Esercitazione 2 Teoria del consumatore

1. Esercitazione 2 Teoria del consumatore

2. x x 2 2 A B x x 1 1 x 2 x 2 C D A x 1 x 1 Esercizio 1

3. part 2 Step 1: set up the Lagrangean:

4. part 2 First-order conditions: Solve to get: Offer curve is a vertical line at 10a

5. part 3 • Points to watch: • Use your knowledge of the “shape” of the solution • For example we can get the solution to type A by adapting part 2 • Types B-D follow by using the diagrams in Part 1

6. part 3, type A x2 20 We can use the demand function from part 2. Income is 20 now (instead of 10r) so solution must be: 20[1-a] x1 Offer curve is horizontal line at 20[1-a]

7. part 3, type B x2 x' Solution must be on one or other axis unless r=b: indifference curve b x1 x'' Offer curve is line segment with kink at x''

8. part 3, type C x2 x' Solution must be on one or other axis:  g x1 x'' Offer curve is blob at x' and line segment from x''

9. part 3, type D x2 Solution must lie on corner of the indifference curve where x2=dx1. Using this fact and the budget constraint x2+rx1=20 we have: d x1 Offer curve is line through all the corners

10. Esercizio 2: Rationing Part 1: standard demand functions

11. Utility maximisation Maximise x1x2x3x4 subject to The Lagrangean is Differentiating, the FOC is which implies Ordinary demand function Using the budget constraint we get l = 4/M. So we have

12. Derive related functions Indirect utility function Substitute optimal demands in utility function Cost function Rearrange to get M as a function of u Compensated demand function Differentiate:

13. Ordinary elasticity Take the ordinary demand function Take logs and differentiate with respect to log p1 and with respect to log pj

14. Compensated elasticity Take the compensated demand function Again take logs and differentiate with respect to log p1 and with respect to log pj

15. Consumption and rationing Part 2: introduce a side constraint

16. Modify the problem • x4 is now fixed at A4 • Define M' := M – p4 A4 • Problem is equivalent to maximising x1x2x3A4 subject to budget with adjusted income M' . • Use results from part 1 applied to 3-good economy Errore: manca A4-1/3

17. Ordinary elasticity again Take the ordinary demand function An income effect Take logs and differentiate with respect to log p4 Take logs and differentiate with respect to log p1 and with respect to log pj

18. Compensated elasticity again Take the compensated demand function Errore: manca A4-1/3 Closer to 0 than before Again take logs and differentiate

19. Consumption and rationing Part 3: more constraints

20. More constraints Reapply the method with one fewer commodity: Errore: manca un pezzo contenente A4 e A3 Closer to 0 than before Differentiate again:

21. Consumption and rationing Part 4: Interpretation • Model illustrates the comparative statics of someone who is subject to a quota ration. • But not rich enough to determine which commodities are consumed at a conventional equilibrium and which will be constrained by the ration. • Parts 2 and 3 show clearly how the compensated demand becomes “steeper” the more external constraints are imposed

22. Points to remember • Modify the problem where appropriate to get a more tractable equivalent. • Re-use the solution to one part of the problem to build the next.

23. Esercizio 3: part 1(a) . These are easy – parabolic contours Even easier – fixed money income First steps: • Sketch indifference curves • Write down budget constraint • Set out optimisation problem

24. x2 . x1 Slope is vertical here We could have x2=0

25. Budget constraint: • Substitute this into the utility function: • We get the objective function: • FOC for an interior solution:

26. FOC for an interior solution: • Therefore, if positive amounts of the two goods are bought: • But this requires: • Otherwise x2=0 and we get x1 from the budget constraint.

27. Demand functions:

28. Maximised utility gives us the indirect utility function: • Otherwise:

29. For case where both goods are consumed • For the cost function use the relation u=V(p,M) • … and solve for M:

30. Esercizio 3: part 1(b,c) . Method: • Use C function to write down CV • (Equivalently use V function to write down CV) • Check income effects

31. Compensating variation is • But demand for good 1 has zero income effect • So CV = CS = EV in this case

32. Esercizio 3: part 2(a) . Method: • Find monopolist’s AR from consumer demand using part 1. • Use standard optimisation procedure

33. Aggregate demand over N consumers using part 1: • Rearrange to get AR curve: • Total Revenue is: • Profits are therefore:

34. MC = MR • FOC: • Optimal output: • From AR curve, price at optimum is: Price > MC • …simplified:

35. Esercizio 3: part 2(b) . Method: • Aggregate the CV for each consumer to define L • Use marginal cost and monopolist’s equilibrium price to evaluate L

36. Use definition of CV with p1´ = c • Evaluate L at p1 = 2c

37. Esercizio 3: part 2(c) . Method: • Add bonus B into the expression for profits • Again use standard optimisation procedure

38. Profits are now: • Value of bonus is: • Expressed in terms of quantity: • So profits become:

39. FOC: Price = MC • Resulting equilibrium:

40. Esercizio 3: Points to note • The fact that indifference curves touch the axis does not cause any great problem • Aggregate welfare loss is found from individual CV • Regulation causes monopoly to behave like competitive firm

41. Esercizio 4: setting • Sketch the indifference curves: “shifted” Cobb-Douglas • k is minimum consumption requirement of other goods. • a is share of budget that goes on rice after an amount has been set aside to buy the min requirement

42. part 1 – approach • Work out the budget constraint. • Use the utility function to set out the Langrangean • Find the FOCs for an interior solution • Check whether the solution makes sense • Find the demand functions • Use these to get household supply function

43. part 1 – getting the FOC • The budget constraint is • px1+ x2 y • where • y := pR1+ R2 • The Lagrangean is • a log(x1) + [1–a]log(x2–k) + l [ y– px1– x2 ] • The FOC for an interior maximum are • a • — – pl = 0 • x1 • 1–a • —— – l = 0 • x2–k • y– px1– x2 = 0

44. part 1 – supply function • From the FOC: • a1–a • px1 = — x2 = k + —— • l l • Adding these and using the budget constraint, we have • y= k + 1/l • Eliminating l in the above: • a • x1 = — [y– k] x2 = ak + [1–a ]y • p • Supply of good 1 is given by • S(p) := R1– x1. • Substituting in for y, we have • a • S(p) = [1–a ]R1–— [R2– k] • p Supply increases with price if R2> k

45. part 2 • If ak + [1–a ]y  c nothing changes from previous case. • Otherwise • px1+ c= y • so that • R2– c • x1= R1 +——— • p