3. What we know…
• Production function: (K,L) → Q
• Suppose we know
– w = wage rate of L
– R = rental rate of K
• Then we can determine how much it costs for a
firm to produce an amount Q=f(K,L) assuming
that the firm endeavors to minimize its costs
• That is, we can figure out TC(Q), so all that
remains to solve the firm’s profit maximization
problem is to determine, P(Q), which depends on
market structure
4. Types of costs
• Accounting costs
– Things for which you pay money
– E.g., bills, labor costs, rent
– Other things like depreciation (i.e., lost value of
assets that lose value over time)
• Economic costs
– Opportunity costs of doing business
• Values of highest foregone options of inputs
– Workers (how much they could earn elsewhere)
– Machines (how much they could earn if put to some other use)
– Entrepreneur (how much he/she could earn as a salaried
employee at some other firm)
5. Overlap between Accounting &
Economic Costs
• There is significant overlap between accounting &
economic costs:
– E.g., labor costs=wages=opportunity cost of labor in a
competitive labor market
• Economic costs > accounting costs
– Entrepreneur’s salary < opportunity costs because
entrepreneur receives a large share of firm’s
accounting profits as owner of firm
– Investors’ opportunity costs are not reflected in firm’s
costs (compensated through accounting profits by
way of dividends, capital gains)
13. Depicting Cost Minimization
K
TC3=rK3 All input combinations on the iso-expenditure
TC2=rK2 (iso-cost) lines are input combinations that
result in the same total cost of production
w/r=MRTS
TC1=rK1 Total cost is declining in the southwesterly
direction
TC0=rK0 A
Isoquant
q0
-w/r This firm can produce q0 for less than
L TC3, but it cannot produce q0 for TC1
or TC2.
Notice that for this and every output
level and given input prices (w & r)
Isocost or isoexpenditure lines there is a unique minimum cost of
production.
Our goal is to derive TC(w,r,q)
14. Duality: Output maximization
• For every constrained maximization problem
there exists a constrained minimization
problem yielding the same solution for
appropriate parameter values (“duality”)
• Dual problem of cost minimization: maximize
output subject to expenditure constraint
15. Output Maximization
K
w/r=MRTS Solving the Lagrangian will yield the
E=rK same optimal condition: w/r=MRTS
A
q3
q2
q1
q0 L
E=wL
17. Demand for inputs?
• Can we derive firm’s demand for inputs (K,L)
using solutions to cost minimization problem
(much like we derived demand for goods using
the solution to the consumer choice
problem)?
• No!
• Cost minimization holds output constant, but
firm’s demand for K & L obviously depends on
how much output it chooses to produce.
18. Deriving Total Cost Function
K
q E
TC3=rK3 Cost expansion path: from
this expansion path we q0 TC0
TC2=rK2
can obtain the TC q1 TC1
associated with each TC(w,r,q)
TC1=rK1 q2
output level q TC2
TC0=rK0 q3 TC3
q3
q2
q0 q1 L
20. Example: Cobb-Douglass production
function
• →wL=rK
• Suppose w=r=$4. Then L=K. If L=K & q=40, we
have
• This implies that TC=$4*4+$4*4=$32 (lowest
possible cost of producing 40 units)
• &
• (extra output for last $1 spent on inputs)
24. Graphing the Cost Functions
$
TC(q) Observations:
1. Ceteris paribus invoked (w & r held
constant)
2. MC initially declining through q0
(may not always be true)
-Results from IRS (benefits to
specialization)
-Shape of the cost functions
q depends on the production
q0 q1
function
$/unit MC(q)=dTC/dq 3. MC intersects AC at minimum value
of AC
-MC<AC for q<q1
-MC>AC for q>q1
4. For q>q0, MC is increasing. This is
AC(q)=TC(q)/q the region of DRS in the production
function.
q
q0 q1
26. CRS production functions & cost
functions
CRS production functions have
$ linear TC functions because MC
TC(q)
is constant:
$
TC(q)
IRS CRS DRS
q q
27. CRS production functions & cost
functions
• What is AC when production function exhibits
CRS everywhere? In previous example and in
all such examples, AC(q)=TC(q)/q=c, a
constant.
When f(K,L) exhibits CRS everywhere,
AC(q)=MC(q)=constant.
$/unit
When you see constant marginal costs, you
immediately know something about the
AC=MC productions function. Likewise, when you see
CRS, you immediately know something about
the MC>
q
28. Effects of input price increase
• When the price of one of the two inputs goes
up, what happens if producers wish to
maintain the same level of output?
