1. Drawing the demand functions and supply functions and &ning the equi
librium price and quantity by solving the two equations together
Q
D
=
a
bP;Q
S
=
c
+
dP:
2. Finding the price elasticity of demand and supply
dQ
D
dP
P
Q
D
=
bP
Q
D
;
dQ
S
dP
P
Q
S
=
dP
Q
S
3. Fitting linear equations to demand data:
demand elasticity (at equilibrium):
&
0
:
5
equilibrium price:
P
= $2
equilibrium quantity:
Q
= 12
million metric tons per year
demand curve:
Q
=
a
bP
Finding
a;b
&
0
:
5 =
b
±
P
Q
=
b
±
2
12
;
hence
b
= 3
:
Substitute this value and values of
P
;Q
into the demand equation
Q
=
a
bP
;
we get the value for
a
= 18
:
Therefore demand is given by
Q
= 18
3
P
4. Drawing indi/erence curves for di/erent utility functions through a point
(
x
0
;y
0
)
u
(
x;y
) =
ax
+
by;u
(
x;y
) =
x
a
y
b
;u
(
x;y
) =
a
ln
x
+
b
ln
y;u
(
x;y
) = min(
ax;by
)
;u
(
x;y
) =
ln
x
+
y
5. Drawing a budget constraint
p
x
x
+
p
y
y
=
I:
6. Finding optimal consumption by solving
MRS
=
p
x
p
y
;p
x
x
+
p
y
y
=
I:
(This applies to CobbDouglas, logarithm and quasilinear case with interior
solutions)
(a) CobbDouglas case:
u
(
x;y
) =
x
2
y
MRS
=
2
xy
x
2
=
2
y
x
and more generally when
u
(
x;y
) =
x
a
y
b
MRS
=
ax
a
±
1
y
b
bx
a
y
b
±
1
=
ay
bx
1
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View Full DocumentTherefore, this
MRS
depends only on the ratio
y
x
:
This makes the CobbDouglas
functions popular with economists. A useful formula to remember
x
=
a
a
+
b
I
p
x
;y
=
b
a
+
b
I
p
y
:
Examples:
(a1) suppose
u
(
x;y
) =
xy;
and
p
x
= 2
;p
y
= 3
;I
= 10
:
optimal consumption choice. Set
y
x
=
2
3
Combine with the budget equation
2
x
+ 3
y
= 10
The two equations are solved simultaneously, and we get a unique solution. For
instance, substitute
y
=
2
3
x
into the budget equation, we get
2
x
+ 3
2
3
x
= 10
and we get
x
= 2
:
5
;y
=
5
3
:
Therefore the optimal consumption choice is the
bundle
(2
:
5
;
5
3
)
:
(a2) Finding the Engel curve: Suppose
u
(
x;y
) =
xy;
and
p
x
= 2
;p
y
= 3
;
let income be denoted by
I:
From the optimal consumption, we get the two
equations
y
x
=
2
3
;
2
x
+ 3
y
=
I
Substitute
y
=
2
3
x
into the budget equation, we have
2
x
+ 2
x
=
I
hence
x
=
I
4
;y
=
I
6
:
In other words, half of the income is spent on
X
to purchase
I
4
quantities of the good, and half of the income is spent on
Y
to purchase
I
6
quantities of the good.
The two solutions
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 Spring '07
 Cheng
 Price Elasticity, Supply And Demand, optimal consumption

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