A410 Substituting from A45 into A49 we get Τ Τ Τ Τ A411 Substituting from A410

A410 substituting from a45 into a49 we get τ τ τ

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? ?? + ? ? ?? (A4.10) Substituting from (A4.5) into (A4.9), we get Τ ?𝑈 ?𝐼 = ?? ? ( Τ ?? ?𝐼) + ?? ? ( Τ ?? ?𝐼) = ? Τ ? ? ?? + ? ? ?? ?𝐼 (A4.11) Substituting from (A4.10) into (A4.11), we get Τ ?𝑈 ?𝐼 = ? Τ (? ? ?? + ? ? ?? ? ? ?? + ? ? ??) = ? (A4.12) Thus the Lagrange multiplier is the extra utility that results from an extra dollar of income.
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54 of 50 An Example Cobb-Douglas utility function Utility function U ( X , Y ) = X a Y 1− a , where X and Y are two goods and a is a constant. 𝑈 ?, ? = ?log ? + 1 − ? log(?) The Cobb-Douglas utility function can be represented in two forms: and To find the demand functions for X and Y, given the usual budget constraint, we first write the Lagrangian: Φ = ?log ? + 1 − ? log ? − ?(? ? ? + ? ? − 𝐼) Now differentiating with respect to X, Y, and l and setting the derivatives equal to zero, we obtain Τ 𝜕Φ 𝜕?= Τ ? ? − ?? ? = 0 Τ 𝜕Φ 𝜕Y= Τ (1 − ?) ? − ?? ? = 0 Τ 𝜕Φ 𝜕λ=? ? X+? ? ? − 𝐼 = 0 𝑈 ?, ? = ? 𝑎 ? 1−𝑎
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55 of 50 Combining these expressions with the last condition (the budget constraint) gives us or ? = Τ 1 𝐼. Now we can substitute this expression for λ back into (A4.13) and (A4.14) to obtain the demand functions: The first two conditions imply that ? ? ? = Τ ? ? ? ? ? = Τ (1 − ?) ? (A4.13) (A4.14) Τ ? ? + Τ (1 − ?) ? − 𝐼 = 0 ? = ( Τ ? ? ? )𝐼 ? = [( Τ 1 − ?) ? ? ]𝐼
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56 of 50 Duality in Consumer Theory duality Alternative way of looking at the consumer’s utility maximization decision: Rather than choosing the highest indifference curve, given a budget constraint, the consumer chooses the lowest budget line that touches a given indifference curve. Minimizing the cost of achieving a particular level of utility: Minimize ? ? ? + ? ? ? subject to the constraint that 𝑈 ?, ? = 𝑈 The corresponding Lagrangian is given by Φ = ? ? ? + ? ? ? − ?(𝑈 ?, ? − 𝑈 ) Differentiating with respect to X , Y , and μ and setting the derivatives equal to zero, we find the following necessary conditions for expenditure minimization: (A4.15) ? ? − ?MU ? ?, ? = 0 and ? ? − ?MU ? ?, ? = 0 𝑈 ?, ? = 𝑈
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