Exercise 5.1

1. Exercise 5.1 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

2. Ex 5.1(1) Question • purpose: construct a simple model of household supply and examine how it works • method: build model up step-by-step through the question parts

3. Ex 5.1(1) Preference map x2 other goods • Shift the origin to (0, k) • Draw ICs homothetic to the shifted origin • • indifference curves are “shifted” Cobb-Douglas • k is min requirement of other goods. • a is share of budget of rice after an amount has been set aside to buy the min requirement • u1 • • u0 k x1 0 rice

4. Ex 5.1(2) Question method: • Work out the budget constraint. • Use the utility function to set out the Lagrangean • Find the FOCs for an interior solution • Find the demand functions • Use these to get household supply function

5. Ex 5.1(2) Budget constraint • Use good 2 as unit of value • price of rice (good 1) is p • price of all other goods (good 2) is 1 • The consumer’s income is therefore: y := pR1+ R2 • The budget constraint is px1+ x2 y where y is given by the above

6. Ex 5.1(2) Lagrangean method • 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

7. Ex 5.1(2) Demand functions • From the FOC: a px1* = — l 1–a x2* = k + —— l • Adding these and using the budget constraint, we have y= k + 1/l • Eliminating l in the above: a x1* = — [y– k] p x2* = ak + [1–a ]y

8. Ex 5.1(2) Supply function • 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

9. Ex 5.1(2) Solution x2 other goods • Endowment x2 • Budget constraint • Optimal consumption • Supply of rice • x* R • k p x1 0 rice S

10. Ex 5.1(3) • Let c be the amount of the ration • If ak + [1–a ]y  c nothing changes from previous case • Otherwise px1+ c= y so that R2– c x1= R1 +——— p c– R2 S(p) = ——— p

11. Ex 5.1(3) Modified solution x2 other goods • Original solution x2 • Generous ration • Severe ration c • Generous ration has no effect • x* • Severe ration on other goods affects supply of rice • x** c R • k x1 0 rice S

12. Ex 5.1: Points to remember • Use diagram to understand features of utility function • Model supply as mirror image of demand • Use diagram to see effect of ration