2. Maximize utility U = xy subject to 4x+3y = 60, where x and y are the two consumption goods, 4 is the price of good x, 3 is the price of good y, and 60 is the budget. a) Set up the Lagrangian objective function. How much is the utility-maximizing consumption bundle? b) Show that the second-order condition/s is/are met (Use H₂ to identify whether it is a maximum or a minimum). c) What is the maximum utility?

2. Maximize utility U = xy subject to 4x+3y = 60, where x and y are the two consumption goods, 4 is the price of good x, 3 is the price of good y, and 60 is the budget. a) Set up the Lagrangian objective function. How much is the utility-maximizing consumption bundle? b) Show that the second-order condition/s is/are met (Use H₂ to identify whether it is a maximum or a minimum). c) What is the maximum utility?

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

See similar textbooks

Similar questions

Question 7 A consumer has a utility function 100-(x₁-4)4 -(x2-3)4 and a budget constraint of 9x1+3x2 <200. How much of good 2 do they consume?
1. Consider a consumer who consumes two goods. The utility function is given by u(x1, x₂) /2x1+1+√√√x₂, - where x > 0 denotes the amount of good i = the price of good 1 is 3, the price of good 2 is p2 > 0, and the income is 12. 1,2 consumed. Unless stated otherwise, (1) Calculate the marginal rate of substitution (measuring the value of good 1) at the consumption plan (x₁, x₂) = (3, 4). (2) Calculate the consumption plan that is optimal for the consumer. (3) Let v(p₂) denote the utility level at the optimal consumption plan. Calculate its derivative with respect to p2, i.e., v'(P2), at p2 = 3. (4) Suppose that p2 = 1 and the utility level is 10. Calculate the compensated demand for each good.
Q2. Suppose a consumer seeks to maximize the utility function U (x, y) = (x + 2) (y + 1), where and y represent the quantities of the two goods consumed. The prices of the two goods and the consumer's income are pa, py, and I. Write out the consumer's budget constraint and the Lagrangian function for the problem.
Refer to the information provided in Figure 6.7 below to answer the questions that follow. Units of Y B Units of X QUESTION #25: Refer to Figure 6.7. Along budget constraint AB, the price of good X is Php10 and theprice of good Y is Php12. If the price of X increases to Php15, the budget constraint will A) pivot in at Point B. B) pivot out at Point A. C) shift in parallel to AB. D) pivot in at Point A. QUESTION #26: Refer to Figure 6.7. Along budget constraint AB, the price of good X is Php12 and theprice of good Y is Php10. If the price of Y decreases to Php8, the budget constraint will A) pivot in at Point B. B) pivot out at Point A. C) shift in parallel to AB. D) pivot in at Point A.
A consumer has a budget of £12 to split between two goods: good 1 has a price of 2, good 2 has a price of 3. Write the consumer’s budget constraint algebraically. Convert this budget constraint into the formula for the budget line. Show this line on a suitably labelled graph. The consumer’s budget increases to £24. Show the effect of this change graphically. A consumer dislikes good 1, and dislikes good 2. Show these preferences on a suitably labelled graph with an indifference curve. Label the graph: which areas would be preferred to those on the line?
A consumer consumption bundle contains good X and Y. Suppose the price of good X goes up and a consumer goes on consuming the exact same amount of X as before. Check the correct statement(s).A) Both X and Y are inferior. B) X is an inferior good.C) In the special case where X is Leisure, and Y is "all other goods", X is an inferior good.D) The consumption of Y decreases.E) It is impossible that the consumption of X remains unchanged when the price of × goes up.F) The substitution effect is negative
Assume, as in Exercise 22.1, that a consumer has utility function F or fruit and chocolate. Determine the consumer's demand functions q1(P1, P2, M) and q2(P1, P2, M). Determine also It* in terms of P1, P2 and M. Find the indirect utility function and show that It* = 8Vj8M. Suppose, as before, that fruit costs $1 per unit and chocolate $2 per unit. If the income is raised from $36 to $36.5, determine the precise value of the resulting change in the indirect utility function. Show that this is approximately equal to (O.5)λ*, where λ* is evaluated at P1 = 1,P2 = 2 and M = 36. Exercise 22.1 A consumer purchases quantities of two commodities, fruit and chocolate, each month. The consumer's utility function is For a bundle (X1, X2) of X1 units of fruit and X2 units of chocolate. The consumer has a total of $49 to spend on fruit and chocolate each month. Fruit cost $1 per unit and chocolate costs $2 per unit. How many units of each should the consumer buy…
4. Maximize utility U = xy subject to 4x+3y=60, where x and y are the two consumption goods, 4 is the price of good x, 3 is the price of good y, and 60 is the budget. a) Set up the Lagrangian objective function. How much is the utility-maximizing consumption bundle? b) Show that the second-order condition/s is/are met (Use |Ħ₂ to identify whether it is a maximum or a minimum). c) What is the maximum utility?
2) Which of the following utility functions represent the same preferences? Explain. a) U (x₁, x₂) = X₁ X₂ b) W (x₁, x₂) = 5lnx₁ +5lnx₂ c) V (x₁, x₂) = x₁¹/3x₂ ¹/3 - 0.8 d) Z(x₁, x₂) = 0.5x₁ + 0.5x₂
Consider a consumer with M = $120 to spend (income) and faces prices PX = $15 and PY =$5 for goods X and Y. Her utility function is U(X, Y) = X1/2 + Y1/2.a Carefully. express the consumer’s choice problem, using the given information (this is whereyou write out the max operator, the choice variables, the objective function, and the budget constraint). b. Compute the absolute value of the consumer’s marginal rate of substitution, and inspect it todetermine the shape of the consumer’s indifference curves: C-shaped, linear, )-shaped, or some other shape.To show your work, neatly use the arrow argument, increasing X (↑) and decreasing Y (↓) to see whether|MRS(X, Y)| is diminishing along an indifference curve. c. If the indifference curves are C-shaped write out the budget line and the equal slopes conditionthat characterize an interior solution to the consumer’s choice problem. Use the particulars for the givenconsumer. Solve these conditions to find the interior solution. On the other…
i. Write down the Lagrangian form and solve the utility maximization problem. ii. Assume that the income increases from 100 to $101. What is the increase in the optimal value of u(x,y)?
Part A . Consider an economy with the following features. • There are 100 identical consumers that derive utility from consuming three different goods: software, computers, and good m. • Each consumer decision utility function is given by U (c, 8) = 4c¹/48¹/4+m, where e denotes the amount of computers that she consumes, s denotes the amount of software that she consumes, and m denotes the amount of good m that she consumes. cand s must be non-negative, but m can take any real value. • Computers are produced by 20 identical competitive firms with a total cost function given by 10c². • Software is produced by 40 identical competitive firms with a total cost function given by 20s². . QUESTION 1: What are the equilibrium prices for software and comput- ers in equilibrium? Part B • Suppose that there is a positive technology shock in the software industry so that that the new cost function of the software firms becomes 10s². Let C and S denote, respectively, the aggregate level of…