Question

3. Carolina loves one ice cream cone (c) as much as one bowl of frozen yogurt (y). Her income is $400.

   1. (a) Find her Marshallian demand for ice cream and frozen yogurt. Find her Hicksian demand for ice cream and frozen yogurt. What is the quantity demanded and utility level when the price of ice cream, p_c, is $4 and the price of frozen yogurt, p_y, is $8.

   2. (b) Still assume the price of ice cream (p_c) is $4 and the price of frozen yogurt (p_y) is $8. Compute the total effect, income effect and substitution effect on ice cream when there is a small change in the price of ice cream.

   3. (c) Now instead of small change, the price of ice cream rises dramatically from $4 to $10. What is the total effect, income effect and substitution effect of the price change on frozen yogurt?

Answer 1

A) preferences are perfect substitutes

So U(c,y) = c + y

So marshallian demand functions

As BC: c*Pc + y*Py= M

M is income = 400

So at eqm ,

MRS_{​​​​c,y}= Pc/Py

As MRS = MUc/MUy = 1

So if MRS > Pc/Py, then only C is consumed

(1> Pc/Py)

If MRS < Pc/Py , only y is consumed

If MRS = Pc/Py, any combination of two goods can be consumed

So for c:

C* = M/Pc, if Pc < Py

= [ 0, M/Pc] , if Pc= Py

= 0 , if Pc > Py

similarly for Y

y* = M/Py, if Py< Pc

= [ 0, M/Py], if Py = Pc

= 0, if Py > Pc

Hicksian Demand functions

Now indirect utility function

V = M/Pc , if Pc< Py

= M/Py, if Pc > Py

Expenditure functions

e = u*Pc, if Pc < Py

= u*Pyar, if Pc> Py

So Hicksian Demand curve

As derivatives of expenditure function wrt price

( C^{​​​​​h}, Y^{​​​​​​h} ) = ( u, 0), Pc < Py

= (0, u) , Pc > Py

b) now as MRS = 1 , & Pc / Py= .5

As MRS >1

So only C is consumed , c*= 400/4 = 100

Now if there is small change in Pc ,

So with small change, new Price, P`c will be < Py

C*` = M/Pc`

So again still only c will be consumed

so total effect will be only income effect , & no substitution effect

so TE = IE = C*` - C*

SE = 0

C) now Pc` = 10

So after price change, Py < Pc`

So now MRS < 1

So only y is consumed

so total effect is substitution effect, no income effect

IE = 0

TE = SE = Y*` - Y*

= 50 - 0

= 50

As Y*` = M/Py = 400/8 = 50