1. MICROECONOMICS Principles and Analysis Frank Cowell
Exercise 2.9
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2007

2. Ex 2.9(1): Question
purpose: demonstrate relationship between short and long run
method: Lagrangean approach to cost minimisation. First part can be solved by a “trick”

3. Ex 2.9(1): Long-run costs
Production function is homogeneous of degree 1
increase all inputs by a factor t > 0 (i.e. z → tz)…
…and output increases by the same factor (i.e. q → tq)
constant returns to scale in the long run
CRTS implies constant average cost
C(w, q) / q = A (a constant)
so C(w, q) = Aq
differentiating: Cq(w, q) = A
So LRMC = LRAC = constant
Their graphs will be an identical straight line

4. Ex 2.9(2): Question
method:
Standard Lagrangean approach

5. Ex 2.9(2): short-run Lagrangean
In the short run amount of good 3 is fixed
z3 = `z3
Could write the Lagrangean as
But it is more convenient to transform the problem thus
where

6. Ex 2.9(2): Isoquants Sketch the isoquant map
Isoquants do not touch the axes
So maximum problem must have an interior solution z2 z1
Isoqnta

7. Ex 2.9(2): short-run FOCs Differentiating Lagrangean, the FOCS are
This implies
To find conditional demand function must solve for l
use the above equations…
…and the production function

8. Ex 2.9(2): short-run FOCs (more)
Using FOCs and the production function:
This implies
where
This will give us the short-run cost function

9. Ex 2.9(2): short-run costs By definition, short-run costs are:
This becomes
Substituting for k:
From this we get
SRAC:
SRMC:

10. Ex 2.9(2): short-run MC and ACq
marginal
cost
average
cost

11. Ex 2.9(3): Question method: Draw the standard supply-curve diagram
Manipulate the relationship p = MC

12. Ex 2.9(3): short-run supply curve
average cost curve
marginal cost curve q p
minimum average cost
supply curve p q

13. Ex 2.9(3): short-run supply elasticity
Use the expression for marginal cost:
Set p = MC for p ≥ p
Rearrange to get supply curve
Differentiate last line to get supply elasticity

14. Ex 2.9: Points to remember Exploit CRTS to give you easy results
Try transforming the Lagrangean to make it easier to manipulate
Use MC curve to derive supply curve