Solution to 201 Assessed Test No. 1 29th October 2001
(a)
The problem asks us to consider the case where consumers’ tastes for X and Y are given by,
U  X 1/ 2 Y 1/ 2
.
The consumers’ budget constraint is, Px X  Py Y  M .
To solve the problem we set up the Lagrange Equation
L

X 1/ 2 Y 1/ 2   M  Px X  Py Y

The First Order Conditions for this problem involves taking the partial derivative of this
equation with respect to the three variables we are interested in X, Y and  .
Applying the differentiation rules to the Lagrangian we get:
1)
2)
3)
1 (1/ 2 )  1 1/ 2
1
X
Y   Px  X 1/ 2 Y 1/ 2   Py
2
2
1
1
Ly  X 1/ 2 Y (1/ 2 )  1   Py  X 1/ 2 Y  1/ 2   Py
2
2
L   M  Px X  PyY
Lx 
Alternatively, we can write the three equations (1 - 3) above more simply as
1 U
L 
 P
x 2 X
x
1U
L 
 P
y 2 Y
y
L  M  Px X  PyY
In order to find a maximum we need to set these three equations equal to zero, and then solve
for X, Y and  . [Recall the conditions for a maximum from Econ 106D last year]
1.1)
2.1)
3.1)
1 U
 P  0
x
2 X
1U
 P 0
y
2 Y
M  Px X  PyY  0
[15 Marks]
1
(b)
Equations (1.1), (2.1) and (3.1) above are three equations in three unknowns X, Y and  , and
thus we can use the three equations to solve for each of the variables we want. [Note that the
values for Px, Py and M will be known and we can substitute these in once we have found the
general solution to the problem]
So we solve the equations for  , Y and X in turn by substituting between equations
By re-arranging equation 1 we can get an expression for 
 
4)
1 U
.
2 Px X
We can substitute this into equation 2.1 for  to get
1U 1 U

P
5)
2 Y 2 Px X y
In 5) we can cancel the U’s on both sides and the 1/2’s on each side and we are left with
1 Py 1

Y Px X

Px
Y

Py
X
By re-arranging this expression we can get X in terms of Y ( and prices which will be known)
Py
6)
[5 Marks]
 X  Y
Px
(c)
If we take the total derivative of the utility function,
dU 
U  X 1/ 2 Y 1/ 2
, we get
U
U
1U
1U
dX 
dY  MU x dX  MU y dY 
dX 
dY
 X
Y
2X
2Y
If we are considering changing X and Y in such a way that we are staying on the same
indifference curve then dU=0. Therefore,
U
U
dX 
dY  0
 X
Y
U
U

dX  
dY
 X
Y
dU  0 
2
U
 X dY


 U dX
Y
U
1U
dY
MU x
Y
 X
2X





U
dX
MU y 1 U
X
Y
2 Y
Px
represents the slope of
Py
the budget constraint. Thus, the economic interpretation of the condition in equation(5) derived
in section (b) above, is that the slope of the indifference curve must equal the slope of the
budget constraint in equilibrium or on other words the rate at which a consumer is willing to
substitute Y for X must be the same as the rate the market is prepared to exchange Y for X in
equilibrium.
[5 Marks]
In other words Y/X is the slope of the utility function. Similarly
(d)
We can now substitute (6) back into (3) and get :
M  Px X  PyY
(3)
 Py 
 M  Px  Y   PyY
 Px 

Substituting in (6)
PyY  PyY  M
Cancelling the Px’s
 2PyY  M
7)
Y
and re-arranging
1M
2 Py
This final expression is the demand curve for Y which we wished to derive, Y = Y(Py, Px ,M).
Note that in this special case Px does not enter the expression in equation 7. In other words the
demand for y depends only on its own price and income.
Next we can substitute (7) back into (3) in order to get the demand curve for X.
Px X  PyY  M
(3)
3
1 M 
 Px X  Py 
 M
 2 Py 
1
 Px X  M  M
2

8)
Px X  M 
X
Substituting in equation (7)
Cancelling out the Py’s
1
1
M M
2
2
Re-arranging
1M
2 Px
This expression is the demand function for the good X, X = X(Px, Py ,M). Note that once again,
in this special case Py does not enter the expression. In other words the demand for X depends
only on its own price and income.
[10 Marks]
(e)
Part e requires that we work out the share of x in expenditure , its own-price elasticity and
income elasticity for X
Share of X in Expenditure
The share of X in expenditure is found by dividing total expenditure on X (PxX) by Income M.
Sx 
Px X
M
But we have found an expression for the demand for X in part d, so
Px
Sx 
1M
1
M
2 Px
1
2


