By Virginia Rose,2014-03-02 21:45
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    5 DEMAND

     The consumer’s demand functions:

    x=x(p, p, m) 1112

    x=x(p, p, m) 2212

    How does a consumer’s demand change as his income changes?

    How does demand change when price of a good changes?

    Comparative statics (比较静态) :

    Statics ---- not concerned with any adjustment process involved in moving from one choice to another.

    Comparative ---- to compare two situations: before and after the change in the economic environment.

    Comparison of demands under different prices

    Comparison of demands under different income levels

    Income Offer Curves and Engel Curves

    Hold the prices fixed, and examine only the change in demand due to the income change.

    An increase in income shifts the budget line in a parallel fashion.

    Income offer curve?收入提供曲线?(income consumption curve):

    the bundles of goods that are demanded at the different levels of income.

     Good 2 income offer curve

    O good 1

    Engel curve?恩格尔曲线?: a graph of the demand for one of the

    goods as a function of income, with all prices being held constant.

     Income Engel curve



    O x x good 1 1 1

Some Examples

    Perfect Substitutes

    u(x, x) = x + x 1212

     Good 2

     Income offer curve

    O good 1

Income offer curve: x = 0, if p < p212

    x = 0, if p < p121

Engle curve for good 1: x = m/p, if p < p 1112

    x= 0, if p < p 1 21


     Engel curve (p < p) 12

    O good 1

Perfect Complements

    u(x, x) = min {x, x} 1212

     Good 2

    O Good 1

Income offer curve: x = x12

    mEngle curve for good 1: x1p;p12

    Cobb-Douglas Preferences aa1u(x, x) =, xx1212

    Income offer curve:

    11aaaxxp121= aa(1a)xxp122


    p1a1x=x 21pa2

    am. Engle curve for good 1: x= 1p1

    Normal and Inferior Goods

    Normal good (正常商品): The demand for it would increase when income increases:

    x1 0m

     good 2

     O good 1

    Luxury good?奢侈品?: the demand for a good goes up by a greater proportion than income.

    Necessary good?必需品?: the demand for the good goes up by a lesser proportion than income.

    ----they are both normal goods.

    Inferior good(低档商品): an increase of income results in a reduction in the consumption of it.

    x1 0m

     Good 2

    O good 1

    The Price Offer Curve and the Demand Curve

    Suppose that we decrease the price of good 1 and hold the price

    of good 2 and money income fixed.

    Price offer curve(价格提供曲线): the connection of the bundles demanded and prices for good 1.

     Good 2

    O good 1

    Demand curve: the plot of the demand function x(p, p, m), 112

    holding p and m fixed at some predetermined values. 2


     p 1

     p 1

    O x x x 1 11

    Some Examples

    Perfect Substitutes

    u(x, x) = x + x 1212

     Good 2

     Price offer curve

    O good 1

    Price offer curve: x = m/p x, if x m/p112 22

    x = 0, if x > m/p 212

Demand curve for good 1: x= m/p, if p < p 1112

    0 x m/p, if p = p 1112

    x= 0, if p > p 1 12


     p= p 12

    O m/p x 2

Perfect Complements

    u(x, x) = min {x, x} 1212

     Good 2 Price offer curve

    O Good 1

Price offer curve: x = x (x m/p) 1212

    Demand curve for good 1:

    m x1p;p12

     p 1

    O m/p x 21

Cobb-Douglas Preferences aa1u(x , x) =xx1212


    11aaaxxp121MRS = = aa(1a)xxp122

    apx22we have px= 11(1a)

    Substitute it into the budget line px+ px = m, we have price 1122

    offer curve

    (1a)mx= 2p2

     Good 2

    O good 1

    amDemand curve for good 1: x= . 1p1A Discrete Good Suppose that good 1 is a discrete good.

    Good 2 is a composite good. p = 1 2

    Consumer buy 1 unit when price falls to r : 1

    u(0, m) = u(1, m r). 1

     Good 2

    O Good 1 Reservation price (保留价格) r: the price at which the consumer n

    is just indifferent to consuming n 1 units or consuming n units of

    the good.

    If the price falls to r, he is willing to buy another unit, and r 22


    u(1, m r) = u(2, m 2r). 22

Consider quasilinear utility function

    u(x, x) = v(x) + x, 1212


    v(0) + m = v(1) + (m r)1.

    We can solve for r 1

    r= v(1) v(0). 1


    v(1) + m r= v(2) + m 2r. 2 2

    This expression becomes

    r= v(2) v(1). 2

In general,

    v(n-1) + m (n-1)r= v(n) + m nr. n n

    We have

    r= v(n) v(n-1) n

    rMUpn11We have MRS = = = MUp122


     r 1


     r 4

     1 2 3 4 5 quantity

    By assumption of diminishing MRS, the sequence of reservation prices must decrease:

    r> r> r 1 2 3

    Ordinary Goods and Giffen Goods

     good 2

     O good 1

    Ordinary good(普通商品): the quantity demanded of good 1 should increase when its price decreases.

    x1 0p1

    Giffen good?吉芬商品?: decrease in the price of good 1 leads to

    a reduction in the demand for it.

    x1 0p1

     Good 2

    O good 1

    Substitutes and Complements

The demand function for good 1

    x(p, p, m) 112

    xx21Substitute: good 1 is a substitute for good 2 if and 00pp22

     Good 2

    O Good 1

    x1Complement: good 1 is a complement to good 2 if and 0p2

    x2 0p2

     Good 2

    O Good 1


    The Substitution Effect

    Given (p, p, m), Consumer chooses the best bundle X satisfying 12

    m = px+px1122

    Suppose p decreases to p, consumer chooses Z satisfying 11

    m = px+px1122

     Good 2

     x X Z 2


    O x good 1 1

    We break the price movement into two steps:

    ----first let the relative prices change and adjust money income so as to hold purchasing power constant,

    ----then let purchasing power adjust while holding the relative

    prices constant.

    The pivoted line:

    m = px+px1122

    ----the same slope as the new budget line.

    ----the original bundle of goods is just affordable at the new

    pivoted line.

    m : the amount of money income that will just make the original

    consumption bundle affordable.

    Subtracting the second equation from the first one at (x, x) 12



    m = m + x?p11

    Y : the optimal purchase on the pivoted budget line.

    Effects of the price changes:

    ----the rate at which you can exchange one good for another changes the substitution effect(替代效应).

     sThe substitution effect on good 1, : the movement from X to x,1

    Y (compensated demand). s xx(p,m)x(p,m).11111

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