1. MICROECONOMICS Principles and Analysis Frank Cowell
Exercise 2.10
MICROECONOMICS
Principles and Analysis
Frank Cowell
November 2006

2. Ex 2.10: Question
purpose: to derive and compare short-run and long-run responses.
method: derive AC, MC, supply in original and modified models

3. Ex 2.10(1): Preliminary steps
Put the production function in a more manageable form
A quick check on the isoquant for m = 2:
Clearly isoquants do not touch the axes
Solution cannot be at a corner z 1 2

4. Ex 2.10(1): Cost minimisation
The Lagrangean:
Differentiate w.r.t. zi to find the FOCs
Rearrange to get:
l (the Lagrange multiplier) is an unknown
We need to eliminate it

5. Ex 2.10(1): Finding l Use the production function
And substitute in for zi:
where
From this we find that

6. Ex 2.10(1): The cost function
l can be simplified to
Substitute into expression for zi; get optimal input demands
So minimised costs expressed as a function of w and q are
This can be written as gBq1/g where
Differentiating this w.r.t. q, MC is
So MC is increasing in q if g < 1

7. Ex 2.10(2): Preliminary
In the “short run” the amounts of inputs k+1,…,m are fixed
So, define the term
(constant in the short run)
The production function can be written:
This is the only part that is variable in the short run.
We see that the problem has exactly the same structure as before
but with different parameters.
Therefore the solution has the same structure as before

8. Ex 2.10(2): Short-run input demand
We can proceed by analogy with the long-run case
Cost-minimising input demands must be:
where we have defined
Multiplying each input demand by wi and summing will give short-run variable costs

9. Ex 2.10(2): Short-run costs Define short-run fixed costs
the amounts of inputs k+1,…,m are fixed
Then short-run total costs are given by
Substituting in for zi* costs in the short run are:
Clearly this expression has the form:
Differentiate costs w.r.t. q and we find short-run MC:

10. Ex 2.10(3): short run supply
From the SRMC we get the short-run supply curve
The condition “MC = price” gives
Solving this for q the supply function is
The elasticity of supply is
Clearly the elasticity falls if gk falls
By definition of gk it must fall if k is reduced

11. Ex 2.10: Points to remember Get the constraint into a convenient form
Get a simple view of the problem by deriving ICs
Use a little cunning to simplify the FOCs
Re-use your solution for other problems that have the same structure