Theory of Cost

1
Theory of Cost

• Chapter 8

2
Introduction

• Classifications of costs
• Implicit or explicit
• fixed or variable
• Develop family of cost curves
• long run (all factors variable)
• short run (all factors except one fixed)
• Cost minimization problem
• Use isoquants from production theory, set tangent
  to isocost line (input price ratio)
• Cost curves incorporate technology used for
  producing an output
• input prices given (supply is perfectly elastic,
  firms are price takers)
• long-run cost curves and returns to scale
• How do cost curves shift when input prices change
  or new technologies are introduced.

3
Explicit And Implicit Costs

• Explicit (or expenditure) costs
• Costs of employing additional inputs not owned by
  firm
• Includes all cash or out-of-pocket expenses
  incurred in production
• Accounting cost for purchased inputs whether they
  are fixed or variable
• Implicit costs (also called nonexpenditure,
  imputed, or entrepreneurial costs)
• Costs charged to inputs that are owned by firm
• An opportunity cost of using an input in
  production of a commodity
• Loss in benefits that could be obtained by using
  these inputs in another activity
• Owners of firm also have an implicit cost
  associated with time devoted to a particular
  production activity

4
Fixed And Variable Costs

• Fixed
• Costs that do not vary with changes in output
• Variable
• Costs associated with variable inputs and do vary
  with output
• Note Explicit and implicit costs may contain
  both fixed and variable costs
• Variable
• Explicit electricity to run machine, cans for
  beer.
• Implicit Opp. cost of time owner spends
  overseeing workers.
• Fixed
• Explicit long term lease on machinery
• Implicit Opp. cost of not selling a new
  technology invented for production
• Total cost (TC) fixed variable

5
Profits

• Normal
• Minimum total return to the inputs necessary to
  keep a firm in a given production activity
• Also called necessary, ordinary, or
  opportunity-cost profit
• Equals implicit cost
• Pure
• Total return above total cost
• Also called economic profit
• In short run, possibility of earning a pure
  profit exists but firms will only earn a normal
  profit in long run
• In long run, firms have ability to enter or exit
  an industry
• Will not operate at a loss or earn a pure profit

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Normal vs. Pure Profits
7
Cost Minimization

• Cost Function A mathematical relationship
  showing the lowest economic cost for each
  possible level of output.
• Cost minimization is done by constrained
  optimization
• Identify lowest cost mix of inputs to achieve a
  given level of output
• Two inputs are used in production
• Perfectly competitive input and output markets
• Firm takes input and output prices as fixed
• Input supply curves are horizontal and perfectly
  elastic
• No barriers to entry or exit
• Producers have perfect information about prices.

8
Cost Minimizing Decision Rule

• Isoquant Technological possibilities based on
  production function (last chapter)
• Isocost line Market possibilities for
  substituting one input for another
• Least cost combination Occurs where the firms
  MRTS (technological possibility for substituting
  inputs holding output constant) equals the rate
  at which inputs can be traded in markets.

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Cost Minimization
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Long-run Costs Isocost

• Isocost equation is
• TC wL rK
• w is wage rate of labor
• r is per-unit input price of capital (book uses
  v)
• Solving isocost equation for K
• K -w/rL TC/v
• Results in a linear equation with TC/v as the
  capital (K), intercept
• -w/r as slope

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Example Find LRTC function
Production Function
Cost Minimization Problem Min TC w0L
r0K where w0 fixed wage r0 fixed
cap. cost s.t.
Solve
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First Order Conditions
(1)
(2)
(3)
? dTC/dq ? is the change in total cost of
production when output is increased by one unit.
Under our assumption that output prices are
fixed, the last unit produced sells for the
market price, Pq. Therefore, ? Pq Value
of the marginal product (VMPL or VMPK) VMPL w0
?MPL PqMPL VMPK r0 ?MPK PqMPK
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Using equations (1) and (2) gt
Expansion Path
Substitute this result into equation (3)
Optimal Level of L, L
Substitute L into expansion path to get K
Optimal Level of K, K
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Find LRTC Function
Know TC w0L r0K and K,L. Substitute
K and L into TC
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Examplew0 5 r0 20 q0 100
Optimal Levels of L and K
Tangency MRTS w/r
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Constructing LRTC

• Vary Level of Output and connect tangency points
  along long-run expansion path

LRTC for CRS Cobb Douglas Example
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Long-Run Input Demand FunctionsDerived Demand
Functions
L, and K were derived assuming q0 was constant
gt
Can determine L, K combinations for any level
of output by varying q gt
Generalized conditional long-run demand functions
gt
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LR Average and Marginal Cost
LRATC LRTC/q gt
LRMC (d/dq)LRTC gt
In this case LRATC LRMC
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Costs and Returns to Scale

