1. The Firm: Optimisation
Prerequisites
Almost essential
Firm: Basics
The Firm: Optimisation
MICROECONOMICS
Principles and Analysis
Frank Cowell
Note: the detail in slides marked “ * ” can only be seen if you run the slideshow
July 2015

2. Overview... Approaches to the firm’s optimisation problem
Firm: Optimisation
The setting
Approaches to the firm’s optimisation problem
Stage 1: Cost Minimisation
Stage 2: Profit maximisation
July 2015

3. 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
July 2015

4. The optimisation problem
Objectives
Constraints
Method
Profit maximisation?
-Technology; other
- 2-stage optimisation
July 2015

5. Construct the objective function
Use the information on prices… wi
price of input i p
price of output
…and on quantities… zi
amount of input i q
amount of output
How it’s done
…to build the objective function
July 2015

6. The firm’s objective function
S wizi
i=1
Cost of inputs:
Summed over all m inputs
Revenue: pq
Subtract Cost from Revenue to get m
S wizi
i=1
Profits:
pq –
July 2015

7. Optimisation: the standard approach
Choose q and z to maximise m
S wizi
i=1
P := pq –
...subject to the production constraint...
Could also write this as zZ(q)
q £ f (z)
..and some obvious constraints:
You can’t have negative output or negative inputs
q ³ 0
z ³ 0
July 2015

8. A standard optimisation method
If  is differentiable…
Set up a Lagrangean to take care of the constraints
L (... )
Write down the First Order Conditions (FOC)
necessity
 L (... ) = 0 ¶z
Check out second-order conditions
sufficiency ¶2
 L (... )
¶z2
Use FOC to characterise solution
z* = …
July 2015

9. 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...
July 2015

10. 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…
July 2015

11. 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
July 2015

12. Overview...
Firm: Optimisation
The setting
A fundamental multivariable problem with a brilliant solution
Stage 1: Cost Minimisation
Stage 2: Profit maximisation
July 2015

13. Stage 1 optimisation S wi zi 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
July 2015

14. 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
July 2015

15. Iso-cost lines increasing cost z2 w1z1 + w2z2 = c" w1z1 + w2z2 = c'
Draw set of points where cost of input is c, a constant z1 z2
Repeat for a higher value of the constant
increasing cost
Imposes direction on the diagram...
w1z1 + w2z2 = c"
w1z1 + w2z2 = c'
w1z1 + w2z2 = c
Use this to derive optimum
July 2015

16. Cost-minimisation S wizi z* q minimise subject to (z)  q z2 z1
The firm minimises cost... z1 z2 q
Subject to output constraint
Defines the stage 1 problem
Solution to the problem
minimise m
S wizi
i=1
subject to (z)  q
Reducing cost z*
But the solution depends on the shape of the input-requirement set Z
What would happen in other cases?
July 2015

17. Convex, but not strictly convex Z
Any z in this set is cost-minimising
An interval of solutions
July 2015

18. Convex Z, touching axis z* z2 z1
Here MRTS21 > w1 / w2 at the solution
Input 2 is “too expensive” and so isn’t used: z2* = 0 z*
July 2015

19. Non-convex Z z* z** z2 z1 There could be multiple solutions
But note that there’s no solution point between z* and z**
z**
July 2015

20. * Non-smooth Z z* z2 z1 z* is unique cost-minimising point for q
MRTS21 is undefined at z*
z* is unique cost-minimising point for q z*
True for all positive finite values of w1, w2
July 2015

21. Cost-minimisation: strictly convex Z
Minimise
Lagrange multiplier
Use the objective function
...and output constraint m
S wi zi
i=1
...to build the Lagrangean
+ l[q – f(z)]
q  f (z)
Differentiate w.r.t. z1, ..., zm ; set equal to 0
... and w.r.t l
Denote cost minimising values by *
Because of strict convexity we have an interior solution
A set of m+1 First-Order Conditions
l f1 (z ) = w1
l f2 (z ) = w2
… … …
l fm(z ) = wm
* * *
one for each input ü ý þ
q = (z )
output
constraint
July 2015

22. If isoquants can touch the axes...
Minimise m
Swizi
i=1
+ l [q – f(z)]
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 ü ý þ
Interpretation
q = f(z*)
Can get “<” if optimal value of this input is 0
July 2015

