Question

Jennifer buys two goods, food (F) and clothing (C), with the utility function U(F,C) = FC. Assume initially that she has an income of $72, the price of clothing is PC = $1 per unit, and the price of food is initially PF1 = $9 per unit and that the price subsequently falls to PF2 = $4 per unit. Use this information and the accompanying graph to answer the following questions.

. . . . . . . . . . . . . . . . . BL2

(a) Find the equation for budget line (BL1) when the price of food is $9. (b) Find the initial consumption bundle A when the price of food is $9.

(c) Find Jennifer’s utility (U1) at consumption bundle A.
(d) Find the equation for budget line (BL2) when the price of food is $4.

(e) Find the consumption bundle C when the price of food is $4. Find Jennifer’s utility (U2) at consumption bundle C.

(f) Find the equation for budget line (BLd) when the price of food is $4 and income is (Id).

(g) Find the consumption bundle B using budget line (BLd) and U1 = FC

(h) Find the income and substitution effects and show that

Price Effect (PE) = Income Effect (IE)+ Substitution Effect (SE).

(i) Find algebraic form of Jennifer’s demand curve for food.

Answer 1

A) Equation for BL1 : C + 9F = 72 where C = units of cloth and F = unit of food

B) At the initial optimum Consumption bundle ;

MRS = Price Ratio ==> C/F = 9 ==> C = 9F

Putting it in BL1 ;. 9F + 9F = 72 ==> F = 72/18 = 4 and. C = 36

C) U1 = (36)(4) = 144

D) BL2 : C +4F = 72

E) At optimal bundle with Bl2 ; MRS = new price ratio = 4

C/F = 4 ==> C = 4F

Putting this in Bl2 ; 4F + 4F = 72 ==> F = 9 and C = 36

U2 = (36)(9) = 324

F) BLd is obtained in such a way that utility remains U1 at the new price ratio ;

Optimal condition with new price ratio is C = 4F ( from part e)

U = FC = 144 and C= 4F

==> 4F2 = 144 ==> F​​​​​​2 = 36 ==> F = 6 and C = 24

Now expenditure required for above Consumption bundle is 6(4)+24(1) = 48 . Hence consumer's oncome must be 48 at BLd

therefore BLd is 4F +C = 48

G) from above part we can see that bundle is (6,24)

H) Substitution Effect is when we separate the effect of income change ( which changes because of price change) from total effect of price change such that only the effect of price change alone remains.

It is calculated above by calculating the budget line that would leave the consumer at the same utility level at the new price ratio. The difference in original and bundle at BLd gives you Substitution Effect .

==> 6-4 = 2

Income effect is the effect of change in income alone. As we move from BLd to BL2 , the increase in income brought about doesn't include the impact of price change and therefore Income Effect is 9-6 = 3

TOTAL EFFECT is Final bundle minus initial bundle of food ( BL2 ans BL1 respectively) : 9-4 = 5

And Substitution Effect + Income Effect = 2+3=5

Hence PE = IE + SE .

I) let price of food be Pf and price of cloth be Pc ;

Optimality Condition guves C/F = Pf/Pc

==> PcC = PfF

Putting it in Budget Constraint : PfF + PcC = M

==> PfF + PfF = M ==> F = M /(2Pf)