Answered: 1. Use the method of Lagrange… | bartleby

Microeconomic Theory

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter4: Utility Maximization And Choice

Section: Chapter Questions

Problem 4.11P

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1. Use the method of Lagrange multipliers with inequality first-order conditions to determine if
corner solutions arise with the following utility functions. If not, explain why not. If so,
outline the condition under which a corner solution arises and find the Marshallian demands
in these cases.
(a) U(1, y) = (x+1)y
(b) U(x, y) = Vr+ V
%3D
2. Find the Marshallian demands for r and y for the utility function in part (a) of the previous
question when an interior solution exists. Use your solution to confirm the condition that
you found previously for a corner solution to arise.

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2. Consider a small island economy where the government is contemplating imposing a tax on bananas to raise revenue for public projects. The islanders' utility from consuming bananas (B) and coconuts (C) is given by the Cobb-Douglas utility function: U(B, C)=B°C1-a where a = 0.3. (a) Write down the consumer's optimization problem and derive the demand functions for bananas and coconuts. (b) Suppose the initial prices are PB $2 and Pc = = $4, and the consumer's income is I = $100. Calculate the initial quantities of bananas and coconuts consumed. (c) Calculate the compensating variation (CV) and equivalent variation (EV) for a tax rate of t = $0.50. Interpret your results in terms of the impact on consumer welfare. (d) Determine the tax revenue per consumer after the tax is imposed, assuming supply is perfectly elastic so the tax falls entirely on consumers. (e) Compare the impacts on consumer welfare between a flat tax equal to the revenue found in part (d) and the per-unit tax.
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2. Consider the two-good model of the utility maximization program subject to a budget constraint. The utility function U of a hypothetical rational consumer and his/her budget constraint are given, respectively, by: U = x1x2, (U) B = p1x1 + p2x2, (B) where xi = the consumer’s demand for consumption good i (i = 1, 2), pi = the price of consumption good i (i = 1, 2), and B = the (exogenously given) budget of the consumer. In this maximization program, assume the following data: B = 240, p1 = 10, p2 = 2. (a) Using the Lagrangian function L, derive the first-order (necessary) conditions for a (local) maximum of the utility function. (b) Compute the optimal values of all choice variables, i.e., x*1 , x*2, and λ* , in the program, where λ signifies the Lagrange multiplier. (c) Using the information of the bordered Hessian matrix H¯ , verify the second order (sufficient) condition for a (local) maximum of the utility function. Note:- Do not provide handwritten solution. Maintain accuracy…
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3. Consider the following utility function, u (21, x2) = min V#1, Varz), where a > 0 Derive the Marshallian demand functions. (Explain your derivation in details.) Does the Marshallian demand increase with price? Are the two consumption goods normal goods? Show two different ways to derive the Hicksian demand functions. (b) Does the Hicksian demand increase with price?

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