M
M
2
So we spend exactly half our income on X in every period regardless of its price. The price
merely determines how much X we can buy with 50% of our income.
[3 Marks]
Own-Price Elasticity
This is the relative change in quantity demanded divided by the relative change in price.
9)
Ex P
x
dX
P dX
 X  x
dPx
X dPx
Px
For this we need to find the derivative of the demand curve with respect to Px :
4
8)
X 
1 M
2P
x
1 M 
dX
d 


P
dPx dPx  2 x 
Differentiating
10)
dX
1
1 M
  M Px 2  
.
dPx
2
2 Px2
->
Ex P  
x
Px
X
1 M 

2
 2 Px 
Substituting in equation (10)
1 M 


 1 M   2 Px2 


 2 Px 
1 M 


Px  2 Px 

Px  1 M 


 2 Px 
Px
->
E x Px 
->
E xPx
->
E xPx  1
Substituting equation (8) in for X
Rearranging
Cancelling out common terms
[4 Marks]
Income Elasticity
For Income elasticity we need the change in demand for X as Income changes.
This is
so substituting for
dX
1 1

dM 2 Px
dX
and X
dM
 x
in the usual definition of income elasticity yields:
M dX
M 1 1 


 1
X dM  1 M   2 Px 


 2 Px 
[3 Marks]
Total marks for section e [10 Marks]
5
(f)
In lectures we noted that the Slutsky equation implied that the Pure Substitution Effect
E xP  S x x had to be non-positive. Given the results derived in section (e) above this is now
x
trivial to check (so trivial it isn't really worth 8 marks!!).
If we substitute the results we have just found into this equation we observe that
-1+ 0.5 (1) = -1 +0.5= -0.5.
and thus this demand function satisfies the Slutsky Condition.
[8 Marks]
Note that here
The share of x in total expenditure is 
Px X
P 1 M  1
 x

M
M  2 Px  2
Similarly the share of y in total expenditure is =1/2 and is constant. This is a special feature of
this utility function and we would not expect it to hold in general. The Cobb-Douglas utility
function is a member of a special class of functions with this property. In general, we would
expect that the amount spent on a good would vary as income rose. For example, Engel showed
that the share of expenditure on food falls as income rises.
There is a simple modification we could make to the Cobb-Douglas function.
U  ( X  A) 1/ 2(Y  B)1/ 2
which says that an agent has to have a basic amount of X (at least A) and Y (at least B) in order
to survive. Now as long as PxA and PyB are different the expenditure shares will no be constant
as Income rises. This is called a Stone-Geary Utility Function after Richard Stone the British
Nobel Prize Winner and R.C. Geary, a famous Irish Statistician/Economist. We can think of the
expenditure on A and B as expenditure on essentials. In this case the share of income (I leave it
to you to work out the details):

1 1  Px A  Py B 
 

2 2
M

and so the share of x rises or falls as income rises depending on the relative share of A and B in
the essential bundle that must be purchased.
6
(g)
The demands for X and Y are given by:
X
1 M 1 40

 10
2 Px 2 2
and utility
and
Y
1 M 1 40

 20
2 Py 2 1
1/ 2
1/ 2
1/ 2
U  X 1/ 2 Y 1/ 2  10  20   200  10 2  1414
. .
[4 Marks]
(h) To solve this part we simply substitute the new prices (Px=4, Py=1) and the old bundle
back into the budget constraint
Px1 X  PyY  4. X  1.Y  4 .10  1.20  M1 . Do the new level of income required to
purchase the old bundle is 60.
[3 Marks]
(i)
To answer this part of the question we need 2 pieces of information. We need to know the
demand for the good X at the new prices when income equals 40 (the original level of income),
and we need to calculate the demand for X when the agent has been compensated for the
increase in the price of X by giving him/her sufficient income such that they can purchase the
original consumption bundle.
The demand for X after its price has risen to 4 is
X
1 M 1 40

5
2 P1 2 4
x
This is the new Marshallian (or ordinary level of) demand
In part (h) above we calculated the income required to purchase the old bundle at the new set of
prices. This is precisely the income we need to calculate the Slutsky income effect, call it Ms.
So if we substitute this level of income and the new prices into our Cobb-Douglas demand
function then we can calculate the demand for X on the Slutsky demand curve.
1 M s 1 60

