Unit 7.

Analyses of LR Production and Costs as Functions of Output

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

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?

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?

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 1. Produce Q where MR = MC

2. Minimize cost of producing Q

optimal input combination

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)

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.

Typical Isoquant

Technological Progress

SR Production in LR Diagram

Indifference Curve & Isoquant Slopes

Indiff Curve Isosquant

- slope = MRS

= rate at which consumer is willing to exch Y for 1X in order to hold U constant

= inverse MU ratio

= MUX/MUY

For given indiff curve, dU = 0

Derived from diff types of U fns:

1) Cobb Douglas U = XY

2) Perfect substitutes U=X+Y

3) Perfect complements U = min [X,Y]

- slope = MRTS

= rate at which producer is able to exch K for 1L in order to hold Q constant

= inverse MP ratio

= MPL/MPK

For given isoquant, dQ = 0

Derived from diff types of production fns:

1) Cobb Douglas Q = LK

2) Perfect substitutes Q=L+K

3) Perfect complements Q = min [X,Y]

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

K

L

Q

K

Q

L

K

L

MP

MPL

K

Cobb-Douglas Isoquants

Inputs are not perfectly substitutable

Diminishing marginal rate of technical substitution

Most production processes have isoquants of this shape

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

Linear Isoquants

Capital and labor are perfect substitutes

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

Leontief Isoquants

Capital and labor are perfect complements

Capital and labor are used in fixed-proportions

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

Two ways to calculate MRTS (= - slope of isoquant):

1. = inverse MP ratio = MPL/MPK (calculated given production function)

2. = -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)

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

= 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

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.

TC

r

w

rL1

K

L

Increasing Isocost

Changing Input Prices

Different Ways (Costs) of Producing q1

Cost Minimization (graph)

LR Cost Min (math) - slope of isoquant = - slope of isocost line

MP

MP

w

r

M P

MPr

w

rr

M P r

M P MPwMP

r

M P

w

MP

MC MC

L

K

L

K

L

K L L

K L

K L

( ) ( )

( )( ) ( )

1 1

Reducing LR Cost (e.g.)

LK

MCMC

MP

w

MP

r

LK

LK

,

SR vs LR Production

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 = 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

Two Different costs of q = 100

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

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 = 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 = 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

Given q = 10K1/2L1/2

Q K L TC=40K+10L

40* 2* 8* 160*

100* 5* 20* 400*

40 5 3.2 232

100 2 50 580

* LR optimum for given 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 qfrom 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 qfrom equal-slopes condition L*=f(k), sub k* as

fn of q for f(k) deriving L* as fn q

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 Kq

q* . 2 0

0 5

L* as fn q

L* = 4K*

= 4(.05 q)

L* = .2q

LR TC = rk* + wL* = 40(.05q)+10(.2q)

= 2q + 2q

= 4q

Graph of SRTC and LRTC

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

$

Q1

Q of output

TCA

TCB

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?

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 outputm = new input level as proportion of old input

levelTypes of Returns to Scale:

1) Increasing q2 > mq1

2) Constant q2 = mq1

3) Decreasing q2 < mq1 output ↑ < input ↑

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

Pq TC TC

d

dq

d Pq

dq

d TC

dq

M R MC

d

dq

d Pq

dq

dTC

dq

M R MC

T

T

T

1 2

1 1

1

1

1

2 2

2

2

2

0

0

m ax

( ) ( )

( )

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 q1 = q2 = 20?

= TR

- TC1

- TC2

= (50)(40)- [100 + 1.5(20)2]- [300 + .5(20)2]

= 2000 – 700 – 500 = $800