Analyses of LR Production and Costs as Functions of Output

1
Unit 7.

• Analyses of LR Production and Costs as Functions
  of Output

2
Palladium is a Car Makers Best Friend?

• Palladium is a precious metal used as an input in
  the production of automobile catalytic
  converters, which are necessary to help
  automakers meet governmental, mandated
  environmental standards for removing pollutants
  from automobile exhaust systems. Between 1992
  and 2000, palladium prices increased from about
  80 to over 750 per ounce. One response at Ford
  was a managerial decision to guard against future
  palladium price increases by stockpiling the
  metal. Some analysts estimate that Ford
  ultimately stockpiled over 2 million ounces of
  palladium and, in some cases, at prices exceeding
  1,000 per ounce. Was this a good managerial
  move?

3
This Little Piggy Wants to Eat

• Assume Kent Feeds is producing swine feed that
  has a minimal protein content () requirement.
  Two alternative sources of protein can be used
  and are regarded as perfect substitutes. What
  does this mean and what are the implications for
  what inputs Kent Feeds is likely to use to
  produce their feed?

4
How Big of a Plant (i.e. K) Do We Want?

• Assume a LR production process utilizing capital
  (K) and labor (L) can be represented by a
  production function Q 10K1/2L1/2. If the per
  unit cost of capital is 40 and the per unit cost
  of L is 100, what is the cost-minimizing
  combination of K and L to use to produce 40 units
  of output? 100 units of output? If the firm
  uses 5 units of K and 3.2 units of L to produce
  40 units of output, how much above minimum are
  total production costs?

5
Q to Produce at Each Location?

• Funky Foods has two production facilities. One
  in Dairyland was built 10 years ago and the other
  in Boondocks was built just last year. The newer
  plant is more mechanized meaning it has higher
  fixed costs, but lower variable costs (including
  labor). What would be your recommendation to
  management of Funky Foods regarding 1) total
  product to produce and 2) the quantities to
  produce at each plant?

6
LR ? Max ?

• 1. Produce Q where MR MC
• 2. Minimize cost of producing Q
• ? optimal input combination

7
Examples of LR Cost Concerns (in the news
recently)

• New production technology reduced Saturn
  machinery costs 30
• GM labor costs per vehicle about 2x greater than
  for Toyota, 1/3x greater than for Ford
• Southwest Airlines costs lower than competitors
  by one-half to one-third
• Sears, K-Mart, Target trying to cut costs to
  compete with Wal-Mart
• Work teams, quality circles, profit sharing,
  computer integrated mfg, computer aided design,
  remanufacturing, etc. are relatively new industry
  buzz words
• Production restructuring has resulted in many
  companies shutting down some plants and expanding
  the operation of other plants (some to 24
  hrs/day, 7 days/wk)

8
Isoquant

• The combinations of inputs (K, L) that yield the
  producer the same level of output.
• The shape of an isoquant reflects the ease with
  which a producer can substitute among inputs
  while maintaining the same level of output.

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Typical Isoquant
10
Technological Progress

11
SR Production in LR Diagram

12

13
Indifference Curve Isoquant Slopes
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MRTS and MP

• MRTS marginal rate of technical substitution
• the rate at which a firm must substitute one
  input for another in order to keep
  production at a given level
• - slope of isoquant
• the rate at which capital can be exchanged
  for 1 more (or less) unit of labor
• MPK the marginal product of K
• MPL the marginal product of L
• ?Q MPK ?K MPL ?L
• ?Q 0 along a given isosquant
• ? MPK ?K MPL ?L 0
• ? inverse MP ratio

15
Cobb-Douglas Isoquants

• Inputs are not perfectly substitutable
• Diminishing marginal rate of technical
  substitution
• Most production processes have isoquants of this
  shape

16
Perfect Substitute Inputs (Examples)

• Liquid vs. dry fertilizer
• Ethanol vs. regular gasoline
• Soy protein vs. other (fish ?) protein
• U.S. soybeans vs. Brazilian soybeans
• Truck vs. rail transportation
• Sugar vs. aspartame

17
Linear Isoquants

• Capital and labor are perfect substitutes

18
Perfect Complement Inputs? (Examples)

• 1 oz. wine 4 oz. 7-Up wine cooler
• 1 bun ¼ lb. beef burger
• 1 bu. soybean hexane solution ? 12 lb. SBO
• Other recipe products

19
Leontief Isoquants

• Capital and labor are perfect complements
• Capital and labor are used in fixed-proportions

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Deriving Isoquant Equation

• Plug desired Q of output into production function
  and solve for K as a function of L.
• Example 1 Cobb Douglas isoquants
• Desired Q 100
• Production fn Q 10K1/2L1/2
• gt 100 10K1/2L1/2
• gt K 100/L (or K 100L-1)
• gt slope -100 / L2
• Exam 2 Linear isoquants
• Desired Q 100
• Production fn Q 4K L
• gt 100 4K L
• K 25 - .25L
• gt slope -.25

21

• Two ways to calculate MRTS ( - slope of
  isoquant)
• inverse MP ratio MPL/MPK (calculated given
  production function)
• -dK / dL (calculated given isoquant equation)
• Two ways to calculate MRS ( - slope of
  indifference curve)
• 1. Inverse MU ratio MUX/MUY (calculated given
  utility function)
• 2. - dy / dx (calculated given indifference
  curve equation)

22
Budget Line

• maximum combinations of 2 goods
• that can be bought given ones income
• combinations of 2 goods whose cost
• equals ones income

23
Isocost Line

• maximum combinations of 2 inputs
• that can be purchased given a
• production budget (cost level)
• combinations of 2 inputs that are
• equal in cost

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Isocost Line Equation

• TC1 rK wL
• ? rK TC1 wL
• ? K
• Note slope inverse input price ratio
• rate at which capital can be exchanged for
• 1 unit of labor, while holding costs
  constant.

