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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

• 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.

• 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:

method:

• Simple sketches in (x1, x2)-space

• 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

• 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

• Low value of a

x2

• High value of a

• U = x2 whenx1 = 0

x1

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

• 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

method:

• Compute consumer’s utility-maximising equilibrium

• Marginal rate of substitution is:

• Interior solution where MRS = m

• So at interior solution we have:

• This implies:

• 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

• 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

• 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