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Frank Cowell: Microeconomics Exercise 5.1 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

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

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Frank Cowell: Microeconomics 0 Ex 5.1(1) Preference map x2x2 x1x1 l 00 l 11 l l k 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. is share of budget of rice after an amount has been set aside to buy the min requirement rice other goods

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

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Frank Cowell: Microeconomics Ex 5.1(2) Budget constraint Use good 2 as unit of value 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: The consumer’s income is therefore: y := pR 1 + R 2 T The budget constraint is px 1 + x 2 y where y is given by the above

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Frank Cowell: Microeconomics Ex 5.1(2) Lagrangean method T The Lagrangean is log(x 1 ) + [1– ] log(x 2 –k) + [ y – px 1 – x 2 ] T The FOC for an interior maximum are — – p x 1 * 1–1– —— – x 2 * –k y – px 1 * – x 2 *

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Frank Cowell: Microeconomics Ex 5.1(2) Demand functions From the FOC: px 1 * — 1– x 2 * k + —— Adding these and using the budget constraint, we have y k + 1/ Eliminating in the above: x 1 * — [y – k] p x 2 * k + [1– ] y

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Frank Cowell: Microeconomics Ex 5.1(2) Supply function Supply of good 1 is given by S(p) := R 1 – x 1 * Substituting in for y, we have S(p) [1– ]R 1 – — [R 2 – k] p Supply increases with price if R 2 > k

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Frank Cowell: Microeconomics 0 Ex 5.1(2) Solution x2x2 x1x1 k Endowment Budget constraint rice other goods x* x2x2 p S R l l Optimal consumption Supply of rice

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Frank Cowell: Microeconomics Ex 5.1(3) Let c be the amount of the ration If k + [1– ] 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

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Frank Cowell: Microeconomics 0 Ex 5.1(3) Modified solution x2x2 x1x1 k Original solution rice other goods x* x2x2 S R l l Generous ration Severe ration x** l Generous ration has no effect Severe ration on other goods affects supply of rice c c

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Frank Cowell: Microeconomics Ex 5.1: Points to remember Use diagram to understand features of utility function Use diagram to understand features of utility function Model supply as mirror image of demand Model supply as mirror image of demand Use diagram to see effect of ration Use diagram to see effect of ration

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