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IV.C. Functional Structure and Related Issues 1.Elasticity of Substitution Read: Blackorby and Russell, Will the Real Elasticity of Substitution Please Stand Up, AER, 79(1989): 882-8. Chambers, Applied Production Econ, pp. 93-100. Review Homogeneity and Homotheticity on your own.

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a.Hicks (2-goods or inputs case only) -The range of σ is [0,]. - Cobb-Douglas example (see notes) -Alternative formula from Silberberg pp. 287-8.

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b. Allen Elasticity of Substition AES = The Allen measure is equivalent to the Hicks when there are 2-goods. Allen extended the concept of elasticity of substitution to the multiple good case.

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Lets try to reconcile Allens measure with Hicks using the 2-good case. where F is the bordered Hessian of the function f(x) [bordered by its own gradient].

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The 1,2 cofactor of this matrix is f 1 f 2 and its determinant is - f 11 f 2 2 + 2f 12 f 1 f 2 - f 22 f 1 2 Note: 1+2=3 so co-factor switches sign We can rewrite this in terms of the cost/expenditure function. focs from cost min problem: r i – λf i = 0 i y – f(x) = 0

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Differentiate focs wrt to r 1

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Restack in matrices:

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Because b is a unit vector, z just extracts the first column of (F*) -1. Thus, if we abstract from r 1 to r j

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Notes: i.AES is symmetric ii.Each good cannot be an Allen complement for all other goods (at least one Allen substitute for every good). iii.AES obtains its sign from the cross-price slope of demand. iv.AES does not conform to our usual concept of elasticity (see BPR paper)

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IV.C.2. Separability Read: Blackorby, Primont, and Russell. J. of Econometrics, 5(1977): 195-209. Pope and Hallam, AJAE, 70(1988): 142-52. Paris, Foster, and Green, AJAE, 72(1990): 499-501. Pollak and Wales, Demand System Specification and Estimation, Oxford Univ. Press, 1992 (pp. 36-53).

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a. What is Weak Separability Weak Separability says that we can write the separable function F(x) as follows: F(x)=F[f 1 (x 1 ),f 2 (x 2 ), …, f m (x m )] The implications of this for us are that it implies that two stage budgeting or conditional optimization are allowable concepts.

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Suppose F(x) is a utility function and x 2 represents a subset of goods broadly classified as food, then if F(x) is weakly separable we need only examine the optimization of f 2 (x 2 ). The optimization of f 2 (x 2 ) is constrained by not spending more on x 2 than M 2 which is the income allocated to total food consumption in the first budgeting stage (or the income remaining after optimization of expenditures on non-food goods).

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The useful result is that we can write the system of food demand equations as functions of the prices of food goods and the expenditure on food goods. That is, X 2 = X(P 2,M 2 ) Does this mean that non-food good prices dont matter in food demand? No, they would enter through their impact on M 2 in the first stage budgeting. There is a constraint, however, because two prices from the same separable sub group would have the same effect on x 2.

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Recall the Quadratic Flexible Functional Form F(x) = а 0 + Σа i f i (x i ) + ½ΣΣβ ij f i (x i )f j (x j ) where β ij = β ji i j Weak Separability (i,j I r from k I r )

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Notes: i. Weak Separability says that the slope of the i-j space level curve is invariant to x k. It does not mean that x k wont shift the level curve in the i-j space. ii. Implications for testing with QFFFs. BPR, Pope and Hallam, and PFG. iii. Strong Separability i I r, j I s, and k I r I s F(x) = G(Σf i (x i ))

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b. Weak Separability and Substitution (read: Berndt and Christensen, RESTUD and J. of Econometrics) Consider weak separability of the cost function with respect to some partitions of prices.

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So, Which we know implies that:

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We can algebraically rearrange the previous condition for weak separability of the cost function to get: Now, if we multiply both sides by C/C k we get the following condition on the Allen Elasticity of Substitution for weakly separable groups.

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What is the important conclusion? It is that weak separability of the cost (or expenditure) function r i and r j from r k implies equality of the Allen Elasticities of Substitution. Note, however, that this refers to substitution between inputs or goods in the partition I r and those not in I r, but it says nothing about substitution between inputs or goods that are both in I r.

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