MICROECONOMICS Principles and Analysis Frank Cowell 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 Isoquant1-4 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(2): Indifference curves Low value of a High value of a U = x2 when x1 = 0 Isoquant1-4 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 A typical IC Reservation IC IC where MRS = MRT = m R2 l U(x*) > U(0, R2) Isoquant1-4 x* l x1
Ex 13.3(3): max utility, low a U(0, R2) > U(x*) Attainable set A typical IC Reservation IC IC where MRS = MRT = m R2 l U(0, R2) > U(x*) Isoquant1-4 x* l For formal comparisons of two utility levels see next part 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 Indifference Curves Interior optimum A fee schedule R2 l F Two part tariff: Fixed charge F price per unit m Isoquant1-4 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