Appendi: The theory of consumer s behavior preference, utility, indifference curve, budget constraint, optimal consumption plan, demand curve 1
1. Preference ordering and utility function! Objects to be selected x, y, z! Consumption set consumption goods bundle x =(,,...,x n ) (Consumption vector)! Preference relation (binary relation)! Preference ordering! Value judgment about the alternatives! xry x is strictly better than y or both are indifferent (x is at least as good as y)! xpy x is strictly better than y (prefer x to y)! xiy x and y are indifferent 2
Nature of the preference relation! Reflectivity: xrx holds for any x.! Completeness: xry or yrx holds for any x and y.! Transitivity: If xry and yrz, then xrz holds for any x, y, and z.! Preference ordering = reflectivity + completeness + transitivity rational preference (relation) 3
Utility function! Utility function preference ordering u=u(x), u=u(,,...,x n )! The function gives a large value to the desirable alternatives! Indifference curve utility function! The curve shows the group of alternatives to have the same desirableness 4
y u=u(, ) Indifference curve 5
u 0 <u 1 u=u 1 0 u=u 0 6
2. The budget constraint! Price (p 1,p 2,...,p n )>0! Income M>0! The quantity of purchase plan (The quantity demanded) (,,...,x n ) 0! The budget constraint budget set! Expenditure Income! p 1 +p 2 + +p n x n M! The budget line (budget constraint line)! p 1 +p 2 + +p n x n =M 7
The budget set! The budget set budget constraint! The set of the possible consumer goods bundles which satisfy a budget constraint (The opportunity set)! p 1 +p 2 + +p n x n M! 0, 0,,x n 0! The inside and the boundary of the triangle which was surrounded by the budget line, the vertical axis, the horizontal axis! The budget line : =-(p 1 /p 2 ) +M/p 2 8
M/p 2 =-(p 1 /p 2 ) +M/p 2 The budget set 0 M/p 1 9
M'/p 2 M/p 2 The parallel shift of the budget line M"<M<M' =-(p 1 /p 2 ) +M'/p 2 M"/p 2 =-(p 1 /p 2 ) +M/p 2 =-(p 1 /p 2 ) +M"/p 2 M"/p 1 M/p 1 M'/p 1 10
The turn of the budget line p" 1 <p 1 <p' 1 =-(p" 1 /p 2 ) +M/p 2 M/p 2 =-(p 1 /p 2 ) +M/p 2 =-(p 1 /p' 2 ) +M/p' 2 M/p' 2 =-(p' 1 /p 2 ) +M/p 2 0 M/p' 1 M/p 1 M/p" 1 11
3. The optimal consumption plan! The indifference curve + budget constraint The optimal consumption plan ( *, *)! The additional assumption about preference Monotonicity: The more, the better. Convexity (The law of diminishing marginal rate of substitution) The more, the better, and the less the worse. People like a mean better than the extremes. 12
marginal rate of substitution MRS=-d /d =u 1 /u 2 E d u=u 0 0 d 13
Explanation of MRS=u 1 /u 2! Differentiate u=u(, ) du=u 1 d +u 2 d! On the same indifference curve du=0 -d /d =u 1 /u 2! Since MRS=-d /d MRS=u 1 /u 2 14
A The law of diminishing marginal rate of substitution MRS A >MRS B >MRS C The indifference curve swelled up for the original point B 0 C u=u 0 13
! The optimal consumption plan ( *, *)! Inner point C of the budget set not optimal Utility increases in moving to the boundary.! Point A on the boundary not optimal Utility increases in moving to the direction of point E on the budget line.! Point B on the boundary not optimal Utility increases in moving to the direction of point E on the budget line.! Point E is optimal the optimal consumption plan 14
To decide the optimal consumption plan A C E monotonicity: u 0 <u 1 0 B u=u 1 u=u 0 15
The optimal consumption plan! The budget line and the indefference curve touch.! price ratio = marginal rate of substitution p 1 /p 2 =MRS=u 1 /u 2! It satisfies the budget constraint with the equal sign. p 1 + p 2 =M 16
4. The derivation of the demand curve! The optimal consumption plan change of the price! The new optimal consumption plan A wide range of prices! The demand curve =d 1 (p 1,p 2,M)! The curve (function) shows how the demand plan changes when the price of the good is changed, making the prices of other goods and the income constant. 19
The derivation of the demand curve The price-consumption curve E 1 E 0 E 2 u=u 2 u=u 0 u=u 1 0 1 0 2 M/p 1 0 M/p 1 2 M/p 1 1 20
p 1 The demand curve of the 1st good p 1 1 p 1 0 =d 1 (p 1,p 2,M) p 1 2 1 0 2 21
5. The income effect and the substitution effect! The effect of income change The change of the quantity demanded when the income is changed The effect by the shift of the budget line 22
M'/p 2 The normal good M/p 2 M"/p 2 The income-consumption curve M"/p 1 M/p 1 M'/p 1 23
The intermediate good The inferior property 24
The effect of the price change! The change of the price i) The change of the real income (The income effect) ii) The change of the relative price (The substitution effect)! The compensation variation with income (The compensation income) How much income is needed after the price change in order to maintain utility level before the price change? 25
The income effect and the substitution effect 2 1 0 E 1 E 2 E 0 u=u 1 u=u 0 0 x 1 1 x 2 1 x 0 1 income effect substitution effect 26