Question #51590

A consumer’s utility function is given as
U(x,y) = In (x+2y- *
+
,
)
Where x and y are two goods of consumption.
(a) Find the indirect utility function of the consumer.
(b) Examine if Roy’s law is satisfied by the consumer’s demand function for y.

Expert's answer

(a) A consumer's indirect utility function v(p, w) gives the consumer's maximal utility when faced with a price level p and an amount of income w. It represents the consumer's preferences over market conditions. This function is called indirect because consumers usually think about their preferences in terms of what they consume rather than prices. A consumer's indirect utility v(p, w) can be computed from its utility function u(x) by first computing the most preferred bundle x(p, w) by solving the utility maximization problem; and second, computing the utility u(x(p, w)) the consumer derives from that bundle. The indirect utility function for consumers is analogous to the profit function for firms.

(b) Roy's identity (named for French economist René Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma relates the ordinary (Marshallian) demand function to the derivatives of the indirect utility function. Specifically, where V(P,Y) is the indirect utility function, then the Marshallian demand (dX1) is a function of the price of X1, the price of X2 (assuming two goods) and the level of income or wealth (m):

X*=dX1(PX1, PX2, m)

(b) Roy's identity (named for French economist René Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma relates the ordinary (Marshallian) demand function to the derivatives of the indirect utility function. Specifically, where V(P,Y) is the indirect utility function, then the Marshallian demand (dX1) is a function of the price of X1, the price of X2 (assuming two goods) and the level of income or wealth (m):

X*=dX1(PX1, PX2, m)

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