The Economics of Production

Make or Buy Decisions

Capacity Expansion

Learning-Curves

Break-even Analysis

Production Functions

Make or Buy?

Let c1 = unit purchase price c2 = unit production cost (c2 < c1) K = fixed cost of production x = number of units required

Produce if K + c2 x c1x

or x K / (c1 - c2 )

Make or Buy Example

• It costs the Maker Bi Company $20 a unit to purchase a critical part used in the manufacture of their primary product line – a thing-um-a-jig

• It is estimated that the part could be produced internally at a unit cost of $16 after incurring a fixed cost of $20,000 for the necessary equipment.

• What to do?

a thing-um-a-jig

What to do?

Let c1 = $20 c2 = $16 K = $200,000 x = number of units required

Produce ifx K / (c1 - c2 ) = 20,000 /(20 – 16) = 5,000

Nonlinear Cost Function

Let c1 = unit purchase price c2 = K + axb where K, a, b > 0 x = number of units required

If c1 = 20 and c2 = 20,000 + 100x.7 , then

More on that Nonlinear Cost Functionx Make Buy1900 $39,730 $38,0001925 $39,911 $38,5001950 $40,092 $39,0001975 $40,272 $39,5002000 $40,451 $40,0002025 $40,630 $40,5002035 $40,701 $40,7002050 $40,808 $41,0002075 $40,985 $41,5002100 $41,162 $42,0002125 $41,338 $42,5002150 $41,513 $43,0002175 $41,688 $43,5002200 $41,862 $44,0002225 $42,036 $44,5002250 $42,209 $45,0002275 $42,381 $45,500

Strategic Decisions Capacity Expansion

• Capacity Growth Planning– when to construct new facilities

– where to locate facilities

– how large to size a facility

• Economies of scale– advantage of expanding existing facilities

– share plant, equipment, support personnel

– avoid duplication at separate locations

Capacity Expansioncompeting objectives: maximize market share

maximize capacity utilization

time

number units

demand

capacity leads demand

time

number units

demand

capacity lags demand

We need a modellet D = annual increase in demand x = time interval (in yrs) between capacity increases r = annual discount rate, compounded continuously f(y) = cost of expansion of capacity y

assume y = xD, then

cost = C(x) = f(xD) [1 + e-rx + (e-2rx ) + (e-3rx ) + …]= f(xD) [1 + e-rx + (e-rx )2 + (e-rx )3 + …]= f(xD) / [1 – e-rx]

assume f(y) = kya , then

find the x that minimizes C(x)

( )( )

1

a

rx

k xDC x

e

A Diversion - the Geometric Series

2

0

2

0

11 ... ...

1

11 ... ...

1

n n

n

n nrx rx rx rxrx

n

y y y yy

e e e ee

You see? It does

converge.

Discounting – another diversion

Consider the time value of money$1.00 today is worth more than a $1.00 next yearHow much more is it worth?

If r = annual interest rate, then it is worth (1+r) $1.00After two years, it is worth (1+r)2 $1.00 (compounded)

Compounded quarterly for 1 yr =

Compounded continuously for one year =

After t years =

41 / 4r

lim 1 /n r

nr n e

lim 1 /t

n rt

nr n e

More diversionary discounting

A stream of costs: C1, C2, …, Cn incurred at times t1, t2,…, tn has a present value of:

1

i

nrt

ii

C e

Why can’t you show us an example?

For an infinite planning horizonwhere x is the time between expansions:

1 1irx

rxi

CCe

e

The Example

• Chemical firm expanding at a cost ($M) of

– where y is in tons per year.

• Demand is growing at the rate of D = 5,000 tons per year and future costs are discounted at a rate of r = 16 percent

• Find x that minimizes

.62( ) .0107f y y

.62

.16

.0107(5000 )( )

1 x

xC x

e

Capacity Expansion Solution

C(x) - $M

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12

years

.62

.16

.0107(5000 )( )

1 x

xC x

e

5 10.356535.1 10.350425.2 10.34565.3 10.342035.4 10.339645.5 10.338395.6 10.338235.7 10.339115.8 10.340985.9 10.34382

6 10.34757

alternately set C’(x) = 0solve for x.

