Production and Cost in the Long Run Overheads. The long run In the long run, there are no fixed...

Post on 17-Dec-2015

219 views 0 download

Tags:

Transcript of Production and Cost in the Long Run Overheads. The long run In the long run, there are no fixed...

Production and Cost in the Long Run

Overheads

The long run

In the long run, there are no fixed inputs or fixed costs; all inputs and all costs are variable

The firm must decide what combination of inputs to use in producing any level of output

Cost minimization assumption

For any given level of output,the firm will choose the input combinationwith the lowest cost

The cost minimization problem

C(y, w1 ,w2 , ) minx1,x2 , ,xn

Σn

i 1wixi such that y f(x1 ,x2 , xn)

Pick y; observe w1, w2, etc;choose the least cost x’s

Why not just pick 0 for all the x’s?

For any output level, there are are usually several different inputcombinations that can be used

Each combination will have a different cost

Consider the hay problem

x1 x2 TPP APP A MPP MPP TFC TVC TC AFC AVC ATC AMC MC8.0 1.0 1536.0 192.00 262.00 256.00 20.0 48.00 68.00 0.013 0.031 0.044 0.023 0.0239.0 1.0 1782.0 198.00 246.00 234.00 20.0 54.00 74.00 0.011 0.030 0.042 0.024 0.02610.0 1.0 2000.0 200.00 218.0 200.00 20.0 60.00 80.0 0.010 0.030 0.040 0.028 0.03011.0 1.0 2178.0 198.00 178.0 154.00 20.0 66.00 86.0 0.009 0.030 0.039 0.034 0.03912.0 1.0 2304.0 192.00 126.0 96.00 20.0 72.00 92.0 0.009 0.031 0.040 0.048 0.06313.0 1.0 2366.0 182.00 62.0 26.00 20.0 78.00 98.0 0.008 0.033 0.041 0.097 0.23114.0 1.0 2352.0 168.00 -14.0 -56.00 20.0 84.00 104.0 0.009 0.036 0.044

4.0 2.0 1345.0 336.25 406.00 424.00 40.0 24.00 64.00 0.030 0.018 0.048 0.015 0.0145.0 2.0 1783.0 356.60 438.00 450.00 40.0 30.00 70.00 0.022 0.017 0.039 0.014 0.0136.0 2.0 2241.0 373.50 458.00 464.00 40.0 36.00 76.00 0.018 0.016 0.034 0.013 0.013 7.0 2.0 2707.0 386.71 466.00 466.00 40.0 42.00 82.00 0.015 0.016 0.030 0.013 0.0138.0 2.0 3169.0 396.13 462.00 456.00 40.0 48.00 88.00 0.013 0.015 0.028 0.013 0.0139.0 2.0 3615.0 401.67 446.00 434.00 40.0 54.00 94.00 0.011 0.015 0.026 0.013 0.01410.0 2.0 4033.0 403.30 418.0 400.00 40.0 60.00 100.0 0.010 0.015 0.025 0.014 0.01511.0 2.0 4411.0 401.00 378.0 354.00 40.0 66.00 106.0 0.009 0.015 0.024 0.016 0.01712.0 2.0 4737.0 394.75 326.0 296.00 40.0 72.00 112.0 0.008 0.015 0.024 0.018 0.02014.0 2.0 5185.0 370.36 224.0 144.00 40.0 84.00 124.0 0.008 0.016 0.024 0.027 0.04216.0 2.0 5281.0 330.06 48.0 -56.00 40.0 96.00 136.0 0.008 0.018 0.02618.0 2.0 4929.0 273.83 -176.0 -304.00 40.0 108.00 148.0 0.008

There are many ways to produce2,000 bales of hay per hour

Workers Tractor-Wagons Total Cost Average Cost10 1 80 0.046.45 1.66 71.94 .035975.48 2 72.8658 0.03643.667 3 82.0015 0.0412.636 4 95.8167 0.04791.9786 5 111.872 .0559

Long run total cost

By minimizing total cost of production forvarious output levels with all inputs variable,the firm determines thelong run total cost of production