 7.5
2 P1 2 4
x
Hence the total effect of the price increase is the fall in demand from 10 to 5
while the Slutsky substitution effect is the fall from 10 to 7.5
and the (Slutsky) income effect is the fall from 7.5 to 5
X 
[10 Marks]
7
(j)
In this part of the question we are essentially deriving the Hicksian compensation measures
corresponding to the Slutsky measures in (i) above. The quick way to do is shown in the CVEV
exercise on the network. The more technically correct way to do it is important to learn,
however, as it has other applications which you will see both later in this course and in
ECON302 next year.
As the question suggests we want to find the income required to enable you to reach the
original level of utility (which we saw in part g was 14.4). That is we need to figure out the X
and Y bundle which gives you U  X 1/ 2 Y 1/ 2  14.4 and therefore the income, MH, required
to buy it
M
H
 Px1 X  Py Y

As the question suggests you need to write the Lagrangian
L
Px X  Py Y   U  X 1/ 2 Y 1/ 2 
The First Order Conditions for this problem involves taking the partial derivative of this
equation with respect to the three variables we are interested in X, Y and  .
Applying the differentiation rules to the lagrangian we get:
1 (1/ 2) 1 1/ 2
1
Lx  Px   X
Y  Px   X 1/ 2 Y 1/ 2
2
2
1 1/ 2 ( 1/ 2 )  1
1
Ly  Py   X Y
 Py   X 1/ 2 Y 1/ 2
2
2
L   U  X 1/ 2 Y 1/ 2
Alternatively, we can write these three equations more simply as
1 U
L  P 
0
j.1)
x
x
2 X
1U
j.2)
L  P 
0
y
y
2 Y
j.3)
L
 U  X 1 / 2Y 1 / 2  0
and setting them equal to zero since we are trying to find a minimum. [Recall the conditions for
a minimum from Econ 106D last year.]
Now equations j.1), j.2) and j.3) are three equations in three unknowns X, Y and  and thus we
can use the three equations to solve for each of the variables we want. [Note that once again we
will know the values for Px, Py and U (which we have already seen must equal 14.4) once we
have found the general solution to the problem, so we are treating these as known]
So we solve the equations for  , Y and X in turn by substituting between equations
8
By re-arranging equation j.1 we can get an expression for 
j.4)
 
2 Px X
.
1 U
We can substitute this into equation j.2 for  to get
2 Px X 1 U
 Py
j.5)
1 U 2Y
In j.5) we can cancel the U’s and the 2’s and we are left with
Px Y

Py
X
By re-arranging this expression we can get Y in terms of X ( and prices which will be known)
P
j.6)
 Y x X
Py
This is precisely the same condition that we obtained in part (a). But the next step is a bit
different. Next we substitute j.6 into the utility function given by j.3.
U  X 1/ 2Y 1/ 2
P 
 X 1/ 2  x X 
 Py 
1/ 2
P
 X 1/ 2 X 1/ 2  x 
 Py 
1/ 2
P
 X  x 
 Py 
1/ 2
Re-arranging this last expression gives us X as a function of prices and the person’s level of
utility
1/ 2
 Py 
X   U
 Px 
Similarly, by inverting j.6 and substituting for Y we get:
U  X Y
1/ 2
1/ 2
 Py 
  Y
 Px 
1/ 2
Y
1/ 2
 Py 
 
 Px 
1/ 2
1/ 2
Y Y
1/ 2
 Py 
 
 Px 
1/ 2
Y
and once again re-arranging this expression gives us Y as a function of prices and the person’s
level of utility
1/ 2
P
Y   x  U
 Py 
[15 Marks]
9
(k) The demand functions above tell us the demand for X
and Y given different utility levels,
or in other words they are Hicksian utility functions.
[5 Marks]
(l) The cost of the bundle we need to buy to get a given level of utility, say U0 =14.4 can now
be found by substituting these demands into the original budget constraint.
 P  1/ 2 
 Py  1/ 2 
M H  Px X  PyY  Px   U   Py  x  U 
 Py 

 Px 




 
 
M H  Px1/ 2 Py1/ 2U  Px1/ 2 Py1/ 2U  2 Px1/ 2 Py1/ 2U

Substituting in the relevant values for Px, Py, and U we get:


M H  2 Px1/ 2 Py1/ 2U  2 41/ 211/ 2 (14.4)  2[2(14.4)]  57.6
[5 Marks]
(m)
The income required to purchase a bundle on the original indifference curve (utility level) at
the new prices is 57.6. This represents the Hicksian income compensation effect and it is less
than the income required for the Slutsky income effect- the cost of purchasing the original
bundle at the new set of prices which we found in part (h) was 60. Thus the difference in cost is
the difference between the Hicksian and the Slutsky income effects. We can see on the diagram
below that the Hicksian effect must be less when the price of X has risen. The dotted line
represents the income that must be returned to the consumer in order to reach the original
indifference curve whereas the dotted line is the income s/he must be given in order to enable
them to purchase the original bundle.
10
Y0
U0
X0
[5 Marks]
11