Question

2. Suppose an individual has quasi-linear preferences goods forxandydescribed by U (x, y) = 4√x+y. Recall this means that MUx=2√x and MU y= 1. Suppose that Px=1,Py= 2 and I= 24.

(a) Plot the income consumption curve associated with a change in income from I= 24 to I= 34.(To do this problem, you would need to calculate the optimal consumption associated with eachof these incomes. Or you could look at the solutions to HW #3.)

(b) What shape would the Engel curve have?

(c) Now we are back to considering only an income of I= 24. What utility level is associated with that at the old prices.

(d) Suppose the price of x increases to 4. Find the optimal level of x and y at the old utility level but the new prices.

(e) Now find the optimal level of x and y at the new price level.

(f) Having done this, calculate for good x the substitution effect, the income effect, and the total effect of the price change.

Answer 1

2. Quasi-Linear preferences functions are linear in one variable and non-linear in other. In our case, it is  U (x, y) = 4√x+y. So it is linear in y-good and the and non-linear in x. The property of such kind of function is that the Marginal rate of Substituition (MRS) (MUx/MUy) depend on good-x only.

(a)The income consumption curve (ICC) associated with a change in income from I= 24 to I= 34.

The ICC is drawn by joining all the points of equilibrium at different levels of income. At equilibrium, we have

(MUx/MUy) = (Px/Py)

2√x /1 = 1/2

Soving, √x = 1/4 or x = 1/16

We see that at equilibrium quantity of x stays the same (it is independent of level of income). To find the equilibrium quantity of y, we put the value of x in budget line P_x*x+P_y*y=I i.e

1*1/4 + 2*y=24

1/4+2y=24

Solving it for y, y = 11.8 = 12

At new level of income I = 34, 1*1/4 + 2*y=34

Solving it for y, y=16.8 = 17

So, although x is independent of income level, y increases as the income is increased. Hence the ICC will have a vertical shape. It is shown below in the image:

(b). Engel curve shows different level of a commodity that is purchased at different level of income (but constant prices). It is drawn in Income-quantity space and is derived from ICC. Engel curve for commodity-x is shown below:

(c). Utility at income leve; I=24. This is obtained by putting equilibrium values of x and y at I=24 in utility function. At I=24, we obtained  √x = 1/4 and   y = 11.8 = 12, So utility,

  U (x, y) = 4√x+y.

U (x, y) = 4*1/4+12 = 13

(d). Price of x increases to 4, the new equilibrium condition will be,

(MUx/MUy) = (New Px/Py)

   2√x /1 = 4/2

2√x = 2

√x =1 or x= 1

To find y, Let's put the value of equilibrium x in old utilty function, 4√x+y = 13

4*1+y=13

y=9

(e).To find the optimal level of x and y at the new price level, we put the equilibrium value of x obtained in previous part in budget line 4x+2y=24

We get, 4*1+2y =24

2y = 20 , so y=10

(f). Price effect = Quantity of x at new price-quantity of x at old price

= 1-1/16 = 15/16 = 0.93

Also, Price effect = substitution effect + the income effect

We saw in the first part that income effect in case of quasi-linear good is 0. So total price effect will be due to substituition effect.

Substitution effect = 0.93

Income effect = 0