Economics II: Micro
Exercise session 2
1
Fall 2009
VŠE
Deriving demand function
Assume that consumer’s utility function is of Cobb-Douglass form:
(1)
U (x; y) = x y
To solve the consumer’s optimisation problem it is necessary to maximise (1)
subject to her budget constraint:
px x + py y
(2)
m
To solve the problem Lagrange Theorem will be used to rewrite the constrained
optimisation problem into a non-constrained form:
max L (x; y; ) = x y + (m
px x
(3)
py y)
The …rst order (necessary) conditions will result in:
1
x
x y
=
px
(4)
=
py
(5)
m = px x + py y
(6)
y
1
Combining (4) and (5) will result in:
(7)
py y = px x
which, combined with (6) will give:
(m
(8)
px x) = px x
and …nally, after some rearrangements becomes:
m
x=
+ px
(9)
This is the demand function for the good x. When the price of the good x; px ,
is …xed then (9) is the Engel curve for the good x: It is easy to see that this was
an example of homothetic preferences: It is enough to check the income elasticity
to be equal to unity:
"xm =
m @x
=
x @m
m
/
@
m
/ @m
m
( + )p
( + )p
1
=
( + )p
( + )p
=1
Re-writing (9) as:
px =
m
x
(10)
+
gives the Inverse Demand function!
1.1
Quasi-linear preferences
Remark 1 Quasi-linear utilities have the form u (x1 ; x2 ) = x1 + v (x2 )!
Suppose the agent is maximising the following utility function:
p
U (x; y) = x + y
(11)
subject to standard budget constraint (2). Assuming that a rational agent will
spend all her money on purchasing the goods (more rigorous alternative is to set
up Lagrangian function), the optimisation problem will beocome:
max
y
py
p
y+ y
px
m
px
(12)
The …rst order (necessary) condition after rearranagements reads:
px
2py
y=
2
(13)
This is the demand function for the good y. It is independent on the income level,
i.e. the agent is going to consume exactly the same amount of the good y as long
as the prices remain constant. On the other hand the agent is spending all her
‘leftover’money on purchasing good x: From (2) and (13) the demand function is:
x=
m
px
px
4py
(14)
which is of the usual form: x = x (px ; py ; m) :
Q: Are x and y substitutes or compliments?
2
2.1
Exercises
True/False
Claim 1 If the Engel curve for a good is upward sloping, the demand curve for
that good must be downward sloping.
2
TRUE: Upward sloping Engel curve
Slutsky
Normal good
(negative income e¤ect
) downward sloping demand curve
Claim 2 If the demand function is q = 3m
p (m is the income, p is the price), then
the absolute value of the price elasticity of demand decreases as price increases.
@q
q
3m
= 1q 3m
FALSE: The elasticity is: dp @p
= pq
1: Thus has
p =
q =
p2
constant elasticity equal to unity. Note: Any utility function of the form q = Ap"
has constant elasticity equal to ":
Claim 3 An increase in the price of Gi¤en good makes the consumers better o¤.
FALSE: Increase in price of any good makes the consumer poorer and thus
worse o¤. (A graphical representation may be helpful!)
Claim 4 The demand function q = 1000 10p. If the price goes from 10 to 20,
the absolute value of the elasticity of demand increases.
TRUE: The elasticity of demand is: " =
1
20
1
1
1
9 ; "p=20 = 10 1000 200 =
4:
4 >
9
10 pq : "p=10 =
10 100010 100 =
Claim 5 In case of perfect complements, decrease in price will result in negative
total e¤ect equal to the substitution e¤ect.
FALSE: In case of perfect compliments there is no substitution e¤ect, and the
total e¤ect is equal to the income e¤ect.
Claim 6 When all other determinants are held …xed, the demand for a Gi¤en good
always falls when income is increased.
TRUE: To prove the claim we need to show that Gi¤en good is always an
inferior good.
We are going to use the version of Slutsky equation that we had in class and
ilustrated in Figure 3 (Note: the …gure is illustrative and does not explain Gi¤en
good). Thus:
x = xs + xm
Slutsky
x = Xold Xnew
total e¤ect
s
x = Xold Xintm substitution e.
xm = Xintm Xnew
income e.
3
Figure 3
Xold
Xnew
Xintm
BC(po,mo)
BC(pn,mi)
BC(pn,mn)
Total effect
Substitution effect
Income effect
As we can see, the …gure illustrates a case when the price of the good went
down, viz.
p = po pn > 0
and we can rewrite the Slutsky equation as
xs
+
p
x
=
p
xm
p
and check for the signs. We know that substition e¤ect is always negative. We
also know that for the Gi¤en good the total e¤ect is positive. Thus the income
e¤ect should be positive:
xm
sgn
=1
(15)
p
In order to prove the claim we need to show that
sgn
xm
m
=
1
or (same as)
xm
< 0
m
that is the demand falls when income increases. Thus we need to see that
sgn [ m] =
1 sgn [ p]
4
(16)
From the budget constraint we know that when the price goes down, the agent
‘gets’richer, i.e. p = po pn > 0 =) mn mo > 0 as the shift of the budget
constraint is paralel to right. Thus we have
m = mo
(17)
mn < 0
Again from (15) and (17) directly follows (16). Q.E.D.
Claim 7 If the goods are substitutes, then an increase in the price of one of them
will reduce the demand for the other.
False: According to the de…nition!
2.2
Problems
Problem 1 Demand functions for beer is given:
qb = m
30pb + 20pc
where m is the income; pb and pc are the prices of beer and cake, respectively; qb
is the demanded quantity.
1. is beer a substitute or compliment for cake? (A:
@qb
@pc
= 20 > 0 =) substitute)
2. assume income is 100, and cake costs 1, what is the demand function? (A:
qb = 120 30pb )
1
30 qb )
3. write the inverse demand function. (A: pb = 4
4. at what price would 30 beers be bought? (A: pb = 4
1
30 30
= 3)
5. Draw the inverse demand. (Hint: It’s a linear function)
6. Draw the inverse demand when pc = 2: (Hint: It’s parallel to the above, but
higher.)
Problem 2 Suppose the demand function is q = (p + a) ; a > 0;
1. Find the price elasticity of demand.
A:
p @q
q @p
=
p
q
(p + a)
1
=
p
(p+a)
(p + a)
1
=
p
p+a
2. Find the price level for which the elasticity is equal to -1?
A:
p
p+a
=
1:p=
a
+1
5
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