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Ex 13.3(1): Question

- purpose: a simple model of choice in the presence of non-convexity
- method: carefully describe attainable set; find possible equilibria for consumers; compare them using Pareto criterion.

Ex 13.3(1): Production possibilities

- x1 units of good 1 must cost F + mx1 of good 2
- F is fixed cost
- m is marginal cost

- There are R2 units of good 2 available in the economy
- So the maximum possible amount of good 1 is:

Gas: production possibilities

- Commodity space

x2

- Endowment of “other goods”

- Fixed cost of gas production

- Attainable set

- Max possible amount of gas

R2

l

F

- Constant MC of gas production

m

l

x1

[R2F]/m

Ex 13.3(2): Preferences

- From the utility function
- We can check the 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

Ex 13.3(3): max utility, high a

- Attainable set

x2

- A typical IC

- Reservation IC

- IC where MRS = MRT = m

R2

l

- U(x*) > U(0, R2)

x*

l

x1

Ex 13.3(3): max utility, lowa

- Attainable set

x2

- A typical IC

- Reservation IC

- IC where MRS = MRT = m

R2

l

- U(0, R2) > U(x*)

x*

- For formal comparisons of two utility levels see next part

l

x1

x1

Ex 13.3(4): interior solution

- Marginal rate of substitution is:
- Interior solution where MRS = m
- So at interior solution we have:
- This implies:

Ex 13.3(4): utility

- Maximised utility at interior solution:
- a[1 - exp (x1*)]+x2* :

- Substituting in the value of (x1*, x2*) utility is:
- Utility at corner (0, R2) is just R2
- So (x1*, x2*) represents a global maximum if
- U(x1*, x2*) >U (0, R2)
- which implies

Ex 13.3(4): implement interior solution

- Attainable set

x2

- Indifference Curves

- Interior optimum

- A fee schedule

R2

l

F

- Two part tariff:
- Fixed charge F
- price per unit m

x*

l

x1

Ex 13.3(4): points to remember

- Fixed cost implies nonconvexity of attainable set
- Nonconvexity implies two possible solutions
- which is relevant?
- depends on preference parameter

- Need to check utility levels to find PE
- Two-part tariff can be used to implement
- induces consumer to choose optimum
- covers production costs

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