Output Workers Tractor-Wagons Cost Average Cost500 3.70 1.07 43.62 0.0871,000 4.91 1.27 54.89 0.0551,500 5.78 1.47 63.99 0.0432,000 6.45 1.66 71.94 0.035972,500 7.03 1.85 79.14 0.031653,000 7.54 2.03 85.78 0.028594,000 8.42 2.37 97.90 0.024485,000 9.16 2.70 108.89 0.02177817,000 10.38 3.32 128.61 0.0183710,000 11.85 4.17 154.54 0.015454320,000 15.30 6.67 225.13 0.011256430,000 17.77 8.85 283.60 0.0094531750,000 21.51 12.73 383.71 0.0076741675,000 25.13 17.11 493.00 0.00657338100,000 28.18 21.22 593.50 0.00593498150,000 33.48 29.17 784.20 0.00522799200,000 38.41 37.36 977.58 0.00488791244,000 42.99 45.52 1168.26 0.00478795245,000 43.10 45.72 1173.06 0.00478798250,000 43.67 46.77 1197.51 0.00479003275,000 46.86 52.80 1337.08 0.00486212290,000 49.39 57.69 1450.07 0.00500025

300,000 52.13 63.14 1575.65 0.00525218301,000 52.64 64.17 1599.25 0.00531311

Long run average cost of productionLRATC

LRATC LRAC LRTCy

C(y, w)y

Examples

LRATC LRTCy

71.942,000

0.03597

LRATC LRTCy

593.5100,000

0.005935

y = 2000

y = 100000

Graphically we can plot LRATC (LAC)as

Long Run Average Cost

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 50000 100000 150000 200000 250000 300000

Output - y

Co

st

LAC

Long run costs are less than or equalto short run costs for any given output level

Why?

If we are free to vary all inputs in the long run, we can match any short run least cost combination

Consider the following data where the short run costs hold wagons fixed at the long run least cost level

Output LAC AC - 1000 AC - 5000 AC - 50000500 0.0872333 0.088091,000 0.05488 0.054881,500 0.0426627 0.042962,000 0.0359713 0.03893 0.039292,500 0.03165 0.033513,000 0.02859 0.029593,500 0.02629 0.026784,000 0.02448 0.024674,500 0.023003 0.023055,000 0.0217783 0.0217786,000 0.0198439 0.0200187,000 0.01837 0.01920210,000 0.0154543 0.02766820,000 0.0112564 0.014988530,000 0.00945317 0.010774440,000 0.00839201 0.0087287450,000 0.00767416 0.0076741652,500 0.00752835 0.00757569

Consider long and short run average cost whenwagons are at the 50,000 bale minimum cost

Long And Short Run Average Cost

00.010.020.030.040.050.060.070.080.09

0 10000 20000 30000 40000 50000 60000

Output - y

Co

st

LAC

AC - 50000

Consider long and short run average cost whenwagons are at the 5,000 bale minimum cost

Long and Short Run Average Cost

0.021

0.023

0.025

0.027

3400 3800 4200 4600 5000

Output - y

Co

st

LAC

AC - 5000

Consider long and short run average cost whenwagons are at the 1,000 bale minimum cost

Long and Short Run Average Cost

0.03

0.04

0.05

0.06

0.07

0.08

0.09

400 600 800 1000 1200 1400 1600 1800 2000 2200

Output - y

Co

st

AC - 1000

LAC

LAC

Because non-integer values for wagons are not typicallyfeasible, we might consider alternative wagon levels instead

0.02

0.03

0.04

0.05

0.06

0.07

500 1500 2500 3500 4500 5500

Output - y

Co

st

AC 2 Wagons

Consider 1, 2 and 3 wagons

00.010.020.030.040.050.060.070.080.09

500 1500 2500 3500 4500 5500 6500

Output - y

Co

st

LAC

AC 1 Wagon

AC 2 Wagons

AC 3 Wagons

Consider 1, 2, 3 and 5 wagons

LAC

AC 1 Wagon

AC 2 Wagons

AC 5 Wagons

00.010.020.030.040.050.060.070.080.09

500 5500 10500 15500 20500 25500 30500

Output - y

Co

st

AC 3 Wagons

Now add 10 wagons

LAC

AC 1 Wagon

AC 2 Wagons

AC 5 Wagons

00.010.020.030.040.050.060.070.080.09

500 5500 10500 15500 20500 25500 30500

Output - y

Co

st

AC 3 Wagons

AC 10 Wagons

Output per period

$ Long-runaverage costATC1

ATC2

ATC3

The long run average total cost curve (LRATC) is an envelope curve that touches all the short run average total cost curves (SRATC) from below.

