# IV.C. Functional Structure and Related Issues 1.Elasticity of Substitution Read: Blackorby and Russell, Will the Real Elasticity of Substitution Please.

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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.

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.

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.

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].

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

Differentiate focs wrt to r 1

Restack in matrices:

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

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)

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).

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.

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).

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.

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 )

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 ))

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.

So, Which we know implies that:

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.

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.