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Unit 7. Analyses of LR Production and Costs as Functions of Output. Palladium is a Car Maker’s Best Friend?.

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Unit 7
Unit 7.

  • Analyses of LR Production and Costs as Functions of Output


Palladium is a car maker s best friend
Palladium is a Car Maker’s 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?


This little piggy wants to eat
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?


How big of a plant i e k do we want
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?


Q to produce at each location
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?


Lr max
LR  Max 

  • 1. Produce Q where MR = MC

  • 2. Minimize cost of producing Q

     optimal input combination


Isoquant
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.




Mrts and mp
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



Cobb douglas isoquants
Cobb-Douglas Isoquants

  • Inputs are not perfectly substitutable

  • Diminishing marginal rate of technical substitution

  • Most production processes have isoquants of this shape


Linear isoquants
Linear Isoquants

  • Capital and labor are perfect substitutes


Leontief isoquants
Leontief Isoquants

  • Capital and labor are perfect complements

  • Capital and labor are used in fixed-proportions


Deriving isoquant equation
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

    • => 100 = 10K1/2L1/2

    • => K = 100/L (or K = 100L-1)

    • => slope = -100 / L2

  • Exam #2 – Linear isoquants

    • Desired Q = 100

    • Production fn: Q = 4K + L

    • => 100 = 4K + L

    • K = 25 - .25L

    • => slope = -.25


Budget line
Budget Line

  • = maximum combinations of 2 goods

    that can be bought given one’s income

  • = combinations of 2 goods whose cost

    equals one’s income


Isocost line
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


Isocost line equation
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.






Lr cost min math
LR Cost Min (math)

  •  - slope of isoquant = - slope of isocost line




Assume a production process
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


Given q 10k 1 2 l 1 2
Given q = 10K1/2L1/2

* LR optimum for given q


  • Given q = 10K1/2L1/2, w=10, r=40

  • 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


Optimal k for q 40 given l 4k
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


  • 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


Optimal k for q 100 given l 4k
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


Sr tc for q 100 if k 2
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



Lrtc equation derivation i e lrtc f q
LRTC Equation Derivation[i.e. LRTC=f(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 L*=f(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 L*=f(k), sub k* as fn of q for f(k) deriving L* as fn q


Lrtc calculation example
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



Optimal size plant i e k to min tc of q 1
Optimal* Size Plant (i.e. K)(*=> to min TC of q1)





Returns to scale
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 > mq1

  • Constant  q2 = mq1

  • Decreasing  q2 < mq1

     output ↑ < input ↑


$

TCA

TCB

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

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?


Multiplant production strategy
Multiplant Production Strategy

  • Assume:

    P = output price = 70 - .5qT

    qT = total output (= q1+q2)

    q1 = output from plant #1

    q2 = output from plant #2

     MR = 70 – (q1+q2)

    TC1 = 100+1.5(q1)2 MC1 = 3q1

    TC2 = 300+.5(q2)2 MC2 = q2


Multiplant max
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


If q 1 q 2 20
 If q1 = q2 = 20?

  •  = TR

    - TC1

    - TC2

    = (50)(40)

    - [100 + 1.5(20)2]

    - [300 + .5(20)2]

    = 2000 – 700 – 500 = $800


Multi plant profit max alternative solution procedure
Multi Plant Profit Max(alternative solution procedure)

  • 1. Solve for MCT as fn of qT

    knowing cost min  MC1=MC2=MCT

     MC1=3q1 q1 = 1/3 - MC1 = 1/3 MCT

    MC2 = q2 q2 = MC2 = MCT

     q1+q2 = qT = 4/3 MCT

     MCT = ¾ qT

  • 2. Solve for profit-max qT

     MR=MCT

     70-qT = ¾ qT

    7/4 qT = 70

     q*T = 40  MC*T = ¾ (40) = 30


Multi plant profit max alternative solution procedure1
Multi Plant Profit Max(alternative solution procedure)

  • 3. Solve for q*1 where MC1 = MC*T

     3q1 = 30

    q*1 = 10

  • 4. Solve for q*2 where MC2 = MC*T

     q*2 = 30


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