# Frank Cowell Microeconomics November 2006 Exercise 5 1

Frank Cowell: Microeconomics November 2006 Exercise 5. 1 MICROECONOMICS Principles and Analysis Frank Cowell

Ex 5. 1(1) Question Frank Cowell: Microeconomics n n purpose: construct a simple model of household supply and examine how it works method: build model up step-by-step through the question parts

Ex 5. 1(1) Preference map §Shift the origin to (0, k) other goods Frank Cowell: Microeconomics x 2 §Draw ICs homothetic to the shifted origin l u 1 l l u 0 k 0 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 § l x 1 rice

Ex 5. 1(2) Question Frank Cowell: Microeconomics method: n Work out the budget constraint. n Use the utility function to set out the Lagrangean n Find the FOCs for an interior solution n Find the demand functions n Use these to get household supply function

Ex 5. 1(2) Budget constraint Frank Cowell: Microeconomics n Use good 2 as unit of value u u n price of rice (good 1) is p price of all other goods (good 2) is 1 The consumer’s income is therefore: y : = p. R 1 + R 2 n The budget constraint is px 1 + x 2 y where y is given by the above

Ex 5. 1(2) Lagrangean method Frank Cowell: Microeconomics n The Lagrangean is a log(x 1) + [1–a ] log(x 2–k) + l [ y – px 1 – x 2 ] n The FOC for an interior maximum are a — – pl = 0 x 1* 1–a —— – l = 0 x 2*–k y – px 1* – x 2* = 0

Ex 5. 1(2) Demand functions Frank Cowell: Microeconomics n From the FOC: a * px 1 = — l 1–a * x 2 = k + —— l n Adding these and using the budget constraint, we have y = k + 1/l Eliminating l in the above: a x 1* = — [y – k] p n x 2* = ak + [1–a ] y

Ex 5. 1(2) Supply function Frank Cowell: Microeconomics n Supply of good 1 is given by S(p) : = R 1 – x 1*. Substituting in for y, we have a S(p) = [1–a ]R 1 – — [R 2 – k] p n Supply increases with price if R 2 > k n

Ex 5. 1(2) Solution §Endowment other goods Frank Cowell: Microeconomics x 2 §Optimal consumption §Supply of rice l x* l R p k 0 §Budget constraint x 1 S rice

Ex 5. 1(3) Frank Cowell: Microeconomics n n n Let c be the amount of the ration If ak + [1–a ] y c nothing changes from previous case Otherwise px 1 + c = y so that R 2 – c x 1 = R 1 + ——— p c – R 2 S(p) = ——— p

Ex 5. 1(3) Modified solution §Original solution other goods Frank Cowell: Microeconomics x 2 §Generous ration §Severe ration c §Generous l x* c effect §Severe ration on other goods affects supply of rice l x** l R k 0 ration has no x 1 S rice

Ex 5. 1: Points to remember Frank Cowell: Microeconomics n n n Use diagram to understand features of utility function Model supply as mirror image of demand Use diagram to see effect of ration

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