– Producers will substitute the now relatively
cheaper input for some of the input whose price
increased
– Total costs definitely do not decrease following an
input price increase. If they do, the producer
could not have been minimizing costs to start with
30. Partial elasticity of substitution
between inputs
• Partial elasticity of substitution (s) measures
how firms change input mix in response to
price changes
• High s → firms change input mix (K/L ratio)
substantially in response to small changes in
relative input prices
• Low s → firms change input mix (K/L ratio)
little in response to changes in relative input
prices
31. Long vs. Short Run Costs
• Short run = period of time during which at least
one input cannot be changed
• Long run = any time horizon during which all
input can be changed
• Variable input = input that can be changed in the
short run
• Fixed input = input that cannot be changed in the
short run
– Some inputs take time to be delivered or made; this
time-to-delivery defines the short run
• All inputs are variable in the long run
34. Enveloping the Short-run cost curves
• Envelope of the STC: The set of lowest costs of
production on any STC (i.e., for any fixed level
of K) for every possible output level
• The envelope of the STC curves defines the TC
curve
• Similarly, the envelope of the SATC curves
defines the AC curve
35. Enveloping the Short-run cost curves
TC
TC(w,r,q)
Imagine that each STC curve
corresponds to the firm’s cost
schedule with a different
number of plants. How many
plants should the firm build to
produce q1, q2, q3, and q4 units
of output?
q
q1 q2 q3 q4
This TC curve is linear. What does that tell us?
36. Enveloping the Short-run cost curves
TC
TC The envelope of the
STC curves need not
be linear.
q
q1 q2 q3 q4
37. Enveloping the Short-run cost curves
Cost per
unit
AC=MC
q
This is a CRS production function
How are we deriving the MC curve from the SMC curves?
38. Enveloping the Short-run cost curves
Not CRS
Cost per
unit
MC
AC
At MES, AC reaches
its minimum values
and
AC=MC=SATC=SMC
q
MES=minimum efficient scale
For each output level, find the SMC curve corresponding to the SATC just tangent
to the AC curve. From this SMC curve we derive the MC (in the LR) of producing
that output level.
40. Example: Short run Cobb Douglass
Costs
• If in the short run you happen to be using the capital level
that minimizes the costs of producing output level q in the
long run (K1*), then you must also be minimizing costs in
the short run, meaning that the derivative of the STC at this
capital level is zero. Use this fact to solve for K1*:
• Plug this back into the STC. You are now effectively using
the K level that minimizes LR costs while using the L-level
that minimizes costs for any given K-level. Thus you must be
minimizing costs in the LR, so this is the TC function:
• You can check that this is indeed the same as the TC we
derived earlier for this Cobb Douglass production function
Editor's Notes
What is the lowest cost at which a firm can produce every level of output
K=capital
Economic costs are the opportunity costs of doing business. These are workers, machines(ciukd have been rented out to other firms) and entrepreneur. Economic costs ALWAYS exceed accounting costs.
When you have positive accounting profits, you can have zero or negative economic profit. When the economic cost is greater you make more profit and are able to pay entrepreneur’s
Economic profits = TR-EC. When you have zero economic profits you have positive accounting profits and are either reinvested in the firm or paid back to shareholders and just compensating them for the opportunity costs of indulging in the business.
Derive the total cost function.
R= rental rate of capital.
Diminishing marginal products and complementarity between two inputs.
E=Budget
Isocost line always have the same slope. All you are doing is finding the points of tangency.
It’s the lowest possible cost of producing 40 units when the capital and labour equals to 4 dollars.
Step 1: set up and solve the lagrangianStep 2: solve for k* and l* using all the FOCStep 3: substitute k* and l* into the cost identity
K and L do not appear in the TOTAL COST FUNCTION.
INCREASING RETURNS TO SCALE DECLINING MARGINAL COSTS.When the marginal cost is below the average cost, the average cost is decreasingWhen the marginal cost is higher than the average, the average is increasing.The point at which a firm achieves its lowest cost per unit of production.To calculate average costTake a ray from the origin and run it up to the total cost curve, the slope of that ray is the average cost curve.
With constant returns to scale production functions, you get constant marginal cost.
Firm A has constant marginal costs = 10. it’s easier when constant marginal costs present.
In response of an input price increase, the firm can reduce the expensive input. If wage rate goes up, the firm reduces labor and increases capital.If input prices go up, there is no way the total costs could go down
Sigma is a characteristic of a production technology and the other (s) describes firm behavior (the behavior of firms minimizing their cost of production)
How responsive is the capital to labor mix. If labor becomes more expensive than capital, the firms would make dramatic shifts from labor intensive input combination to capital intensive input combinations
The short run is not a fixed amount of time, it is the run which is dependent on a fixed input.
L* is the amount of labor you need to hire in order to get that amount of output that you need produced. You must pay the short run fixed cost (SFC)
Suppose capital could take any value, what would be the lowest possible cost of producing output level Q? the value you choose would be the value used in the long run.
It is linear tells us that it is constant returns to scale
You find the point of tangency from the SATC and the LRAC and trace it down to the SMC. How to derive the long run MC.Minimum efficient scale is that output in the long run, cost per unit is minimized. In the long run, firms operating in perfect competition would all produce at their minimum efficient scale.
It would have a linear cost function and have constant AC AND MC in the long run. No matter how much you produce, R doesn’t change.Q= 10K^.5 L^.5 L^1/2= Q/10K^5L*= Q^2/100K1VARIABLE COST IS WxL*
He is showing us a way of deriving the long run variable cost without using the lagrangian.
