production and cost in the long run overheads. the long run in the long run, there are no fixed...
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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