Question

Imagine a representative consumer, whose utility for apples (X) and all other goods (Y) can be represented in a Cobb-Douglas form.

1) Please graphically represent consumer indifference curves, given prices Px and Py and the budget constraint M.

2) What will happen to consumer utility and optimal bundle if consumer income (budget) increases and apples are a necessity good? Please show graphically and explain the intuition.

3) How would the Engel curve look like for point #2?

Answer 1

Solution-

(1) given-

Consumer's utility For apples (X) and all other goods (Y) can be represented as Cobb-Doublas form.

Thus utility is written as

U=K. X^a Y^b, where a and b are utility elasticities of apples and other goods respectively. K is constant. Also budget constraint of the consumer is given by M=px.X+ py.Y ,where px is price of apples and py is price of all other goods.

VERY = kxdy, k is constant L & B are utility elasticities of Apples & other goods respectively Suppose there is constant Returns to scale to utility, so L + B = 1 B = 1-d. So, utility can be re written as Un,y) = kxx y 1-2 MRS ( slope of In difference were of Utility) MRS= MUM = = kxxx-11lt muy su K XX (1-2) Y MRS at diff point.

Other IR Product ICA (U4) - IC (UB) I C₂ (U2) -IG (U) M Apples > PX Optimal Bundles (x , y ) It is obtained by equalisation of slope of Budget fine and Slope of I-C shume i Slope of B = Slope of IC . Page = MRSxy A higher Indifference Cewere represents a higher level of Utility / Satisfaction to the consumer.

2) Income elasticity of necessity goods is positive but less than 1. This means that due to a proportional rise in income, proportional rise in quantity demanded of necessity goods is less than proportional rise in income. Thus consumption of apple would rise but not by a greater percentage due to rise in income. It can be shown graphically as

Other PY Budget line shifts rightward due to rise in Income from Mito M.. New Eau point is Ez Ja X* X * Mi Me Apples % change in Apple <% change in Income

3) Engel curve is the curve showing the relationship between quantity demanded of any particular good with income. As apple is a necessity good, it's income elasticity would be positive but less than 1. That is, the Engel curve will be inelastic for Apple, depicting the proportional rise in quantity demanded of apple is less than proportional rise in income.

(Engel Curve . Income (M2) Y₂ (M) 7 I EL Q, Q2 Quantity demanded