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Increasing Isocost

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Changing Input Prices

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Different Ways (Costs) of Producing q1

28
Cost Minimization (graph)

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LR Cost Min (math)

• ? - slope of isoquant - slope of isocost line

?
?
?
?
?
30
Reducing LR Cost (e.g.)
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SR vs LR Production

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Assume a production process

• Q 10K1/2L1/2
• Q units of output
• K units of capital
• L units of labor
• R rental rate for K 40
• W wage rate for L 10

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• Given q 10K1/2L1/2, w10, r40
• Minimum LR Cost Condition
• ? inverse MP ratio inverse input P ratio
• ? (MP of L)/(MP of K) w/r
• ? (5K1/2L-1/2)/(5K-1/2L1/2) 10/40
• ? K/L ¼
• ? L 4K

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Two Different costs of q 100

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SR TC for q 100? (If K 2)

• Q 100 10K1/2L1/2
• ? 100 10 (2)1/2(L)1/2
• ? L 100/2 50
• ? SR TC 40K 10L
• 40(2) 10(50)
• 80 500 580

36
Optimal K for q 100? (Given L 4K)

• Q 100 10K1/2L1/2
• ? 100 10 K1/2(4K)1/2
• ? 100 20K
• ? K 5
• ? L 20
• ? min SR TC 40K 10L
• 40(5) 10(20)
• 200 200 400

37

• SR TC for q 40? (If K 5)
• q 40 10K1/2L1/2
• ? 40 10 (5)1/2(L)1/2
• ? L 16/5 3.2
• ? SR TC 40K 10L
• 40(5) 10(3.2)
• 200 32 232

38
Optimal K for q 40? (Given L 4K)

• q 40 10K1/2L1/2
• ? 40 10 K1/2(4K)1/2
• ? 40 20K
• ? K 2
• ? L 8
• ? min SR TC 40K 10L
• 40(2) 10(8)
• 80 80 160

39
Given q 10K1/2L1/2

? LR optimum for given q
40
LRTC Equation Derivationi.e. LRTCf(q)

• ? LRTC rk wL
• r(k as fn of q) w(L as fn of q)
• To find K as fn q
• from equal-slopes condition Lf(k), sub f(k)
  for L into production fn and solve for k as fn
  q
• To find L as fn q
• from equal-slopes condition Lf(k), sub k as
  fn of q for f(k) deriving L as fn q

41
LRTC Calculation Example

• Assume q 10K1/2L1/2, r 40, w 10
• L 4K (equal-slopes condition)
• K as fn q
• q 10K1/2(4K)1/2
• 10K1/22K1/2
• 20K ?

• LR TC rk wL 40(.05q)10(.2q)
• 2q 2q
• 4q

• L as fn q
• L 4K
• 4(.05 q)
• L .2q

42
?

43
Graph of SRTC and LRTC

44

• Assume a firm is considering using two different
  plants (A and B) with the corresponding short run
  TC curves given in the diagram below.

TCA
TCB
Q of output
Q1

• Explain
• 1. Which plant should the firm build if neither
  plant has been built yet?
• 2. How do long-run plant construction decisions
  made today determine future short-run plant
  production costs?
• 3. How should the firm allocate its production
  to the above plants if both plants are up and
  operating?

45
Returns to Scale

• ? a LR production concept that looks at how the
  output of a business changes when ALL inputs are
  changed by the same proportion (i.e. the scale
  of the business changes)
• Let q1 f(L,K) initial output
• q2 f(mL, mK) new output
• m new input level as proportion of old input
  level
• Types of Returns to Scale
• Increasing ? q2 gt mq1
• Constant ? q2 mq1
• Decreasing ? q2 lt mq1
• ? output ? lt input ?

46
Multiplant Production Strategy

• Assume
• P output price 70 - .5qT
• qT total output ( q1q2)
• q1 output from plant 1
• q2 output from plant 2
• ? MR 70 (q1q2)
• TC1 1001.5(q1)2 ? MC1 3q1
• TC2 300.5(q2)2 ? MC2 q2

47

48
Multiplant ? Max

• (1) MR MC1
• (2) MR MC2
• (1) 70 (q1 q2) 3q1
• (2) 70 (q1 q2) q2
• from (1), q2 70 4q1
• Sub into (2),
• ? 70 (q1 70 4q1) 70 4q1
• ? 7q1 70
• ? q1 10, q2 30
• ? TR TC1 TC2
• (50)(40)
• - 100 1.5(10)2
• - 300 .5(30)2
• 2000 250 750 1000

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? If q1 q2 20?

• ? TR
• - TC1
• - TC2
• (50)(40)
• - 100 1.5(20)2
• - 300 .5(20)2
• 2000 700 500 800