The Firm: Optimisation
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Transcript The Firm: Optimisation
Prerequisites
Almost essential
Firm: Basics
THE FIRM:
OPTIMISATION
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2012
Frank Cowell: Firm Optimization
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Overview...
Firm:
Optimisation
The setting
Approaches to the
firm’s optimisation
problem
Stage 1: Cost
Minimisation
Stage 2: Profit
maximisation
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The optimisation problem
We want to set up and solve a standard optimisation problem
Let's make a quick list of its components
... and look ahead to the way we will do it for the firm
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The optimisation problem
Objectives
-Profit maximisation?
Constraints
-Technology; other
Method
- 2-stage optimisation
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Construct the objective function
Use the information on prices…
wi
•price of input i
p
•price of output
…and on quantities…
zi
q
•amount of input i
•amount of output
…to build the objective function
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How it’s done
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The firm’s objective function
m
Cost of
inputs:
S wizi
Revenue:
pq
•Summed over all m inputs
i=1
•Subtract Cost from Revenue to get
Profits:
m
pq – S wizi
i=1
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Optimisation: the standard approach
Choose q and z to maximise
m
P := pq – S wizi
i=1
...subject to the production
constraint...
q f (z)
..and some obvious constraints:
q 0
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z0
• Could also write this as zZ(q)
•You can’t have negative
output or negative inputs
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A standard optimisation method
If f is differentiable…
Set up a Lagrangean to take care
L (... )
of the constraints
Write down the First Order
necessity
L (... ) = 0
z
Conditions (FOC)
Check out second-order
sufficiency
conditions
Use FOC to characterise solution
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2
2L (... )
z
z* = …
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Uses of FOC
First order conditions are crucial
They are used over and over again in optimisation
problems.
For example:
• Characterising efficiency.
• Analysing “Black box” problems.
• Describing the firm's reactions to its environment.
More of that in the next presentation
Right now a word of caution...
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A word of warning
We’ve just argued that using FOC is useful.
• But sometimes it will yield ambiguous results.
• Sometimes it is undefined.
• Depends on the shape of the production function f.
You have to check whether it’s appropriate to apply the
Lagrangean method
You may need to use other ways of finding an
optimum.
Examples coming up…
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A way forward
We could just go ahead and solve the maximisation problem
But it makes sense to break it down into two stages
• The analysis is a bit easier
• You see how to apply optimisation techniques
• It gives some important concepts that we can re-use later
First stage is “minimise cost for a given output level”
• If you have fixed the output level q…
• …then profit max is equivalent to cost min.
Second stage is “find the output level to maximise profits”
• Follows the first stage naturally
• Uses the results from the first stage.
We deal with stage each in turn
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Overview...
Firm:
Optimisation
The setting
A fundamental
multivariable problem
with a brilliant
solution
Stage 1: Cost
Minimisation
Stage 2: Profit
maximisation
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Stage 1 optimisation
Pick a target output level q
Take as given the market prices of inputs w
Maximise profits...
...by minimising costs
m
S wi zi
i=1
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A useful tool
For a given set of input prices w...
…the isocost is the set of points z in input space...
...that yield a given level of factor cost
These form a hyperplane (straight line)...
...because of the simple expression for factor-cost
structure
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Iso-cost lines
Draw set of points where
cost of input is c, a constant
z2
Repeat for a higher value
of the constant
Imposes direction on the
diagram...
w1z1 + w2z2 = c"
w1z1 + w2z2 = c'
w1z1 + w2z2 = c
z1
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Use this to
derive
optimum
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Cost-minimisation
z2
The firm minimises cost...
Subject to output constraint
q
Defines the stage 1 problem.
Solution to the problem
minimise
m
S wizi
i=1
subject to f(z) q
z*
z1
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But the solution depends on the
shape of the input-requirement
set Z.
What would happen in
other cases?
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Convex, but not strictly convex Z
z2
Any z in this set is
cost-minimising
z1
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An interval of solutions
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Convex Z, touching axis
z2
Here MRTS21 > w1 / w2 at the
solution.
z*
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z1
Input 2 is “too expensive”
and so isn’t used: z2* = 0
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Non-convex Z
z2
z*
There could be multiple solutions.
z**
But note that there’s no solution
point between z* and z**
z1
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Non-smooth Z
z2
MRTS21 is
undefined at z*.
z* is unique costminimising point for q
z*
z1
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True for all positive finite
values of w1, w2
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Cost-minimisation: strictly convex Z
Minimise
m
S wi zi
Lagrange
multiplier
q – ff(z)]
(z)
+ l[q
i=1
Use the objective function
...and output constraint
...to build the Lagrangean
Differentiate w.r.t. z1, ..., zm ; set equal to 0
... and w.r.t l
Because of strict convexity we have Denote cost minimising values by
an interior solution
A set of m+1 First-Order Conditions
l* f1 (z* ) = w1
l*f2 (z*) = w2
… … …
l*fm(z *) = wm
q = f(z*)
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one for
each input
output
constraint
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*
If isoquants can touch the axes...
Minimise
m
Swizi
+ l [q – f(z)]
i=1
Now there is the possibility of corner solutions.
