(5) A consumer has the following utility function: U(x, y) = (x+2)(y+1), where x and y are quantities of two consumption goods whose prices are P, and Py, respectively. The consumer also has a budget of B. (a) Write down the consumer's Lagrangian associated with this primal problem. From the first-order conditions, find expressions for the demand functions x and y. (b) Find the expression for indirect utility function U = U(P, P, B) (namely the maximum value function). (c) Write down the associate dual problem with a target level of utility U. Derive an expression for the expenditure function E = E(P, P, U") of the dual problem. (d) Use Shephard's Lemma to find the Hicksian demands of associated dual problem, i.e., OE/OP, rh, OE/OP, y", and OE/OU= u", where " is the Lagrangian multiplier in this problem.
(5) A consumer has the following utility function: U(x, y) = (x+2)(y+1), where x and y are quantities of two consumption goods whose prices are P, and Py, respectively. The consumer also has a budget of B. (a) Write down the consumer's Lagrangian associated with this primal problem. From the first-order conditions, find expressions for the demand functions x and y. (b) Find the expression for indirect utility function U = U(P, P, B) (namely the maximum value function). (c) Write down the associate dual problem with a target level of utility U. Derive an expression for the expenditure function E = E(P, P, U") of the dual problem. (d) Use Shephard's Lemma to find the Hicksian demands of associated dual problem, i.e., OE/OP, rh, OE/OP, y", and OE/OU= u", where " is the Lagrangian multiplier in this problem.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:(5) A consumer has the following utility function: U(x, y) = (x+2)(y+1), where x and
y are quantities of two consumption goods whose prices are P, and Py, respectively. The
consumer also has a budget of B.
(a) Write down the consumer's Lagrangian associated with this primal problem. From
the first-order conditions, find expressions for the demand functions x and y.
(b) Find the expression for indirect utility function U = U(P, P, B) (namely the
maximum value function).
(c) Write down the associate dual problem with a target level of utility U. Derive an
expression for the expenditure function E = E(P, P, U") of the dual problem.
(d) Use Shephard's Lemma to find the Hicksian demands of associated dual problem,
i.e., OE/OP, rh, OE/OP, y", and OE/OU= u", where " is the Lagrangian multiplier
in this problem.
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