5. P5: Lagrange Multiplier Method a. Consider the following utility function: U = X¹Y4, x, y > 0. Let Px Py = 3 and M = 100. Find the utility-maximizing consumption bundle. b. Find the marginal utilities of the goods, x and y. Are they always positive? c. Sketch their graphs. d. - 2 and By implicit differentiation, find the slope of any difference curve and show that it is given by the ratio of marginal utilities of the two goods. Are the indifference curves always negatively sloped? Sketch their graphs.
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- 4. P4: Functions of two or more independent variables a. An individual's utility function has the Cobb-Douglas form: U=X0.5y2 where U is an index of her utility and X and Y are the weekly quantities of the two goods consumed. Find the marginal utilities of the two goods. Does the consumer ever experience satiation with respect to either good? b. How does the marginal utility of each good vary as consumption of it increases with consumption of the other good held constant? What is the economic significance of the signs of these partial derivatives? c. By differentiation, show that the indifference curve is negatively sloped and convex in the KL plane. Give an economic interpretation to the slope and curvature of the indifference curve. Sketch the graph of the indifference curve. 5. P5: Lagrange Multiplier Method a. Consider the following utility function: U=X¹Y4, x, y> 0. Let Px = 2 and Py = 3 and M = 100. Find the utility-maximizing consumption bundle. b. Find the marginal utilities of…3. A canonical utility function. Consider the utility function u(c) 1-σ where c denotes consumption of some arbitrary good and ơ (Greek lowercase letter sigma") is known as the "curvature parameter" because its value governs how curved the utility function is. In the following, restrict your attention to the region c> (because "negative consumption" is an ill-defined concept). The parameter σ is treated as a constant. Plot the utility function for ơ-0. Does this utility function display diminishing marginal utility? Is marginal utility ever negative for this utility function? Plot the utility function for ơ marginal utility? Is marginal utility ever negative for this utility function? Consider instead the natural-log utility function u(c)=In(c). Does this utility function display diminishing marginal utility? Is marginal utility for this utility function? Determine the value of σ (if any value exists at all) that makes the general utility function presented above collapse to the…Q6) Utility Ratio and Consumption Maximization A consumer downloads 3 movies and 4 apps per week. Suppose the price is $2 per movie and $2 per app, and the marginal utility for this consumption level is 6 for a movie and 8 for an app. Instructions: Round your responses to one decimal place. a. Calculate, show the formula, marginal utility per dollar. ..Per movie ......Per app b. Is this optimal consumption? c. If not, how should they change their consumption to maximize?
- 1 Consider a consumer who consumes only two goods, 21 and z2. His utility function is u (21, 22) = In z1 + 2. where In represents the natural logarithm. 1.a Derive the consumer's MRS of good 1 for good 2 using calculus to calculate his marginal utility from z and his marginal utility from z2. 1.b If the price of z1 is $2 per unit and the price of z2 is $4 per unit, and the consumer's income is $100, What is the equation of this consumer's budget line? 1.c What are the optimal consumption choices, z and z5, for this consumer? Show your work.Suppose you are thirsty and you want to quench your thirst with water. You observe that the first mouthful of water you drink gives you the most satisfaction, but subsequent mouthfulls satisfy you less and less until you finally decide you have had enough water to quench your thirst. We are describing a phenomenon we named __________. A.decreasing consumption. B.marginal analysis. C.diminishing marginal utility. D.utility.3. Suppose Mary enjoys Pepsi and Coke according to the functionU(P;C) = 4C + 5P. What does her utility function say about her MRS of Coke for Pepsi? What do her indi§erence curves look like? What type of goods are Pepsi and Coke for Mary? If Pepsi and Coke each cost $1 and Mary has $20 to spend on these products, how many units of each product should she buy in order to maximize her utility? Show this utility maximiz- ing combination combination of Pepsi and Coke on the graph. how would her consumption and utility maximizing bundle of Coke and Pepsi change if the price of Coke decreases to 50 cents. 4. Vera is an impoverished graduate student who has only $100 a month to spend on food. She has read in a government publication that she can assure an adequate diet by eating only peanut butter and carrots in the Öxed ratio of 2 pounds of peanut butter to 1 pound of carrots, so she decides to limit her diet to that regime. a) If peanut butter costs $4 per pound and carrots cost $2 per…
- . 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, s) = 4c¹/4g¹/4+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?2. Vibha consumes three goods X, (shirts), X, (food in pounds), and X, (shelter in square feet). Vibha's utility function is given by: U (X,, X, X,) = 5ln Xt 3ln X + 2ln X, where In (x) or log. (x) is the natural logarithm. Vibha's weekly income is $100. Let the price of X, = P, = $10, the price of X, = P, = $2, and the price of X, = P; = $4. What is the equation for Vibha's budget constraint? а. b. |How many shirts, pounds of food, and square feet of shelter, will Vibha consume, as a rational consumer?1) Suppose that a person consumes two goods, x and y, in fixed proportions. He or she always consumes 1 unit of x together with 3 units of y no matter what the relative prices are.a) What is the mathematical form for this person's utility function?b) Calculate the Marshallian demand functions for both goods for this person.c) Calculate the indirect utility function and the expenditure function for this person.d) In class we discussed why expenditure functions are concave in prices. Is the expenditure function you calculated in part (c) concave in ???
- 1. Kofi consumes two goods (x) and (y). The following utility function represents his preferences: u(x, y) = 0.4 ln x+0.6 ln y Br Paxt by Y Suppose the prices of the two goods (x and y) are p, and p, and his budget is M. You are required to do the following: a) Specify Kofi's budget set. - b) Derive the Marshallian demand function for each of the two commodities c) Derive the indirect utility function and verify the Roy's identity. d) How many units of each of the two commodities will be consumed by Kofi if his income is GHS100.00 and p. = 1; p, = 2?Utility maximization under constraint, substitution and income effect, CV and EV Josh gets utility (satisfaction) from two goods, A and B, according to the utility function U(A, B) = 4[A-4 + B-4-4 + 40. While Josh would like to consume as much as possible he is limited by his income. a. Maximize Josh's utility subject to the budget constraint using the Lagrangean method b. Suppose PA increases. Show graphically the income, substitution effect and total effect and explain c. Suppose PA increases. Show the graphs for CV and EV and explain. One graph for CV and one graph for EV6. Find the indirect utility function and the expenditure function for the Cobb-Douglas utility function.