The Firm: Optimisation

Report
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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z0
• Could also write this as zZ(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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