Title: Theory of Cost
1Theory of Cost
2Introduction
- 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.
3Explicit 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
4Fixed 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
5Profits
- 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
6Normal vs. Pure Profits
7Cost 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.
8Cost 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.
9Cost Minimization
10Long-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
11Example Find LRTC function
Production Function
Cost Minimization Problem Min TC w0L
r0K where w0 fixed wage r0 fixed
cap. cost s.t.
Solve
12First 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
13Using 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
14Find LRTC Function
Know TC w0L r0K and K,L. Substitute
K and L into TC
15Examplew0 5 r0 20 q0 100
Optimal Levels of L and K
Tangency MRTS w/r
16Constructing LRTC
- Vary Level of Output and connect tangency points
along long-run expansion path
LRTC for CRS Cobb Douglas Example
17Long-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
18LR Average and Marginal Cost
LRATC LRTC/q gt
LRMC (d/dq)LRTC gt
In this case LRATC LRMC
19Costs 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
20Constant Returns to Scale
q(aK, aL) azq(K,L) aq0 where z 1 Costs
increase at the same rate as output
21Decreasing Returns to Scale
q(aK, aL) azq(K,L) lt aq0 where z lt 1 Costs
increase more rapidly than output
22Increasing Returns to Scale
q(aK, aL) azq(K,L) gt aq0 where z gt 1 Costs
increase less rapidly than output
23Cost Curves with Constant, Decreasing, and
Increasing Returns
LMC
LAC
24Average 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
25Short-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
26Short-run cost curves
Note SRMC should Pass through the Minimum of
SAVC SATC
27Example 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
28First Order Conditions
(1)
(2)
? dTC/dq SRMC
SRTC wL rK0
SMC dSRTC/dq
?
SATC SRTC/q
SVC
29Examplew0 5 r0 20 K0 50
- Properties of SR Cost Functions
- SRMC is increasing with Output
- SRAVC will eventually rise
- SRATC is U shaped.
30Some 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.
31Relate Costs to Production
Figure 7.1
Figure 8.9
32Law 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
33SR 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
36Constant Returns to Scale
37Price changes?
Increase in wages? w0 -gt w Shifts isocost
line to left Shifts expansion path to
left Raises LTC
38Price/Fixed Cost Change
39Technology Change?