Please no written by hand solutions 8. Derive the indirect utility function (p. M) by substituting u = v(p, M) in the expen-diture function, setting the expenditure function equal to M, and solving for v(p, M).9. Derive the Marshallian demand functions ri, 2, and r as functions of the pricesp= (P₁, P2, P3) and income M. To do this, use Roy's identity.(p, M) =Əv(p, M)/apiƏv(p, M)/OM10. Are goods ₁, 2, and a normal goods? Compute the appropriate partial derivativesand answer.11. What are the signs of Oz (p, M)/Op,, for i=1,2,3? Answer and back up your answerwithout directly computing the partial derivatives. Problem 1. Expenditure Minimization.In this exercise, we consider an expenditure minimization problem with a utility functionover three goods. The utility function isu(x₁, x2, 3)=√√x1x₂x3.The prices of goods x = (1, 22, 23) are p = (P₁, P2, P3). We denote income by M, as usual,with M > 0. Assume x R (i.e., ₁ > 0, ₂ > 0, 3 > 0), unless otherwise stated.1. Compute du/or and ou/or. Is the utility function increasing in ₁? Is the utilityfunction concave in a₁?2. The consumer minimizes expenditure subject to a utility constraint. Let u representthe minimum level of utility the consumer requires. Write down the expenditure min-imization problem of the consumer with respect to x = (₁, 2, 3). Explain brieflywhy the utility constraint is satisfied with equality.3. Write down the Lagrangian function (use À for the Lagrange multiplier).4. Write down the first order conditions with respect to x = (₁, 2, 3) and A.5. Solve for the Hicksian demand functions hi, h₂, and ha as functions of the pricesp= (P₁, P2, P3) and the minimum required utility u. [Hint: combine the first andsecond first-order conditions, then combine the first and third first-order conditions,and finally plug into the utility constraint.]6. Find 0h (p, ü)/ap2 and 0h; (p. u)/Ops. Based on this, are these goods (net) substitutesor (net) complements?7. Write down the expenditure function as a function of p= (P₁, P2, P3) and u, i.e., whatis e(p, ū)?

Utility function : u (x1, x2 , x3 ) = (x1 , x2, x3 )0.5Price of the goods are : p1,,,p2,, x2 , x3 )…

Problem 1. Expenditure Minimization. In this exercise, we consider an expenditure minimization problem with a utility function over three goods. The utility function is u(x₁, 2, 3)=√√123. The prices of goods x = (1, 2, 3) are p = (P1, P2, P3). We denote income by M, as usual, with M > 0. Assume x E R+ (i.e., z₁ > 0, 2 > 0, 3 > 0), unless otherwise stated.

Problem 1. Expenditure Minimization. In this exercise, we consider an expenditure minimization problem with a utility function over three goods. The utility function is u(x₁, 2, 3)=√√123. The prices of goods x = (1, 2, 3) are p = (P1, P2, P3). We denote income by M, as usual, with M > 0. Assume x E R+ (i.e., z₁ > 0, 2 > 0, 3 > 0), unless otherwise stated.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter4: Utility Maximization And Choice

Section: Chapter Questions

Problem 4.13P

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