(Solved): Problem 2: Utility maximization with 3 constraints There are two goods, x and Y, available in arbitr ...
Problem 2: Utility maximization with 3 constraints There are two goods,
xand
Y, available in arbitrary non-negative quantities (so the consumption set is
R_(+)^(2)). The consumer has a preference relation over consumption bundles that can be represented by the differentiable, strictly increasing, strictly quasi-concave, utility function
u:R_(+)^(2)->Rwith
u(x,y)=xy+\alpha x+\beta ywhere
xis the quantity of good
x,yis the quantity of good
Y, and
\alpha ,\beta >0are a preference parameters. The consumer is planning to purchase a consumption bundle. However, the consumption bundle will need to be delivered to the consumer's home, and the delivery company has space and weight constraints for the shipment. As a result, the consumer has three constraints: a wealth constraint, a space constraint, and a weight constraint. Wealth constraint: The price of good
xis 100 Dirhams/unit, the price of good
Yis 100 Dirhams/unit, and the consumer only has 500 Dirhams of wealth. Space constraint: The volume of good
xis
60c(m^(3))/()unit, the volume of good
Yis
20c(m^(3))/(u)nit, and the delivery company will only ship goods with a total volume of
240cm^(3)or less. Weight constraint: The weight of good
xis
4k(g)/(u)nit, the weight of good
Yis
12k(g)/(u)nit, and the delivery company will only ships goods with a total weight of 48 kg or less. (a) In an appropriate diagram, illustrate the map of indifference curves for the consumer. (b) In an appropriate diagram, illustrate the wealth, space and weight constraints, and indicate the constraint set for the consumer. (c) For what values of
\alpha >0and
\beta >0is it optimal for the consumer to (i) only purchase good
x? (ii) only purchase good
Y? (iii) spend all of her wealth? For each part, clearly show your calculations.
