Exercise 13.3

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# Exercise 13.3 - PowerPoint PPT Presentation

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

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
• 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:
Ex 13.3(2): Question

method:

• Simple sketches in (x1, x2)-space
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
• 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
• Low value of a

x2

• High value of a
• U = x2 whenx1 = 0

x1

Ex 13.3(3): Question

method:

• Check points on each of the two families of indifference curves
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): Question

method:

• Compute consumer’s utility-maximising equilibrium
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