(b) Derive the household’s compensated demand function for goods 1 and 2, i.e., obtain functions of the form Xi = fi (P1, P2, U) , I = 1, 2 where U is the household’s level of utility.
Economics
Consider a household whose preferences are described by the utility function
U(X1, X2) = X1X2
where X1 and X2 are household’s consumption of goods 1 and 2 respectively. Consider that household’s budget constraint is:
P1X1 +P2X2 =I.
(a) Derive the household’s demand functions for goods X1 and X2.
(b) Derive the household’s compensated demand function for goods 1 and 2, i.e., obtain functions of the form
Xi = fi (P1, P2, U) , I = 1, 2
where U is the household’s level of utility.
(c) Assume that in the initial situation the commodity prices, P1 and P2, and the household income level, I, are given by
P1 = $1, P2 = $1 and I = $2.
Sketch the compensated and uncompensated demand curves for good 2 with P1 held constant at the initial level. In the compensated case, U is held constant at the initial level while in the uncompensated case, I is held constant.
(d) By how much must I be increased if P2 increases to $2 (P1 remains at $1) and our household is to maintain its initial level of utility. Be sure to check your answer by examining the area under the compensated demand curve.
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