In most applications that follow we suppress the h superscript of \u03c5 h and \u03c5 h

In most applications that follow we suppress the h

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In most applications that follow we suppress the h superscript of υ h and υ h , and use instead υ and υ i . Given factor rental rates w , the household’s income is given by w · υ , which is used to purchase q j units of consumption good j at market price p j , j = 1 , ..., M. Then, the household’s budget constraint is given by w · υ p · q where q = ( q 1 , ..., q M ) R M ++ . In other words, each household consumes a strictly positive level of each consumption good. Consumer preferences over goods are represented by the util- ity function u : R M ++ R + , defined as u ( q ) . Assumption 1 u ( q ) satisfies the following properties:
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2. The Preliminaries 11 1. u ( q ) is increasing and strictly concave in q , 2. u ( q ) is everywhere continuous, and everywhere twice dif- ferentiable, 3. u ( q ) is homothetic. Assumption 1.1 yields indifference curves that are convex, As- sumption 1.2 ensures Marshallian demands are continuous func- tions, while Assumption 1.3 yields Marshallian demands that are separable in prices and income. Two indirect functions emerge from the consumer’s problem: the indirect utility function and the expenditure function. The indirect utility function gives the household’s maximum attain- able utility given income w · υ , defined as V ( p , w · υ ) max q { u ( q ) : w · υ p · q } The indirect utility function inherits the following properties from the direct utility function (see Cornes, pp. 67–70): V1. Homogeneous of degree zero in p and w · υ ; V ( θ p , θ w · υ ) = V ( p , w · υ ) , θ > 0, V2. V ( p , w · υ ) is convex in p , V3. V ( p , w · υ ) is continuous and differentiable in p and w · υ , V4. V ( p , w · υ ) = v ( p ) w · υ : separable in p and w · υ , By V4, the marginal utility of an additional unit of income is v ( p ) . V5. Given differentiability, Marshallian demands follow from Roy’s identity, q j ( p ) ( w · υ ) = v p j ( p ) v ( p ) w · υ (2.1) where, throughout the text, the subscript on a function indicates a partial derivative, e.g., v p j = ∂v ( p ) /∂p j and v p 1 p 2 = 2 v ( p ) /∂p 1 ∂p 2 .
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12 2. The Preliminaries Since consumers face the same prices and have identical pref- erences, the “community” indirect utility function is given by V = v ( p ) ( w · V ) while the total domestic Marshallian demand for good j is Q j = q j ( p ) ( w · V ) , j ∈ { 1 , · · · , M } (2.2) These functions are the simple aggregation of individual con- sumer welfare and demands. It also follows from V1 that (2.2) is homogeneous of degree minus one in prices p and of degree one in income. The expenditure function gives the minimum cost of achieving utility level q R at given prices p , and is defined as E ( p , q ) min q { p · q : q u ( q ) } The expenditure function inherits from u ( · ) , the following prop- erties: E1. E ( p , q ) > 0 for any p and q > 0 , E2. E ( p , q ) is non-decreasing in p and q, E3. E ( p , q ) is concave and continuous in p , E4. E ( λ p , q ) = λE ( p , q ) , λ > 0 : homogeneous of degree 1 in p , E5. E ( p , q ) = E ( p ) q : separable in p and q, E6. Shephard’s lemma: If E ( p , q ) is differentiable in p , then q j = E p j ( p , q ) = E p j ( p ) q, j = 1 , ..., M E 1 says purchasing a strictly positive consumption bundle is costly. E 2 says, all else equal, (i) if the price of a consump-
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