• ATC AVC AFC

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LR Costs

• In the long-run FC 0.
• Economies of scale is a long-run concept
• increasing if LAC is falling
• decreasing if LAC is rising
• constant if LAC is constant

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One technology may exhibit all three properties
31
Next Week

• Organization of the Firm.
• Project 1 is due next week.
• Midterm Exam next week

32
Multi-Product Production

• Economies of scope lower costs to produce two
  products together.
• C(Q1,0) C(0,Q2) gt C(Q1,Q2)
• Due to shared resources (indivisible inputs).
• Hub-and-spoke airline operations.

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• Cost complementarity production of a second
  product lowers MC of the first.
• MC1(Q1,0) gt MC1(Q1,Q2)
• One product uses byproduct of production of the
  other product.
• Lumber mills generate sawdust and shavings as
  byproduct of sawing and shaving lumber.
• MC of byproduct is zero.