# ps3sol_09-1 - Cornell University Economics 3130 Problem Set...

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Cornell University Economics 3130 Problem Set 3 Solutions 1. Consider the utility function U ( x 1 ,x 2 ,x 3 ) = α ln x 1 + β ln x 2 + γ ln x 3 . Assume the consumer has income I and faces prices p 1 , p 2 , and p 3 . Find the Marshallian demands. We can use the Lagrangian method. L = α ln x 1 + β ln x 2 + γ ln x 3 + λ ( I - p 1 x 1 - p 2 x 2 - p 3 x 3 ) The ﬁrst order conditions (FOC’s) are as follows. x 1 : αx - 1 1 - λp 1 = 0 x 2 : βx - 1 2 - λp 2 = 0 x 3 : γx - 1 3 - λp 3 = 0 λ : I - p 1 x 1 - p 2 x 2 - p 3 x 3 = 0 Using the ﬁrst two FOC’s we ﬁnd that x 2 = βp 1 αp 2 x 1 . Using the ﬁrst and third FOC’s we ﬁnd that x 2 = γp 1 αp 3 x 1 . Substituting into the budget constraint (the fourth FOC), we ﬁnd that x 1 = α α + β + γ I p 1 . It follows that x 2 = β α + β + γ I p 2 and x 3 = γ α + β + γ I p 3 . 2. Consider the two-good CES utility function U ( x 1 ,x 2 ) = 1 (1 - α ) ( x 1 - α 1 + x 1 - α 2 ). Suppose that α can take any value from 0 to inﬁnity. As usual, assume that the consumer has income I and faces prices p 1 and p 2 . (a) What is the marginal rate of substitution? MRS = ( x 2 x 1 ) α (b) What value of α corresponds to perfect substitutes? to Cobb-Douglas? to perfect complements? For perfect substitutes α = 0, for Cobb-Douglas α = 1, and for perfect comple- ments α goes to inﬁnity. (c) In class we calculated the elasticity of substitution as follows: σ = MRS r dr dMRS . We deﬁned r = x 2 x 1 . The answer from (a) above is a simple function of r , so you can write the ﬁrst term using r and not x 1 or x 2 . Then, with the MRS written in terms of r , take the derivative dMRS dr . The inverse of that is the second term in the

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## This note was uploaded on 08/29/2009 for the course ECON 3130 taught by Professor Masson during the Spring '06 term at Cornell University (Engineering School).

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ps3sol_09-1 - Cornell University Economics 3130 Problem Set...

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