The Long Run

1
The Long Run

• Theory of Production Cost
• in the Long-run

2
Objectives of Discussion

• Establish the conditions that will guide the
  optimizing firms choices in the LR
• Show the relationship between LR production
  principles the firms LR costs
• Derive firms LR cost curves
• Examine the factors that influence the shape of
  firms LR cost curves
• Show the relationship between firms Short-run
  Long-run cost curves

3
Long-run Input Combinations

• In SR analysis, K was fixed there was only one
  combination of L K for producing each level of
  Q
• In LR, can vary both K L and thus a given level
  of Q can be produced with several combos of K L

4
Production Isoquants

• The fact that different combos of L K can
  produce same level of output
• Implies possible to substitute K for L, or L for
  K, and maintain same level of output
• If the two inputs are continuously divisible,
  then there are an infinite number of K-L combos
  for each output level
• isoquant (a.k.a. Equal Quantity)
• A line joining all combinations of K L that
  can produce a given level of output
• Each point on an isoquant represents the maximum
  output for the given combination of inputs --i.e.
  it is technically efficient

5
Characteristics of Isoquants(IQ)

• Output is constant along an IQ
• Ratio of K/L varies from point to point on IQ
• Each K-L combo can be on only one IQ and there is
  an IQ for each K-L combo
• IQs are negatively sloping
• IQs are convex to origin
• IQs further from origin represent more output

6
Input Substitution Along an IQ

• Negative slope implies input substitution
• i.e. To maintain the level of Q a reduction in K
  must be offset by an increase in L
• What would IQ look like if substitution not
  possible?
• IQs convex to origin
• Rate at which K can be replaced by L along an IQ
  diminishes as ratio of K/L decreases

7
Marginal Rate of Technical Substitution

• MRTS is rate at which inputs can be substituted
  along an IQ
• Defined as negative of ?K/?L

• MRTS is related to ratio of MPL to MPK as
  follows
• For small moves along an IQ, the ?Q associated
  with ?K can be represented as (MPK)(?K)
• The ?Q associated with ?L can be represented as
  (MPL)(?L)
• Since ?Q must 0 along an IQ, then

8
The Firms Budget Lines--Isocost Curves

• An isocost curve shows all combos of inputs that
  firm can purchase with given cost outlay
• C wL rK is total cost eq.
• Rewriting we have

Three isocost curves--same relative prices,
different total cost outlays

• If r 60 w 40 C 7,200, what is isocost
  eq.?
• Answer K 120 - .67(L)
• Input price ratio gives slope
• Changes in C cause parallel shifts in isocost line

9
The Optimal Input Combination

• Optimal combo of K L
• Produces a given output at least cost, or
  alternatively
• Produces the maximum output for a given cost
  outlay
• Same rule applies in either case-- find combo at
  which

• MRTS is negative of slope of IQ w/r is slope of
  the firms isocost line
• So, optimal combo occurs where slope of isocost
  slope of IQ

Answer 1.66 so over segment A-B, (MPL/MPK) gt
(w/r) (MPL/w) gt (MPK/r)
10
The Expansion Path(EP)

• To move from production to cost we introduce EP
• EP is a line showing cost minimizing input combo
  for any level of output
• Each point on EP represents different level of
  output and cost
• The ratio of input prices are constant
• At each point on EP

Each point on EP is economically efficient
11
Returns to Scale
Unit costs are decreasing as Q increases

• Defined--responsiveness of output to proportional
  change in all inputs
• If all inputs are increased by a factor of ?
  (e.g. 25) and output increases by factor of z
  such that
• f(?L, ?K) z Q
• If z gt ? we have IRTS--e.g. if z 50
• If z lt ? we have DRTS--e.g. if z 10
• If z ? we have CRTS--e.g. if z 25
• Returns to scale have a lot to do with the
  pattern of firms LR cost curve as it expands
  output

Unit costs are increasing as Q increases
12
Long-Run Costs

• Long-run total cost (LTC) for a given level of
  output is given by
• LTC wL rK
• Where w r are prices of labor capital,
  respectively, (L, K) is the input combination
  on the expansion path that minimizes the total
  cost of producing that output

13
Long-Run Costs

• Long-run average cost (LAC) is the cost per unit
  of Q at the optimal input combo
• LAC is U-shaped
• Falling LAC indicates economies of scale (IRTS)
• Rising LAC indicates diseconomies of scale (DRTS)

14
Long-Run Costs

• Long-run marginal cost (LMC) measures the rate of
  change in long-run total cost as output changes
  along expansion path
• LMC is U-shaped
• LMC lies below LAC when LAC is falling
• LMC lies above LAC when LAC is rising
• LMC LAC at the minimum value of LAC

15
Derivation of a Long-Run Cost Schedule
LMC
100
120
1.20
1.20
200
140
0.70
0.20
300
200
0.67
0.60
400
300
0.75
1.00
500
420
0.84
1.20
600
560
0.93
1.40
700
720
1.03
1.60
16
Long-Run Total, Average, Marginal Cost
17
Long-Run Average Marginal Cost Curves in a
Continuous World
18
Various Shapes of LAC
19
Constant Long-Run Costs

• When constant returns to scale occur over entire
  range of output
• Firm experiences constant costs in the long run
• LAC curve is flat equal to LMC at all output
  levels

20
Constant Long-Run Costs
21
SR vs. LR Costs
22
Relationship Between SR LR Cost Curves

• LAC is least costly way of producing a given
  level of Q
• Each point on LAC corresponds to a given point on
  EP
• Once the firm chooses a combo of K L, firm is
  in SR situationSR Costs become relevant
• SR cost curves always lie above LR cost curves
  for any level of output
• SR cost is tangent to LR cost curve at one, or
  more points

23
Long-Run Average Cost as the Planning Horizon
24
Relations Between Short-Run Long-Run Costs

• LMC intersects LAC when the latter is at its
  minimum point
• At each output where a particular ATC is tangent
  to LAC, the relevant SMC LMC
• For all ATC curves, point of tangency with LAC is
  at an output less (greater) than the output of
  minimum ATC if the tangency is at an output less
  (greater) than that associated with minimum LAC

25
Economies of Scope

• Can exist for a multi-product firm
• Joint cost of producing two or more goods is less
  than the sum of the separate costs of producing
  the two goods
• For two goods, X Y, economies of scope exist
  when
• C(X, Y) lt C(X) C(Y)
• Diseconomies of scope exist when
• C(X, Y) gt C(X) C(Y)

26
Summary of LR Cost

• LR cost curves are derived from firms EP
• LAC LTC/Q where LTC is sum of optimal amounts
  of L K multiplied by their respective prices
• LMC ?LTC/ ?Q
• LAC and LMC are u-shaped if firm experiences IRTS
  followed by DRTS
• LMC intersects LAC at the latters minimum point
• At the Q corresponding to minimum point of the
  LAC there will be a SR ATC that is tangent to LAC
  and their minimum points will coincide
• At the least cost output level in the plant
  represented by this ATC SMCATCLACLMC