Marshallian Demand Functions : Some Examples Marshallian Demand Functions : Some Examples

i Fixed Coefficients

If the utility function is

u(x) = min
(ax1,bx2)
then either u/x1 = 0 ( if ax1 > bx2 ), or u/x2 = 0 ( if ax1 < bx2 ), or the utility function is not differentiable ( if ax1 = bx2 ).

If the price of each good is positive, then the consumer's optimal choice of consumption bundle x*(p1,p2,M) must be located at the kink in the L-shaped indifference curve, at which ax1 = bx2. [Why? If ax1 > bx2, for example, then the first-order conditions would be u1 = 0 = lp1 and u2 = b = lp2 : the first condition says l = 0 and the second condition says l > 0. Or, more intuitively, why waste money on good 1, when added consumption of the good does not make you any better off? ]

So the Marshallian demands satisfy the condition ax1 = bx2, and the budget line equation p1x1 + p2x2 = M. The first condition says that

x2 = a
b
 x1
Substituting this definition in the equation of the budget line,
p1x1 + p2 a
b
 x2 = M
or
x1 = b
p1 b + p2 a
 M
which is the Marshallian demand function for good # 1. Therefore the Marshallian demand function for good #2 is
x2 = a
p1b + p2a
 M

ii Perfect Substitutes

If

u(x) = a1x1 + a2x2 + ¼anxn
then
u
xi
 = ai
for each good i. The marginal rate of substitution between any two goods i and j is ai/aj, which is a constant, independent of the quantities consumed of the goods. The indifference curves between any two goods are straight lines. Mathematically, the first-order conditions
u
x1
 = a1 = lp1
and
u
x2
 = a2 = lp2
could both hold only if a1/a2 = p1/p2, which would happen by coincidence. Usually, the consumer will choose to be at a corner solution, spending all her money on the good i for which ai/pi is highest. That is, if
a1/p1 > a2/p2 > ¼an/xn
for example, then she will choose
x1 = M
p1
and xi = 0 for every i > 1. Those would be her Marshallian demands. Only if there was a tie, so that, for example
a1
p1
 = a2
p2
 > a3
p3
 > ¼ an
pn
would she choose to consume positive quantities of more than one good. In this case, the slope of her indifference curve between x-1 and x2 equals the slope of her budget line. Her Marshallian demands are not unique : any (x1,x2,x3,¼,xn) with p1x1 + p2x2 = M, and x3 = x4 = ¼ = xn = 0 would be tied for most preferred among the bundles which she could afford.

iii Quasi-Linear Preferences

(iiia)     u(x1,x2,x3) = x1 + 2   __
Öx2
 
 + lnx3

In this case, the three first-order conditions are

1 = lp1      (1)
1
  __
Öx2
 
= lp2      (2)
1
x3
 = lp3      (3)

Equation (1) can be used to substitute for l in equations (2) and (3) :

1
Öx2
 = p2
p1
      (2¢)
1
x3
 = p3
p1
      (3¢)
which can be re-arranged into
x2 = ( p1
p2
 )2      (2¢¢)
x3 = p1
p3
      (3¢¢)
which are the Marshallian demand functions for goods #2 and #3.

Since

x1 = M - p2x2 - p3x3
p1
therefore
x1 = M
p1
 - p1
p2
 - 1
is the Marshallian demand function for good #1. This expression is only correct if the person's income M is high enough so that M > (p1)2/p2 - p1 ; otherwise x1 would be negative. [ left to the reader : what would Marshallian demands be if income M were lower than this? ]

In this example, the quantities demanded of goods 2 and 3 were independent of the person's income M. Increases in income are all spent on good #1. This property holds whenever a person has quasi-linear preferences. If

u(x) = x1 + f(x2,x3,¼,xn)
then the first-order conditions to the consumer's utility maximization problem are
1 = lp1
f
xi
 = lpi     i = 2,¼,n
If the first equation is used to substitute 1/p1 for l in the remaining n-1 equations, then the first-order conditions for x2,x3,¼,xn are n-1 equations in the n-1 unknowns x2,x3,¼,xn. That means they can be solved without reference to the budget condition, or to the income level M. This property holds only if preferences are quasi-linear.

(iiib)     u(x1,x-2,x3) = x1 + lnx2 +ln(x2+x3)
Here the first-order condition on x1 again implies that
l = 1
p1
so that the other two first-order conditions can be written
1
x2
 + 1
x2 + x3
 = p2
p1
      (2)
1
x2 + x3
 = p3
p1
      (3)
Substitution of equation (3) into equation (2) yields
1
x2
 = p2
p1
 - p3
p1
      (2¢)
or
x2 = p1
p2-p3
      (2¢¢)
which is the Marshallian demand function for good #2. Since equation (3) can be written
x2 + x3 = p1
p3
      (3¢)
then equation (2¢¢) implies that
x3 = p1(p2-2p3)
p3(p2-p3)
      (3¢¢)
which is the Marshallian demand function for good #3. Substitution into the budget constraint then implies
x1 = M
p1
 -2
is the Marshallian demand function for good #1. These demand functions are only valid when M > 2p1 and when p2 > 2p3. [  left to the reader : what happens when these inequalities are not satisfied? ]

Here quantity demanded of good #3 depends on all three prices : but quasi-linearity implies that quantity demanded of good #2 and of good #3 is independent of income M.

iv Cobb-Douglas Preferences

If

u(x1,x2,¼,xn) = x1a1x2a2 ¼xnan
then it is simplest to use the monotonic transformation U(x) = ln[u(x)] to get
U(x) = a1lnx1 + a2 lnx2 + ¼an lnxn
First-order conditions are now
ai
xi
 = lpi     i = 1,¼,n
These equations can then be written
pi xi = ai
l
     i = 1,¼,n
From the budget constraint
p1x-1 + p2x2 + ¼pn xn = a1 + a2 + ¼an
l
 = M
so that
1
l
 = M
a1 + a2 + ¼an
Substituting back in the original first-order conditions,
xi = ai
a1 + a2 + ¼+ an
 M
pi
     i = 1,2,¼,n
which are the Marshallian demand functions in this case. With Cobb-Douglas preferences, quantity demanded of each good does depend on income M : in fact quantity demanded of each good is proportional to income. But quantity demanded of each good depends only on the price of that good, and not on the prices of any of the other goods. In this case, the proportion of her income that the person spends on good i, [(pixi)/ M] equals
ai
a1 + a2 + ¼an
which is a constant - independent of income and of all the prices. If a person had Cobb-Douglas preferences, then the proportion of her income which she spent on food, or on housing, would depend only on her tastes, and would not change with her income, or with the prices of food or housing.

v CES Preferences

Done in the textbook.

vi Stone-Geary Preferences

For simplicity, I choose the utility function

U(x) = b1 ln(x1-s1) + b2ln(x2-b2)+ ¼+bnln(xn-sn)
with
b1 + b2 + ¼+ bn = 1
The first-order conditions are
bi
xi -si
 = lpi     i = 1,2,¼,n
or
pi xi = pi si + bi
l
Using the budget constraint p1x1 + p2x2 + ¼pnxn = M and the fact that the bi's sum to 1, therefore
M = n
å
 j = 1 
 pj sj + 1
l
so that
1
l
 = M - n
å
 j = 1 
 pj sj
which means that
xi = si + bi
pi
 [M - n
å
 j = 1 
 pj sj]
is the Marshallian demand function for good #i. These demand functions are only valid if the person has enough income to pay for her ``required'' consumption levels si : if M ³ åj = 1n pjsj.

File translated from TEX by TTH, version 2.00.
On 21 Jan 2002, 13:53.