Topics covered. ECON6021 Microeconomic Analysis. Convex Combination. Bundle of goods

Transcription

1 Topics covered ECON601 Microeconomic nalsis Consumption Theor I 1. udget Constraint. ioms of Choice & Indifference Curve 3. tilit Function 4. Consumer Optimum 1 undle of goods Conve Combination is a bundle of goods consisting of units of good (sa food) and units of good (sa clothing). ( c, c ) ( t + (1 t), t + (1 t) ) (, ) C is also represented b (, ) (, ) 3 4 1

2 Conve Combination Slope of budget line ( t + (1 t) ) Slopeof C ( t + (1 t) ) fl C is on the st. line linking & t t t t Slope of d d (market rate of substitution) Conversel, an point on can be written as ( t (1 t ), t + (1 t ) ) where t [0,1] + $ per jar nit: loaves per jar (independe nt of $) $ per loaf 5 6 Ta: a $ lev per unit is imposed for each good Eample: jar of beer $4 loaf of bread $ Slope feasible consumption set + t $4 + $ + t $ + $ 3 fl Slope of budget line changes $4 per jar $ per loaf loaves per jar 4 after lev is imposed oth and double, ' ' $8 per jar $4 per loaf loaves per jar No change in market rate of substitution fter doubling the prices

3 ioms of Choice ioms of Choice & Indifference Curve Nomenclature: : is preferred to : is strictl preferred to ~: is indifferent to Completeness (Comparison) n two bundles can be compared and one of the following holds:,, or both (fl ~) Transitivit (Consistenc) If,, C are 3 alternatives and, C, then C; lso If ~, ~C, then ~C ioms of choice Continuit and is sufficientl close to C, then C. Strong Monotonicit (more is better) (, ), (, ) and, with at least one is strict, then >. Conveit If ~, then an conve combination of & is preferred to and to, that is, for all 0 <t <1, (t +(1-t), t +(1-t) ) ( i, i ), i or. If the inequalit is alwas strict, we have strict conveit. 11 Indifference Curve When goods are divisible and there are onl two tpes of goods, an individual s preferences can be convenientl represented using indifference curve map. n indifference curve for the individual passing through bundle connects all bundles so that the individual is indifferent between and these bundles. 1 3

4 roperties of Indifference Curves Negative slopes ICs farther awa from origin means higher satisfaction I II Not preferred bundles referred bundles 13 roperties of Indifference Curves Non-intersection Two indifference curves cannot intersect Coverage For an bundle, there is an indifference curve passing through it. Q 14 roperties of Indifference Curves ending towards Origin It arises from conveit aiom The right-handside IC is not allowed tilit Function

5 tilit Function Level of satisfaction depends on the amount consumed: (,) 0 (,) ll the combination of & that ield 0 (all the alternatives along an indifference curve) V(, 0 ), an indifference curve (,)/, marginal utilit respect to, written as M. 17 (, ) (, ) d d + d 0 (, ) (, ) d0 d + d 0 (b construction) Slope: d / M < 0 d / M (if strong monotonicit holds) 18 MRS > MRS Marginal rate of substitution 0 The MRS is the ma amount of good a consumer would willingl forgo for one more unit of, holding utilit constant (relative value of epressed in unit of ) MRS DMRS: d d / M > 0 d d / M 0 dmrs d constant 0 <

6 Measurabilit of tilit V100 V00 V n order-preserving re-labeling of ICs does not alter the preference ordering. ositive monotonic (orderpreserving) transformation The are called positive monotonic transformation ' '' ''' '''' ositive Monotonic Transformation What is the MRS of at (,)? fl How about? / MRS / ositive Monotonic Transformation IC s of order-preserving transformation overlap those of. However, we have to make sure that the numbering of the IC must be in same order before & after the transformation. ' ' fl MRS' '/ '/ 3 4 6

7 ositive Monotonic Transformation roof Theorem: Let (,) be an utilit function. Let VF((,)) be an orderpreserving transformation, i.e., F(.) is a strictl increasing function, or df/d>0 for all. Then V and represent the same preferences. 5 Consider an two bundles (, ) and (, ). Then we have: (, ) > (, ) F( (, )) > F( (, )) V(, ) > V(, ) V Q.E.D. 6 Consumer Optimum Constrained Consumer Choice roblem references: represented b indifference curve map, or utilit function (.) Constraint: budget constraint-fied amount of mone to be used for purchase ssume there are two tpes of goods and, and the are divisible 7 8 7

8 Consumption problem udget constraint given mone income in $ given price of good given price of good udget constraint: + Or, + (strong monotonicit) d d+ d0 (b construction) d- d D MRS D C D schic willingness to substitute t, m MRS is ver high for. I m willing to substitute - for - D. ut the market provides me more to point D! fl MRS > D 9 30 Consumer Optimum Normall, two conditions for consumer optimum: 0 1 oth & C satisf (1) and () roblem: bending toward origin does not hold. MRS / (1) No budget left unused () C

9 Special Cases Quantit Control coffee Generall low MRS coffee Generall high MRS Ma (,) Subject to (i) I + (ii) R 1 0 tea MRS < for all (, ) MRS > for all (, ) tea n Eample: (,) (1) () (1) Corner at 0 () Interior solution 0<<R (3) corner at R (4) corner at R ma, subject to I + (1) M, M (3) (4) or M MRS market rate of sub. () M I I Hence, ( (1))

10 Other Eamples of tilit Functions D C satisfies (1) but not (), C satisf () but not (1) Onl D satisfies both (1) &() (, ) α + β (, ) min{ α, β} I 1 I 1 I 1 I I/ n application: Intertemporal Choice Our framework is fleible enough to deal with questions such as savings decisions and intertemporal choice. Intertemporal choice problem Income in period C 1600 u(c 1,c )const 500 Slope C 1 Income in period 40 10

11 1000-C 1 S (1) 500+S(1+r)C () Substituting (1) into (), we have 500+(1000-C 1 )(1+r)C Rearranging, we have r-(1+r) C 1 C > C sing C 1 C C, we finall have r 500 C r + r r ÆflC Æ (S ) 41 11