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**MICROECONOMICS Principles and Analysis Frank Cowell**

Exercise 5.1 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

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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

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Ex 5.1(1) Preference map x2 other goods Shift the origin to (0, k) Draw ICs homothetic to the shifted origin l 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 Expanto l u1 l u0 k x1 rice

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**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

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**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

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**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

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**Ex 5.1(2) Demand functions From the FOC: a px1* = — l 1–a x2* = k + ——**

Adding these and using the budget constraint, we have y = k + 1/l Eliminating l in the above: x1* = — [y – k] p x2* = ak + [1–a ] y

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**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

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**Ex 5.1(2) Solution k S x2 Endowment other goods x2 Budget constraint**

Optimal consumption Supply of rice x* l Expanto R l k p x1 rice S

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**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) = ———

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**Ex 5.1(3) Modified solution**

other goods Original solution x2 Generous ration Severe ration c Generous ration has no effect x* l Expanto Severe ration on other goods affects supply of rice c x** l R l k x1 rice S

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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

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