Question

a) The utility function for a consumer utility is U=30Q11/2Q21/2 . If the price per unit of Q1 is Kshs 10 and Kshs 5per unit of Q2, determine quantities Q1 and Q2 that the consumer should have to maximize utility if the consumer budgeted Kshs 350 (7 Marks).

b) Assuming there are two goods X and Y and two persons, analyze the exchange of goods between the two using the Edge worth Box framework indicating the Pareto efficient allocation (16Marks).

c) Clearly describe substitution effect and income effect for a fall in price for a normal good and an inferior good 

Answer 1

a) The utility function for a consumer utility is $U = 30Q_1^{\frac{1}{2}} Q_2^{\frac{1}{2}}$. If the price per unit of Q1 is Kshs 10 and Kshs 5per unit of Q2, determine quantities Q1 and Q2 that the consumer should have to maximize utility if the consumer budgeted Kshs 350 (7 Marks).

$U = 30Q_1^{\frac{1}{2}} Q_2^{\frac{1}{2}}$

Budget line:

$350=10Q_1+5Q_2$

The slope of the budget line equals the slope of the utility function when the bundle is optimal.

$Slope of BL=\frac{-price Of Q_2}{price of Q_1}=\frac{-2}{10}=-0 .5$

$Slope of U =\frac{-MUQ_1}{MUQ_2}$

$MUQ_1=15Q_2$

$MUQ_2=15Q_1$

$Slope of U=\frac{-15Q_1}{15Q_2}$

Taking the slope of the budget line and equating it to the slope of U:

$\frac{-15Q_1}{15Q_2}=-0.5$

$-15Q_1=-7.5Q_2$

$Q_1=0.52Q_2$

By replacing $Q_1$ In the budget line, we have:

$10×(0.5Q_2)+5Q_2=350$

$10Q_2=350$

$Q_2=35$

$Q_1=0.5Q_2$

$Q_1=17.5$

b) Assuming there are two goods X and Y and two persons, analyze the exchange of goods between the two using the Edge worth Box framework indicating the Pareto efficient allocation (16Marks).

Allocating products X and Y to persons 1 and 2 is Pareto efficient if it is difficult to make one person better alone without making the other person worse off. As a result, the edge worth box frame enables the study of exchanges between two persons for various commodities.

We can determine the Pareto-efficient points by first determining Person 1's utility and then determining which point on Person 1's indifference isoquant optimizes Person 2's utility. Any gain in Person 2's utility must come at the price of Person 1, and vice versa; hence, the point is Pareto efficient.

c) Clearly describe substitution effect and income effect for a fall in price for a normal good and an inferior good 

The income effect and the substitution effect of the normal goods both go in a similar way. When the relative price of goods reduces, there is an increase in the quantity demanded of the goods since it is now less expensive as compared to substitute goods. With a lower price, the consumers now have higher total purchasing power which hence increases the total consumption of the product.

For an inferior good the income and substitution effects act in opposing directions. When the price of an inferior commodity falls, the income effect reduces the quantity consumed, but the substitution effect rises the quantity consumed.