Consider a consumer with wealth w who consumes two goods, which we shall call goods A and B. Let the amount of good that the consumer consumes be xe and the price of good l be pe. Suppose that the consumer's preferences are described by the utility function: u(XA,XB) = ailog xA+ a²log XB where the at are some fixed strictly positive parameters and all logs are natural logs. a. Set up the utility maximisation problem and write down the Lagrangian. b. Write down the first order necessary conditions for an interior maximum. c. Solve the first order conditions to obtain the Marshallian demand functions
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- The vending machine in Katherine's office building offers cans of pop and candies. Katherine's utility function is U = 3PC, where P is the amount of pop consumed per %3D week and C is the amount of candy consumed per week. Pop costs $1 and candy costs $0.5 per bag. If Katherine has $10 to spend, she will consume A bags of candy.(b) It is thought that a consumer measures the utility, u(x, y), of possessing a quantity, z, of apples and a quantity, y, of oranges by the formula u(x, y) = rªyla, where a is a constant. Suppose that apples cost p dollars each and oranges q dollars each. If the consumer has M dollars to spend on apples and oranges, use the method of Lagrange multipliers to find the quantity of each that he will buy in terms of a, p, q and M. [You are not required to justify the use of the method of Lagrange multipliers here.] For equally priced apples and oranges when he has one dollar to spend, he is actually observed buying one apple and two oranges. What value of a does this imply?On a given evening, J. P. enjoys the consumption of cigars (c) and brandy (b) according tothe functionU(c, b) = 20c− c²+ 18b − 3b²a. How many cigars and glasses of brandy does he consume during an evening? (Cost isno object to J. P.)b. Lately, however, J. P. has been advised by his doctors that he should limit the sum ofglasses of brandy and cigars consumed to 5. How many glasses of brandy and cigarswill he consume under these circumstances?
- 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.Let say you have 3 products: cake, candy, and bread. The revenue per cake is $8. The revenue per candy is $4. The revenue per bread is $5. Define the number of cake, candy, and bread you are about to produce per day as x, y, and z respectively. The manager wants that the number of candy is at least equal to the total number of cake and bread. What is the correct linear constraint to show this? Group of answer choices z – x – y ≥ 0 x – 2y – z ≥ 0 y – x – z ≥ 0 x – y – z ≥ 0of 55 The vending machine in Katherine's office building offers cans of pop and candies. Katherine's utility function is U = 3PC, where P is the amount of pop consumed per week and C is the amount of candy consumed per week. Pop costs $1 and candy costs $0.5 per bag. If Katherine has $10 to spend, she will consume
- Suppose that the preferences of a consumer regarding the consumption of goods 9₁ and q2 are represented by function: 1-α U(9₁, 92) = 91 92 where q₁ ≥ 0, 9₂ ≥ 0, 0 0 and faces prices p₁ and p2 respectively for goods 9₁ and 92. a. Draw a diagram reporting the indifference curves representing the preferences of the consumer and the budget line. Also, derive and describe the demand functions of the consumer for goods 9₁ and 92. Draw the diagram for the (Marshallian) demand function of good 92. b. Consider a reduction in price p2. Drawing and describing the indifference curves and budget constraint, and referring to the concepts of substitution effect and income effect, report how the quantity demanded of q2 changes when p2 decreases. c. Consider now the dual problem. Specifically, minimise the expenditure function subject to the consumer achieving a level of utility ū> 0 and derive the Hicksian demands for goods 9₁ and 92. Report and describe the expression of the minimised expenditure…Ginny likes to eat Bertie Bott’s Every Flavour Beans (b), and Chocolate Frogs (f). Herpreferences over these lollies are Cobb-Douglas, and can be presented by the utility functionu(b, f) = 1/5(b − 2)(f + 3). Each Every Flavour Bean costs 25 Knuts and each Chocolate Frogcosts 12 Knuts. Ginny has 314 Knuts to spend. Assume that beans and frogs are both perfectlydivisible goods.(a) Find Ginny’s optimal bundle.(b) Suppose that Ginny is mugged by Draco on her way to the lolly cart, so that by the timeshe is there, she is left only with 75 Knuts. What is her optimal bundle now?(c) Now, consider the general case, where Ginny’s income is m, the price of Every FlavourBeans is pb, and the price of Chocolate Frogs is given by pf . In other words, treat ChocolateFrogs as the numeraire. Find Ginny’s demand for each good in terms of m, pb and pf .(d) Find the good on which Ginny will always spend more than half of her income.(e) Find the own-price, cross-price, and income elasticities of good 2.Ginny likes to eat Bertie Bott’s Every Flavour Beans (b), and Chocolate Frogs (f). Herpreferences over these lollies are Cobb-Douglas, and can be presented by the utility functionu(b, f) = 1/5(b − 2)(f + 3). Each Every Flavour Bean costs 25 Knuts and each Chocolate Frogcosts 12 Knuts. Ginny has 314 Knuts to spend. Assume that beans and frogs are both perfectlydivisible goods.(a) Find Ginny’s optimal bundle.(b) Suppose that Ginny is mugged by Draco on her way to the lolly cart, so that by the timeshe is there, she is left only with 75 Knuts. What is her optimal bundle now?(c) Now, consider the general case, where Ginny’s income is m, the price of Every FlavourBeans is pb, and the price of Chocolate Frogs is given by pf . In other words, treat ChocolateFrogs as the numeraire. Find Ginny’s demand for each good in terms of m, pb and pf .(d) Find the good on which Ginny will always spend more than half of her income.(e) Find the own-price, cross-price, and income elasticities of good 2…
- A 2015 report by the music industry estimated the revenue lost to the industry every yearfrom illegal downloading. In this problem we will derive some of the estimates that may havegone into their calculation (approximately).First, start with the individual consumer’s problem. Suppose a typical consumer has a yearlyentertainment budget of I that they can allocate between music downloads (D) and otherforms of entertainment (E). Consumer preferences are characterized by a utility functionU(D, E). a.) Write an expression for the consumer’s budget constraint as a function of their entertainment budget and the prices of music downloads (Pd ) and other entertainment (Pe). (b) Write the consumer’s constrained optimization problem in Lagrangian form. (Note: Youdo not need to solve it or derive first order conditions.)(c) If the preferences are concave, will the consumer ever consume both of the goods together?1. A standard model of choice under risk is Expected Utility Theory (EUT) in which preferences over lotteries that pay monetary prizes (x₁, x2, ..., xs) with probabilities (P1, P2, ..., Ps) with Eps = 1 are represented by the function L S (a) What does it mean to say that a function represents the consumer's prefer- ences? Σpsu(xs) Choice 1 8=1 (b) State and briefly comment on the axioms required for the EUT representation. (c) Consider the following experiment of decision making under risk in which sub- jects are asked which lottery they prefer in each of the following two choices: Lottery B 0 with prob. 0.01 10 with prob. 0.89 50 with prob. 0.10 Lottery D Choice 2 Lottery A 0 with prob. 0 10 with prob. 1 50 with prob. 0 Lottery C 0 with prob. 0.90 10 with prob. 0 50 with prob. 0.10 Suppose that the modal responses are Lottery A in Choice 1 and Lottery D in Choice 2. Assume that utility of zero is equal to zero and illustrate why it is not possible to reconcile these experimental…