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

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

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

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

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

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

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

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

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

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) = ———

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

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