Learning CurvesBased upon the observation that unit labor hours or costs decrease for each additional unit produced

Units produced

Directlabor hrsper unit

Why does this happen?• Employee learning• reduced set-up times• better routing and scheduling of material (WIP)• improved tool design• more efficient material handling equip. (MHE)• reduced lead-times• improved (simplified) product design• production smoothing• quality assurance• revised plant layout• increased machine utilization

Learning Curve(experience curves)

Y(u) = labor hours to produce the uth unit

assume Y(u) = au-b

a = hours to produce the first unitb = rate at which production hours decline

labor comingto work

Learning Curves

Assume hours to produce unit 2n is a fixedpercentage of the hours to produce unit n

Then for an 80 percent learning curve:

3219.2ln

8.ln

80.2)2(

)(

)2(

bor

an

na

nY

nY bb

b ln(Pct/100)

ln 2b

Observe the simple

formula

Learning Curvesleast-squares analysis

Unit Direct LaborNumber - x Hours -Y(x)

20 35.840 30.160 27.380 25.7100 24.1

Fit Y(x) = ax-b

using Excel

Y(x) = 74x-.243

2-.243 = .845or a 84.5% learning curve

Learning CurvesCumulative Cost

x

xTxV

aiiYxT

aiiYx

i

bx

i

b

)()(

)()(

)(

11

hours to produce ith unit

cumulative direct laborhrs to produce x units

average unit hours toproduce x units

Learning CurvesApproximate Cumulative Cost

b

xa

xb

xaxV

b

xadiiadiiYxT

bb

bxb

x

1)1()(

1)()(

1

1

00

ExampleY(x) = 74x-.243

2-.243 = .845 or a 84.5% learning curve

15.420030,8333/20V(2000)

hr.833,30)2000(75.97)2000(

75.97757.

74

1243.

7474)(

757.

757.757.1243.

0

243.

T

xxx

diixTx

X T(x) V(x)100 3192.396 31.92396200 5395.062 26.97531300 7333.268 24.44423400 9117.509 22.79377500 10795.36 21.59072600 12393.02 20.65504700 13926.95 19.89564800 15408.34 19.26043900 16845.28 18.716981000 18243.86 18.24386

Break-Even AnalysisLet x = number of units produced and sold x = S-1(unit selling price) S(x) = unit selling price F = fixed cost g(x) = variable cost to produce x units

then break-even point occurs when revenue = cost; or

S(x) x = F + g(x)

and profit = revenue – cost orP(x) = S(x)x – [F + g(x)]

Sam Evenon a break

Break-Even Analysis

x

$

F

Breakeven pt

loss

profit

Maxprofit

Revenue curve

Cost curve

loss

Diminishingreturns

Break-Even AnalysisDemand Curve

x

S(x)

S(x) = d + e x + f x2 (quadratic)

d, e, and f are constants to be determined

Break-Even AnalysisDemand Curve

x

S(x)

S(x) = d + e x + f x2

d, e, and f are constants to be determined

Approximate as linearS(x) = d + e x

Break-Even AnalysisUnit Cost

Let M = direct material unit cost ($/unit)L = direct labor rate ($/hour)B = factory burden rateY(x) = direct labor hours to produce unit xC(x) = cost to produce unit x

C(x) = M + L Y(x) + L B Y(x)

= M + (1+B) L Y(x)

= M + (1+B) L a x-b

Learningcurve effect

The Factory Burden Diversion

Manufacturing Costs

Factory burdenDirect costs

Direct laborDirect

material

Indirect material

Indirect labor

Indirect expense

-Supervision-Engineering-Maintenance

-Heating-Lighting-Depreciation-Rent & Taxes

-Office &janitorialsupplies -Paint

Factory Burden - exampleCategory annual costIndirect material $ 6,120Indirect labor 42,800Indirect expenses 22,900

total $71,820

Product annual production labor hours rate wages A 100,000 1000 $9/hr $9,000 B 140,000 1400 $7/hr 9,800 C 80,000 1600 $7/hr 11,200

total 4000 $30,000

burden rate = 71,820 /30,000 = 2.394 per direct labor $

Manufacturing Costs

General Overhead Costs

Profit

Selling Price S(x)

