Question

1. A consumer has the following utility function: U(x, y) = x(y+1), where x and y are quantities of two consumption goods whose prices are Px and Py, respectively. The consumer also has a budget of B. Therefore, the consumer's Lagrangian is
x(y +1) + ƛ(B Pxx – Pyy)
a.) From the first-order conditions find expressions for the demand functions. What kind of good is y? In particular what happens when Py > B?
b.) Verify that this is a maximum by checking the second-order conditions. By substituting x* and y* into the utility function, find an expression for the indirect utility function
U* = U(Px, Py, B)
and derive an expression for the expenditure function
E = E (Px Py, U*)
c.) This problem could be recast as the following dual problem
Minimize Pxx+ Pyy
subject to x(y+ 1) = U*
Find the values of x and y that solve this minimization problem and show that the values of x and y are equal to the partial derivatives of the expenditure function ∂E / ∂Px and ∂E / ∂ Py, respectively

Answer 1

Answer:

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Explanation:

The Lagrangian function is given by:

L

=

x

(

y

+

1

)

+

λ

(

B

−

P

x

x

−

P

y

y

)