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Theory of Cost

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Title: 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

6
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.

9
Cost Minimization
10
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

11
Example Find LRTC function
Production Function
Cost Minimization Problem Min TC w0L
r0K where w0 fixed wage r0 fixed
cap. cost s.t.
Solve
12
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
13
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
14
Find LRTC Function
Know TC w0L r0K and K,L. Substitute
K and L into TC
15
Examplew0 5 r0 20 q0 100
Optimal Levels of L and K
Tangency MRTS w/r
16
Constructing LRTC
  • Vary Level of Output and connect tangency points
    along long-run expansion path

LRTC for CRS Cobb Douglas Example
17
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
18
LR Average and Marginal Cost
LRATC LRTC/q gt
LRMC (d/dq)LRTC gt
In this case LRATC LRMC
19
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

20
Constant Returns to Scale
q(aK, aL) azq(K,L) aq0 where z 1 Costs
increase at the same rate as output
21
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
23
Cost Curves with Constant, Decreasing, and
Increasing Returns
LMC
LAC
24
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

25
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

26
Short-run cost curves
Note SRMC should Pass through the Minimum of
SAVC SATC
27
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
28
First Order Conditions
(1)
(2)
? dTC/dq SRMC
SRTC wL rK0
SMC dSRTC/dq
?
SATC SRTC/q
SVC
29
Examplew0 5 r0 20 K0 50
  • Properties of SR Cost Functions
  • SRMC is increasing with Output
  • SRAVC will eventually rise
  • SRATC is U shaped.

30
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.

31
Relate Costs to Production
Figure 7.1
Figure 8.9
32
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

33
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

35
  • 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

36
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
38
Price/Fixed Cost Change
39
Technology Change?
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