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Exercise 13.3

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

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  1. Exercise 13.3 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

  2. 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.

  3. 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:

  4. Ex 13.3(2): Question method: • Simple sketches in (x1, x2)-space

  5. 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

  6. 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

  7. Ex 13.3(2): Indifference curves • Low value of a x2 • High value of a • U = x2 whenx1 = 0 x1

  8. Ex 13.3(3): Question method: • Check points on each of the two families of indifference curves

  9. 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

  10. 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

  11. Ex 13.3(4): Question method: • Compute consumer’s utility-maximising equilibrium

  12. Ex 13.3(4): interior solution • Marginal rate of substitution is: • Interior solution where MRS = m • So at interior solution we have: • This implies:

  13. 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

  14. 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

  15. 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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