• Constant returns to scalegt LRATC constant
• ?LRATC/?q 0
• Long-run average cost does not change for a given
  change in output
• Increasing returns to scale gt LRATC is declining
• ?LRATC/?q lt 0
• Increases in total cost are proportionally
  smaller than an increase in output
• Implies that inputs less than double for a
  doubling of output
• Corresponds to LTC also less than doubling
• Decreasing returns to scale gt LRATC is
  increasing
• ?LRATC/?q gt 0
• Increases in total cost are proportionally larger
  than an increase in output
• Implies that inputs more than double for a
  doubling of output
• Corresponds to LTC more than doubling for a
  doubling of output

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Constant Returns to Scale
q(aK, aL) azq(K,L) aq0 where z 1 Costs
increase at the same rate as output
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Decreasing Returns to Scale
q(aK, aL) azq(K,L) lt aq0 where z lt 1 Costs
increase more rapidly than output
22
Increasing Returns to Scale
q(aK, aL) azq(K,L) gt aq0 where z gt 1 Costs
increase less rapidly than output
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Cost Curves with Constant, Decreasing, and
Increasing Returns
LMC
LAC
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Average and Marginal Cost Relationship

• Marginal cost is not cost of producing last unit
  of output
• Cost of producing last unit of output is same as
  that of producing all other units of output
• Average cost of production
• Marginal cost is increase in cost of producing an
  extra increment of output
• Equal to average cost plus an adjustment factor
• Additional cost to all factors of production
  caused by increase in output
• Marginal cost differs from average cost by
  per-unit effect on costs of higher output,
  multiplied by total output
• If LAC does not vary with output adjustment
  factor is zero

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Short-run Costs

• Short run gt One of the inputs is fixed
• Example Upon signing a 5-year lease for their
  restaurant, owners have committed themselves to
  paying a fixed amount in costs whether they
  operate or not
• Results in a short-run situation where, for
  short-run profit maximization, owners will
  determine lowest TC for a given level of output
  and fixed input
• Lowest TC is called short-run total cost (STC)
• STC short-run total variable cost (STVC) total
  fixed cost (TFC)
• Assuming that capital is fixed at K in short
  run
• STC(K) STVC(K) TFC(K) min(wL vk)
• (wL vk) is isocost equation
• STVC(K) wL and TFC(K) vk
• L denotes level of labor that minimizes costs
  for a given level of output
• Even if firm were to produce nothing, in short
  run it must still pay TFC
• TFC is a horizontal line, showing that at all
  output levels, TFC remains the same

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Short-run cost curves
Note SRMC should Pass through the Minimum of
SAVC SATC
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Example Find SRTC function
Production Function
Cost Minimization Problem Min TC w0L
r0K0 where w0 fixed wage r0 fixed
cap. Cost s.t. K0 fixed capital
Solve
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First Order Conditions
(1)
(2)
? dTC/dq SRMC
SRTC wL rK0
SMC dSRTC/dq
?
SATC SRTC/q
SVC
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Examplew0 5 r0 20 K0 50

• Properties of SR Cost Functions
• SRMC is increasing with Output
• SRAVC will eventually rise
• SRATC is U shaped.

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Some points about AC/MC

• AFC is continually declining as output increases
• As output tends toward zero (infinity), AFC
  approaches infinity (zero)
• SATC and SAVC never intersect
• Approach each other as output increases
• SATC is sum of SAVC and AFC
• SATC is U-shaped due to Law of Diminishing
  Marginal Returns
• Short-run marginal cost (SMC) for a fixed level
  of capital K is defined as

• Due to Law of Diminishing Marginal Returns,
• SMC may at first decline, but will ultimately
  rise.

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Relate Costs to Production
Figure 7.1
Figure 8.9
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Law of Diminishing Marginal Returns

• Law of Diminishing Marginal Returns
• At some point when adding additional variable
  inputs marginal product of variable inputs will
  decline
• A positively sloping SMC represents diminishing
  marginal returns
• A negatively sloping SMC represents increasing
  marginal returns
• Shape of SMC is determined by Law of Diminishing
  Marginal Returns
• As output increases SMC curve will have a
  positive slope at some point

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SR vs. LR
34

• In general, there are an infinite number of
  short-run total cost curves
• One for every conceivable level of fixed input
• LRTC Envelope of all these SR cost-minimizing
  choices
• STC curves for alternative levels of fixed input
  capital completely cover top of LTC curve and
  will not dip below it
• STC will only equal LTC at output level where
  long-run optimal input usage of capital
  corresponds to fixed capital input level
  associated with STC

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• Where STC is tangent to LTC, SATC is also tangent
  to LAC
• SATC curves envelop top of LAC curve
• SATC cannot be less than LAC for a given level of
  output

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Constant Returns to Scale
37
Price changes?
Increase in wages? w0 -gt w Shifts isocost
line to left Shifts expansion path to
left Raises LTC
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Price/Fixed Cost Change
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Technology Change?