Answer to Question #51182 in Microeconomics for zem lotjem
U(x,y) = In (x+2y-y2/2)
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.
(c) Find the expenditure function of the consumer e(p,u) where price of x = 1 and price of y
(d) Find the Hicksian demand function hy (p,u) for commodity y, where the price of x is 1
and the price of y is p.
(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)
(c) Find the expenditure function of the consumer e(p,u) where price of x = 1 and price of y = p.
Formally, the expenditure function is defined as follows. Suppose the consumer has a utility function u defined on L commodities. Then the consumer's expenditure function gives the amount of money required to buy a package of commodities at given prices p that give utility of at least u^*.
(d) Find the Hicksian demand function hy (p,u) for commodity y, where the price of x is 1 and the price of y is p.
A consumer's Hicksian demand correspondence is the demand of a consumer over a bundle of goods that minimizes their expenditure while delivering a fixed level of utility. If the correspondence is actually a function, it is referred to as the Hicksian demand function, or compensated demand function.
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