Problem 2 Suppose UXY min2XAY Suppose Initially Pr a Compute

Problem 2 Suppose U(X,Y)= min(2X,AY). Suppose Initially Pr (a) Compute the optimal bundle and Utility of the consumer from the bun- Now suppose the price of good X changes to Px while I, Py remain (b) Compute the new optimal bundle and Utility of the consumer from the (c) At new prices, what income level wil give consumer the same level of (d)Decompose the change in demand for good X and good Y due to the price dle same bundle. Utility as in Part(a). change into substitution effect and Income effect. Comment on the sign of the two effects. (5,5,5,15 Points)

Solution

U(X,Y)= min(2X,4Y) Px=2 Py=1 I=10

a)This is the case of perfect complements ie the two goods are produced in a fixed proportion , so if we have more of one good its of no use unless we have the required proportion of other good

The budget constraint is given by

PxX+PyY=I

With 2X=4Y (fixed proportion )

X= 2Y

putting the values in budget constraint

2(X)+1(Y)=10

2X+X/2= 10

5X/2=10

X=4

Y= X/2= 2

optimal bundle is (4,2)

the utility function is

U(X,Y)= min(2X,4Y)

putting the values of X and Y in it we get

U= min(8,8)= 8

b) If Px changes to 1, we get

PxX+PyY=I

With 2X=4Y (fixed proportion )

X= 2Y

putting the values in budget constraint

X(1)+X/2=10

3X/2=10

X=20/3

amd Y= 10/3

New optimal bundle is (20/3,10/3)

putting in Utility function

U= min(2*20/3, 4*10/3)

= min(40/3,40/3) = 40/3= 13.34

c) finding the new income level with the same level of utility

As we know that the old utility was 8 which was given by the bundle (4,2)

now that we have new price of X putting it with the optimal bundle found earlier and the original price of Y we can find the new income

Px\'X+PyY=I\'

1(4)+1(2)= 6

d) We had the orginal bundle before the price change as done in part a)

which is (X0,Y0)=(4,2)

With new prices and original utility we have

I\'= 4(1)+2(1)= 6

Now we find the cosnumption bundle with new income and new price levels

(X1,Y1)=[ I\'/(Px\'+Py\'/2), I\'/(2Px\',Py\') ] ( How consumption is impacted by changing the prices and income )

=[ 6/1.5, 6/3]

= [ 4,2 ]

now with the original income and new prices ( ie the change in [purchasing power due to change in prices )

(X2,Y2)= [ I/(Px\'+Py\'/2), I/(2Px\',Py\')]

= [10/1.5, 10/ 3]

= (6.7,3.4) ~ (7,3)

So substitution effect which is how consumption is impacted by changing the prices and income is given by

X1-X0= 4-4= 0 ie we have no substitution effect

in Perfect complements we have no substitution effect since the goods are consumed in fixed proportion and together ie they are not substitutes of each other .

So income effect is the total effect which is

X2-X0= 6.7-4= 2.7~3

y2-Y0= 3.3-2= 1.3~1

Income effect is positive because as the price reduces the number of units purchased will increase given the income is fixed