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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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