Administrative Costs

MarketingCosts

DevelopmentCosts

Demands

Cumulative Cost

g(x) = M x + L (1+B) T(x)

= M x + L (1+B) [a x1-b / (1-b)]

total cost = F + M x + L (1+B) a x1-b / (1-b)where F is a fixed cost to produce product x

Unit cost:C(x) = M + (1+B) L a x-b

1)(

1

b

xaxT

bLearning curve

Break-Even Analysis -Profit

Profit = P(x) = S(x) x - [F + g(x)]

letting S(x) = d + ex, e < 0

P(x) = (d + e x) x - F - M x - L (1+B) a x1-b /(1-b)

= d x + e x2 - F - M x - g x1-b

where g = L (1+B) a /(1-b)

More Break-Even Analysis

P(x) = (d - M) x + e x2 - g x1-b - F

break-even: set P(x) = 0 and solve for x

maximize profit: set and solve for xdP(x)

= 0 dx

2

1

2

( ) 2 (1 ) 0

( ) 2 (1 )

b

b

dP xd M ex b gx

dx

d P xe b b gx

dx

for e < 0, a max point can exist

Break-Even Analysis - example

P(x) = d x + e x2 - F - M x - g x-b+1

where g = (1+B) L a /(1-b)

Data:d = 100e = - .01F = $100,000M = $4B = .5L = $20 / hra = 10b = .60

P(x) = 100 x - .01 x2 – 100,000

- 4 x – (1+.5) (20) (10) x.4 / .4

= 96x -.01x2 –750 x.4 –100,000

2-.6 = 66%

The Math

2 0.4

0.6

1.6

1/1.6

( ) 96x .01x – 750 x –100,000 0

1382

'( ) 96 .02 300 0

* 4706

''( ) .02 180 0

.02296.07

180

P x

x

P x x x

x

P x x

x

The Graph

P(x)

-$150,000

-$100,000

-$50,000

$0

$50,000

$100,000

$150,000

0 1000 2000 3000 4000 5000 6000 7000

x = 1382

x = 4706

Production Functions

A production function expresses the relationship between an organization's inputs and its outputs. It indicates, in mathematical or graphical form, what outputs can be obtained from various amounts and combinations of factor inputs.

In its most general mathematical form, a production function is expressed as:Q = f(X1,X2,X3,...,Xn) where: Q = quantity of output and

X1,X2,X3,...,Xn = factor inputs (such as capital, labor, raw

materials, land, technology, or management)

Production Functions

There are several ways of specifying this function. One is as an additive production function:

Q = a + bX1 + cX2 + dX3,...

where a,b,c, and d are parameters that are determined empirically.

Another is as a Cobb-Douglas production function

Q = f(L,K,M) = A * (Lalpha) * (Kbeta) * (Mgamma)

where L = labor, K = capital, M = materials and supplies, and Q = units of product.

Cobb-Douglas Production Function

Q = f(L,K,M) = A * (Lalpha) * (Kbeta) * (Mgamma)

Properties of the Cobb-Douglas production function:Decreasing returns to scale: alpha + beta + gamma < 1Increasing returns to scale: alpha + beta + gamma > 1

Let CL, CK, and CM = the unit cost of labor, capital, and material, then

C(L,K,M) = CLL + CKK + CM M

is the total cost function

A Little Production ProblemAn interesting problem: Given a monthly budget of $B, how should the

money be spent to obtain a specified output Q?

Find L, K, and M where L = dollars spent on labor, K = dollars spent on facilities and equipment, and M = dollars spent on material

I know I can work this

one.

.3 .2 .4( , , ) 100

Subject to: $1,000,000

3 ; 2( )

Q f L K M L K M

L K M

M L K L M

The Inevitable Example

labor capital materialA alpha beta gamma    

100 0.3 0.2 0.4      L K M Q    $8,333 $66,667 $25,000 794,700 RHSbudget 1 1 1 100,000 100,000   -3   1 0 0  -2 1 -2 0 0

Stop the madness.Optimize your production system!

profits

homework: turn-in breakeven problemtext: Chapter 1- 29, 30, 31, 32, 34, 35

36, 37 ,38, 43, 44