A set of m+1 First-Order
Conditions
l*f1 (z*) w1
l*f2 (z*) w2
… … …
l*fm(z*) wm
q = f(z*)
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Interpretation
Can get “<” if optimal
value of this input is 0
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From the FOC
If both inputs i and j are used
and MRTS is defined then...
fi(z*)
wi
———
= —
*
fj(z )
wj
MRTS =
input price ratio
If input i could be zero then...
fi(z*)
wi
———
—
*
fj(z )
wj
MRTSji input price ratio
“implicit” price = market price
“implicit” price market price
Solution
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The solution...
Solving the FOC, you get a cost-minimising value for each
input...
zi* = Hi(w, q)
...for the Lagrange multiplier
l* = l*(w, q)
...and for the minimised value of cost itself.
The cost function is defined as
C(w, q) := min S wi zi
{f(z) q}
vector of
input prices
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Specified
output level
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Interpreting the Lagrange multiplier
The solution function:
C(w, q) = Siwi zi*
= Si wi zi*– l* [f(z*) – q]
Differentiate with respect to q:
Cq(w, q) = S
i
iwiH q(w, q)
*
i
i fi(z ) H q(w,
– l* [S
Rearrange:
At the optimum, either the
constraint binds or the
Lagrange multiplier is zero
Express demands in terms of (w,q)
q) – 1]
Vanishes because of FOC
l *fi(z*) = wi
Cq(w, q) = Si [wi – l*fi(z*)] Hiq(w, q) + l* Lagrange multiplier in the stage
1 problem is just marginal cost
Cq (w, q) = l*
This result – extremely important in economics – is just an
applications of a general “envelope” theorem.
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The cost function is an amazingly useful
concept
Because it is a solution function...
...it automatically has very nice properties
These are true for all production functions
And they carry over to applications other than the firm.
We’ll investigate these graphically
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Properties of C
z1 *
C
C(w, q+q)
C(w, q)
Draw cost as function of w1
Cost is non-decreasing in input prices .
Increasing in output, if f continuous
Concave in input prices.
°
Shephard’s Lemma
C(tw+[1–t]w,q) tC(w,q) + [1–t]C(w,q)
w1
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C(w,q)
———— = zj*
wj
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What happens to cost if w changes to tw
z2
Find cost-minimising inputs for w, given q
q
Find cost-minimising inputs for tw, given q
So we have:
•
z*
C(tw,q) = Si t wizi* = t Siwizi* = tC(w,q)
The cost function is homogeneous
of degree 1 in prices.
z1
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Cost Function: 5 things to remember
Non-decreasing in every input price
• Increasing in at least one input price
Increasing in output
Concave in prices
Homogeneous of degree 1 in prices
Shephard's Lemma
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Example
Production function: q z10.1 z20.4
Equivalent form:
log q 0.1 log z1 + 0.4 log z2
Lagrangean: w1z1 + w2z2 + l [log q – 0.1 log z1 – 0.4 log z2]
FOCs for an interior solution:
w1 – 0.1 l / z1 = 0
w2 – 0.4 l / z2 = 0
log q = 0.1 log z1 + 0.4 log z2
From the FOCs:
log q = 0.1 log (0.1 l / w1) + 0.4 log (0.4 l / w2 )
l = 0.1–0.2 0.4–0.8 w10.2 w20.8 q2
Therefore, from this and the FOCs:
w1 z1 + w2 z2 = 0.5 l = 1.649 w10.2 w20.8 q2
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Overview...
Firm:
Optimisation
The setting
…using the
results of stage 1
Stage 1: Cost
Minimisation
Stage 2: Profit
maximisation
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Stage 2 optimisation
Take the cost-minimisation problem as solved
Take output price p as given
• Use minimised costs C(w,q)
• Set up a 1-variable maximisation problem
Choose q to maximise profits
First analyse components of the solution graphically
• Tie-in with properties of the firm (in the previous presentation)
Then we come back to the formal solution
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Average and marginal cost
p
increasing
returns
to scale
decreasing
returns
to scale
The average cost curve
Slope of AC depends on RTS
Marginal cost cuts AC at its minimum
Cq
C/q
q
q
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Revenue and profits
A given market price p
Revenue if output is q
Cost if output is q
Profits if output is q
Profits vary with q
Maximum profits
Cq
C/q
p
P
price = marginal cost
q q q
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q q q
q*
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What happens if price is low...
Cq
C/q
p
q* = 0
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price < marginal cost
q
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Profit maximisation
Objective is to choose q
to max:
pq – C (w, q)
From the First-Order
Conditions if q* > 0:
p = Cq (w, q*)
C(w, q*)
p ————
q*
In general:
p Cq (w, q*)
pq* C(w, q*)
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“Revenue minus minimised cost”
“Price equals marginal cost”
“Price covers average cost”
covers both the cases:
q* > 0 and q* = 0
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Example (continued)
Production function: q z10.1 z20.4
Resulting cost function: C(w, q) = 1.649 w10.2 w20.8 q2
Profits:
pq – C(w, q) = pq – A q2
where A:= 1.649 w10.2 w20.8
FOC:
p – 2 Aq = 0
Result:
q = p / 2A
= 0.3031 w1–0.2 w2– 0.8 p
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Summary
Key point: Profit maximisation can be viewed in two
stages:
Review
• Stage 1: choose inputs to minimise cost
Review
• Stage 2: choose output to maximise profit
What next? Use these to predict firm's reactions
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