Another Example

0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30 35

Plant size and economies of scale

Economists often refer to the collection of fixed inputs at a firm’s disposal as its plant

Restaurant

Corn farmer

Dentist

buildingfixtureskitchen items

landmachinerybreeding stock

officedrill

Choosing the optimal plant sizeFor different output levels, different plants are appropriate

Short Run Average Cost

0.03

0.04

0.05

0.06

0.07

0.08

0.09

500 750 1000 1250 1500 1750 2000

Output - y

Co

st

AC 1 Wagon

AC 2 Wagons

Consider plant sizes of 1, 2 and 3 wagons

Short Run Average Cost

00.010.020.030.040.050.060.070.080.09

500 1500 2500 3500 4500 5500

Output - y

Co

st AC 1 Wagon

AC 2 Wagons

AC 3 Wagons

We can add 5, 6 and 7 wagons

Short Run Average Cost

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

500 5500 10500 15500

Output - y

Co

st AC 1 Wagon

AC 2 Wagons

AC 3 Wagons

AC 5 Wagons

AC 6 Wagons

AC 7 Wagons

AC 1 WagonAC 2 Wagons

AC 5 Wagons

AC 7 WagonsAC 10 WagonsAC 15 Wagons

Or 1, 2, 3, 5, 7, 10 and 15 wagons

Short Run Average Cost

00.01

0.020.03

0.040.05

0.060.07

0.08

500 8000 15500 23000 30500 38000 45500

Output - y

Co

st

AC 3 Wagons

AC 2 Wagons

AC 3 Wagons

AC 10 Wagons

AC 15 Wagons

AC 20 Wagons

And all the way up to 40 wagons

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

500 10500 20500 30500 40500 50500 60500 70500

Output - y

Co

st

AC 1 Wagon

AC 7 Wagons

AC 40 Wagons

AC 5 Wagons

7 Wagons

10 Wagons

20 Wagons

Long and Short Run Average Costs

0.000.010.020.030.040.050.060.070.08

0 40000 80000 120000 160000 200000

Output - y

Co

st

5 Wagons

15 Wagons

40 Wagons

LAC

40 wagons is only efficient at over 200,000 bales

Economies of size and the shape of LRATC

We measure the relationship between average cost and output by the elasticity of scale (size)

εS ACMC

If AC > MC, then the cost curve is downwardsloping and S > 1

If MC > AC, then the cost curve is upwardsloping and S < 1

MC

Long Run Average & Marginal Cost Curves

01020304050607080

0 10 20 30 40

LRAC

εS ACMC

AC > MC S > 1

y

LRAC is downward sloping

MC

Long Run Average & Marginal Cost Curves

01020304050607080

0 10 20 30 40

LRAC

εS ACMC

AC < MC S < 1

y

LRAC is upward sloping

Economies of scale (size)

When average cost is falling as output rises, we saythe firm experiences economies of scaleor increasing returns to size

Increasing returns to size AC > MC εS > 1

When long run total cost rises proportionately lessthan output, production is characterized by economies of scale and the LRATC curve slopes downward

MC

Long Run Average & Marginal Cost Curves

01020304050607080

0 10 20 30 40

LRAC

εS ACMC

AC > MC S > 1

y

Economies of Size/Scale

Why do economies of scale occur?

Gains from specialization

More efficient use of lumpy inputs

blast furnace

combine

X-ray machine

receptionist

Diseconomies of scale (size)

When average cost rises as output rises, we saythe firm experiences diseconomies of scaleor decreasing returns to size

Decreasing returns to size MC > AC εS < 1

When long run total cost rises more than in proportionto output, production is characterized by diseconomies of scale and the LRATC curve slopes upward

MC

Long Run Average & Marginal Cost Curves

01020304050607080

0 10 20 30 40

LRAC

εS ACMC

AC > MC S > 1

y

Diseconomies of Size

Why do diseconomies of scale occur?

Changes in the quality of inputs

Supervision and motivation problems

Externalities or congestion in production

Constant returns to scale (size)

When average cost does not change as output rises, we say the firm experiences constant returnsto size or scale

Constant returns to size MC AC εS 1

When both output and long run total cost rise by the same proportion, production is characterized byconstant returns to scale and the LRATC is flat

Why do constant returns to scale occur?

Duplication of processes

Fixed production proportions and replication

Economies and diseconomies balance out

General shape of the LRAC curve

048

1216202428323640

0 5 10 15 20 25 30Output - y

Co

st

LRAC

The End

LAC

AC 1 Wagon

AC 2 Wagons

AC 5 Wagons

00.010.020.030.040.050.060.070.080.09

500 5500 10500 15500 20500 25500 30500

Output - y

Co

st

AC 3 Wagons

AC 10 Wagons