23. From the FOC fi(z*) wi ——— = — fj(z*) wj fi(z*) wi ——— £ — fj(z*) wj
If both inputs i and j are used and MRTS is defined then...
fi(z*) wi
——— = —
fj(z*) wj
MRTS = input price ratio
“implicit” price = market price
If input i could be zero then...
fi(z*) wi
——— £ —
fj(z*) wj
MRTSji £ input price ratio
“implicit” price £ market price
Solution
July 2015

24. The solution... C(w, q) := min S wi zi
Solving the FOC, you get a cost-minimising value for each input...
The solution...
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) ³q}
vector of
input prices
Specified output level
July 2015

25. Interpreting the Lagrange multiplier
The solution function:
C(w, q) = Siwi zi*
= Si wi zi*– l* [f(z*) – q]
At the optimum, either the constraint binds or the Lagrange multiplier is zero
Differentiate with respect to q:
Cq(w, q) = SiwiHiq(w, q)
– l* [Si fi(z*) Hiq(w, q) – 1]
Express demands in terms of (w,q)
Vanishes because of FOC l *fi(z*) = wi
Rearrange:
Cq(w, q) = Si [wi – l*fi(z*)] Hiq(w, q) + l*
Cq (w, q) = l*
Lagrange multiplier in the stage 1 problem is just marginal cost
This result – extremely important in economics – is just an applications of a general “envelope” theorem
July 2015

26. 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
July 2015

27. Properties of C C C(w, q+q) C(w, q) °
z1*
Draw cost as function of w1 w1 C
C(w, q+q)
Cost is non-decreasing in input prices
C(w, q)
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)
C(w,q)
———— = zj*
wj
July 2015

28. What happens to cost if w changes to tw
Find cost-minimising inputs for w, given q z1 z2 q
Find cost-minimising inputs for tw, given q
So we have:
C(tw,q) = i t wizi* = t iwizi* = tC(w,q) z* z*
The cost function is homogeneous of degree 1 in prices
July 2015

29. 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
July 2015

30. Example Production function: q  z10.1 z20.4
Equivalent form: log q  0.1 log z 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 z log z2
From the FOCs:
log q = 0.1 log (0.1 l / w1) log (0.4 l / w2 )
l = 0.1– –0.8 w10.2 w20.8 q2
Therefore, from this and the FOCs:
w1 z1 + w2 z2 = 0.5 l = w10.2 w20.8 q2
July 2015

31. Overview... …using the results of stage 1 Firm: Optimisation
The setting
…using the results of stage 1
Stage 1: Cost Minimisation
Stage 2: Profit maximisation
July 2015

32. 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
July 2015

33. Average and marginal cost
increasing returns
to scale
decreasing returns
to scale p
The average cost curve
Slope of AC depends on RTS
Marginal cost cuts AC at its minimum Cq
C/q q q
July 2015

34. * Revenue and profits p P Cq C/q q q* q q q q q price = marginal cost
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 q q* q q q q P
price = marginal cost q
July 2015

35. What happens if price is low...Cq
C/q p
price < marginal cost
q* = 0 q
July 2015

36. Profit maximisation pq – C (w, q) Objective is to choose q to max:
“Revenue minus minimised cost”
From the First-Order Conditions if q* > 0:
p = Cq (w, q*)
C(w, q*)
p ³ ———— q*
“Price equals marginal cost”
“Price covers average cost”
In general:
covers both the cases:
q* > 0 and q* = 0
p £ Cq (w, q*)
pq* ³ C(w, q*)
July 2015

37. Example (continued) Production function: q  z10.1 z20.4
Resulting cost function: C(w, q) = w10.2 w20.8 q2
Profits:
pq – C(w, q) = pq – A q2
where A:= w10.2 w20.8
FOC:
p – 2 Aq = 0
Result:
q = p / 2A
= w1–0.2 w2– 0.8 p
July 2015

38. Summary Key point: Profit maximisation can be viewed in two stages:
Stage 1: choose inputs to minimise cost
Stage 2: choose output to maximise profit
What next? Use these to predict firm's reactions
Review
Review
July 2015