Question

(b) You consume two goods, good x and good y. These goods sell at prices px = 1 and py = 1, respectively. Your preferences are represented by the following utility function: U(x; y) = x + ln(y): You have an income of m = 100. How many units of x and y will you buy and what will is your utility? If px increases from $1 to $2; figure out the compensating variation (CV) associated with price change.

(c) If instead your utility is U(x; y) = ln(x) + y; figure out the compensating variation (CV) as px increases from $1 to $2:

(d) Are the compensating variations the same for both of the above utility functions? Explain your answer rigorously.

Answer 1

(b) According to the information, you consume good X and good Y. Now, given that Px=$1 and Py=$1.

Utility function is given by U(X,Y)=X+ln(Y).

You have an income of m=$100.

Now, to find out the consumption bundle of (X,Y), we need to maximize the utility subject to budget constraint.

Budget Constraint: Px.X+Py.Y=m

or, X+Y=100........(1)

Hence, Max {X+ln(Y)} subject to X+Y=100.

We will equate the MRS with the price ratio.

MRS=MUx/MUy

Here, MUx=dU/dX=1

and,. MUy=dU/dY=1/Y.

Hence, MRS=1/(1/Y)=Y.

Hence, MRS=Px/Py

or, Y=Px/Py.........(2)

or, Y=1/1

or, Y*=1.

And, from the budget constraint

X+Y=m => X=m - Y => X=m - Px/Py.........(3)

or, X=100 - 1

Hence, X*=99.

Hence, you will buy 99 units of X and 1 unit of Y.

Now, U(X,Y)=X+ln(Y)

or, U(99,1)=99+ln(1)

or, U*=99 {as ln(1)=0}

Hence, your utility is 99.

Now, Px increases from $1 to $2. We have to find out the Compensating Variation (CV).

The CV is the extra income needed to maintain the previous utility level(i.e. 99) at the new price ratio.

From the utility function, we will substitute the value of X by (m-Y) as X+Y=m.

Hence, U(X,Y)=(m-Y)+ln(Y)........(4)

Now, from equations (2) we put the value of Y here in (4)

Hence, U(X,Y) = (m - Px/Py)+ln(Px/Py)........(5)

Now, Px=$2 here and Py=1 and U(X,Y)=99. We will get the value of m for which you will maintain your previous utility level.

Now, from (5)

99 = m - (2/1) + ln(2/1)

or, m = 99+2-ln(2)

or, m' = 100.30

Hence, the compensating variation is given by,

∆m = m'-m = 100.30-100 = 0.3

The compensating variation(CV) is 0.3.

(c) & (d) Nowz if the utility function is given by U(X,Y) = ln(X)+Y

Then, MUx=1/X and MUy=1

Hence, MRS=1/X

Equating MRS with price ratio,

1/X=Px/Py

or, X=Py/Px..........(6)

or, X**=1(as Px=1 and Py=1 initially)

and Y**=m-X=m-Py/Px.........(7)

or, Y**=100-1=99.

Hence, U(X,Y)=ln(X)+Y

or, U**= ln(1)+99=99

Hence, the new utility function gives you U**=99.

Now, if Px increases from $1 to $2, then we have to find out your compensating variation or CV.

Now, from U(X,Y)= ln(X)+Y

we can write, U(X,Y)= ln(Py/Px)+(m-Py/Px)......(8) {from equations (6) and (7)}

Hence, here, Px=2 and Py=1 and U(X,Y)=99. We need to find m for which the previous utility level is maintained.

99 = ln(1/2) + m - 1/2

or, m'' = 99 + 0.5 - ln(0.5)

or, m'' = 100.20(Approx)

Hence, compensating variation(CV) is given by

∆m = m'' - m = 100.20 - 100 = 0.2.

The compensating variation is not same for this utility function. The CV is 0.2 in this case.

The utility functions are not same in (a) and (b). In (a) your utility varies with Y and in (b) your utility varies with X. Hence, due to change in price of X, the effect on the utility functions will not be same, as we can see from equations (5) and (8). Thus you will require different CV for obtaining the same utility level. Thus CV s are not same in the two cases.

Hope the solutions are clear to you my friend.