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(x, y) = (x + 1)y (b) U(x, y) = √x + √y
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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(x, y) = (x + 1)y
(b) U(x, y) = √x + √y
Step by step
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- Create a full optimization problem with constraints and employ the Lagrange multiplier technique. One on the consumer side with constraints (utility maximization) and one on the producer side (either production / cost minimization , and or profit maximization.Consider the following function describing the utility of a consumer: U(x1, x2, x3) = a1*ln(x1) + a2*ln(x2) + a3*ln(x3), where ln = natural logarithm and a1, a2, a3 constants a. Pose the primal problem (using Langrange's method), obtaining the Marshallian demands for each good and the individual's indirect utility function. b. From the results obtained from question a., find the minimum expenditure function and the Hicksian demands.Consider U(q1,q2) = q1 + v(q2) where v' > 0 and v'' < 0. This utility function is called a quasi-linear utility function. Assume q1 is a numeraire. Find the demand function for q2. *What does v mean in this question? Also, could you solve this problem without using Lagrange multipliers? Thank you.
- Suppose the utility function for Njeru is given by U = f(X1, X2) = X10.25 X20.75. Given that the price of commodity X1 is Kshs. 10 per unit and that of community X2 is Kshs. 15 per unit. Using Lagrangian multiplier technique determine units of the two commodities Njeru will buy given an income of Kshs. 40,000 in order to maximize his utility.For a > 0, consider a consumer whose utility function amounts to u(x1, x2) = − exp(−ax1x2). Can you take first order conditions to solve the utility maximization problem? Explain your argument. Next solve the utility maximization problem, and derive Marshallian demands and the indirect utility function. Given your calculations, state and use the duality theorem to find the expenditure function and Hicksian demand Note:- Do not provide handwritten solution. Maintain accuracy and quality in your answer. Take care of plagiarism. Answer completely. You will get up vote for sure.Question 3 Consider the utility function of the form: ?=?1?1?2?2 Given the budget constraint: ?1?1+?2?2=? Show that the implied Marshallian demand curves are: ?1=?1(?1+?2)??1 ?1=?2(?1+?2)??2
- Please Note: I have the question 1 solved and only need answer for question 2 thanks. 1. Use the method of Lagrange multipliers with inequality first-order conditions to determine ifcorner 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 demandsin these cases.(a) U(x, y) = (x + 1)y(b) U(x, y) = √x + √y 2. Find the Marshallian demands for x and y for the utility function in part (a) of the previousquestion when an interior solution exists. Use your solution to confirm the condition thatyou found previously for a corner solution to arise.Sales are the function of advertising in The Dawn and Diva Magazine (X, Y). S = XY2 If the price of advertising in The Dawn and Diva Magazine is Rs.5 and Rs.10 respectively. The total budget for advertising is Rs.105. For maximizing the sales of Dawn and Diva Magazine find out the best combination of advertisements in newspapers and magazines by using the Lagrangian multiplier. please provide a complete solution with all the steps including formulas and proper workingQ1. Derive the Marshallian demand and indirect utility function for ?(?,?)=(0.3?‾‾√+0.7?√)2u(x,y)=(0.3x+0.7y)2. Q2. Derive the Hicksian demand and the expenditure function for ?(?,?)=(0.3?‾‾√+0.7?√)2u(x,y)=(0.3x+0.7y)2.
- Note:- Do not provide handwritten solution. Maintain accuracy and quality in your answer. Take care of plagiarism. Answer completely. You will get up vote for sure. For the utility function given below, derive the Marshallian demand, the indirect utility and the expenditure function. Please be careful, there may be corner solutions in some cases u(x1,x2)=2x1 +3x2Please Use This Bartleby Expert Solution to Create Bordered Hession Matrix (Addition Image Provided): Expert Solution arrow_forward Step 1 For utility maximization the second-order condition we have to show the utility function is convex (having diminishing MRS). So we use Bordered Hessian matrix to check the second-order condition of utility maximization. The Bordered Hessian matrix is H=uxxuxypxuyxuyypypxpy0 The second-order condition is satisfied only when |H|=uxxuxypxuyxuyypypxpy0>0 Here uxx=∂2u∂x2, uyx=uxy=∂2u∂xy uyy=∂2u∂y2 arrow_forward Step 2 Now Utility function: u(x,y)=x+2y+1 .......(1) Differentiate partially from equation 1 with respect to x: ∂u∂x=y+1again differentiate above equation with respect to x∂2u∂x2=0 Similarly ∂u∂y=x+2and ∂2u∂y2=0 and∂2u∂yx=1 arrow_forward Step 3 Putting value in Bordered Hessian matrix: |H|=01px10pypxpy0H=-10-pxpy+pxpy-0H=2pxpy>0 because px and py are positive number So second-order condition is satisfied.You have k20 per week to spend and two possible uses for the money: telephoning friends back home and drinking coffee. Each Hour of phoning costs k2 and each cup of coffee costs k1. Your utility function is U(X,Y)=XY,where X is the hours of phoning you do and Y the number of cups of coffee you drink. What are your optimal choices? What is the resulting utility levels? You can use the standard result on the constrained maximization of such a function, but must state in clearly