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Exercise 13.3. MICROECONOMICS Principles and Analysis Frank Cowell. November 2006. Ex 13.3(1): Question. purpose : a simple model of choice in the presence of non-convexity

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Exercise 13 3

Exercise 13.3

MICROECONOMICS

Principles and Analysis

Frank Cowell

November 2006


Ex 13 3 1 question
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
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:


Ex 13 3 2 question
Ex 13.3(2): Question

method:

  • Simple sketches in (x1, x2)-space


Gas production possibilities
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

[R2F]/m


Ex 13 3 2 preferences
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 2 indifference curves
Ex 13.3(2): Indifference curves

  • Low value of a

x2

  • High value of a

  • U = x2 whenx1 = 0

x1


Ex 13 3 3 question
Ex 13.3(3): Question

method:

  • Check points on each of the two families of indifference curves


Ex 13 3 3 max utility h igh a
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 low a
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 question
Ex 13.3(4): Question

method:

  • Compute consumer’s utility-maximising equilibrium


Ex 13 3 4 interior solution
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
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
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
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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