1. EC212: Worked Example #3.21 Frank Cowell April 2004
This presentation covers exactly the material set out in the file WorkedExamples.pdf, but with the addition of a few graphics and comments
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2. WX3.21: part 1 Marginal cost Fixed cost
x1 units of good 1 must cost F + mx1 of good 2
There are X2 units of good 2 available in the economy
So the maximum possible amount of good 1 is:

3. WX3.21: part 1 x2 X2
Fixed cost F
marginal cost m
gas x1

4. WX3.21: part 2 We can easily check the MRS
Clearly MRS goes to 0 as x1 goes to 
For a given x1 MRS is the same for all x2
MRS is high for high a and vice versa

5. WX3.21: part 2 x1 x2
gas
U=x2 when x1 = 0
high a
low a

6. WX3.21: Part 3

7. WX3.21: part 3 - high a x1 x2 X2
gas B
Utility increases
At A MRS=m A

8. WX3.21: part 3 - high a MRS is: So at A we have: This implies:
Utility at A is
a[1 - exp (x1*)]+x2* :
Utility at B is X2
So UA > UB if

9. WX3.21: part 3, low a x1 x2 X2
gas B
Utility increases A

10. WX3.21: Part 3, implementation
Budget constraint must pass through B
It must be such as to make A utility-maximising
This can be achieved if a non-linear budget constraint can be introduced
A public corporation could do this
Make a fixed charge F and then sell gas at marginal cost m.

11. WX3.21: Part 4 Idea here is simple
(formal analysis can get a bit tiresome)
Obviously a private corporation might market the gas in the same manner as the public corporation
But can gas market be decentralised?
Obviously if p > m (case of simple monopoly) this would be inefficient. But what about a market solution with p = m ?

12. WX3.21: part 4 x1 x2 X2
gas A B
Profit maximisation at price p=m will take economy to B, not A!
price = MC

13. WX3.21: Points to note
Draw a simple picture of each part of the problem
Use the diagram to determine location of PE allocations
Interpret diagram in terms of a budget constraint.
Rest of problem can be understood in similar way. Feasible set is obviously non-convex, so decentralisation may not be possible.