Indicating consumption sider an agent who want to maximize her duty function, 6₁, 62) - ₁2 subject to budge respectively, while p>0 and w>0 are exogenously given parameters, indicating the relative price of good 2 and the relative wealth in term of good 1's price. Decide whether the following claims are true or false. W The value of the Lagrange multiplier is λ = 2p O False O True The value function u(w) is equal to u(w) = w² 4p O True O False (w) measures the approximate change in utility that can be obtained from an increase/decrease in wealth w. O True O False modity and
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- Rafe is optimally choosing to consume 6 apples and 3 bananas. The prices of apples and bananas are p. = 7 and pb - 7. Which of the following utility functions over quantities of apples (a) and bananas (b) could represent Rafe's preferences? = u(a, b)-a4/5 61/5 Ou(a, b) = a¹/5 64/5 Ou(a, b)-a2/3 b1/3 Ou(a, b)-a¹/3 2/3Problem 3 (10 points; 5 points for each part): Richard has a total utility function of: TU = 10q₁q₁²+24q2q2² +9192, where q₁ and q2 are goods. Richard is currently consuming at q₁ = 4 and q₂ = 6, which is also his utility maximizing consumption level. (Richard's optimal point is an interior solution). (a) How much additional utility would Richard receive if he were to increase his consumption of q2 from 6 to 7? (5 points). (b) How much q2 is Richard willing to pay (in terms of q2) in order to consume the 5th unit of q₁? (5 points).You were presented with a utility maximizing rule which states: If you always choose the item with the greatest marginal utility per dollar spent, when your budget is exhausted, the utility maximizing choice should occur where the marginal utility per dollar spent is the same for both goods. That rule is expressed as follows: Group of answer choices (The marginal utility associated with good 1 / the price of good 2) = (the marginal utility associated with good 2 / the price of good 1) % change in price / % change in quantity (The marginal utility associated with good 1 / the price of good 1) = (the marginal utility associated with good 2 / the price of good 2) The marginal utility per dollar of good 1 > the marginal utility per dollar of good 2.
- Suppose the total utility function for the consumer who is consuming two goods, good x and good z is U(x, z) = ln(2x) + 2lnz. The consumer’s income is $16 and is faced with the prices of each unit of good x and good z. The price per unit of good x is $4 and the price per unit of good z is $1. How many units of good x and good z should the consumer consume so as to maximize his/her utility? Also, what will be the total utility earned from the consumption of that many units of good x and good z?There are two goods, coffee and mineral water, available in arbitrary nonnegative quantities (so the consumption set is R2+). A consumer has preferences over consumption bundles that are represented by the following utility function:u(c, m) = min{c, m} + c + m, where c is the quantity of coffee (in grams) and m is the quantity of mineral water (in liters). The consumer has wealth in Dirhams of w > 0. The price of coffee is p > 0 (in grams/Dirham) and the price of mineral water q > 0 (in liters/Dirham). (a) In an appropriate diagram, illustrate the consumers map of indifference curves. Make sure you label the diagram clearly, and include as part of your answer any calculations about the slopes of the indifference curves. (b) Formulate and solve the consumer’s utility maximization problem. Your final answer should describe the consumer’s demand for coffee and mineral water as a function of the wealth w and the prices p and q, as well as the consumer’s indirect utility…There are two goods, coffee and mineral water, available in arbitrary nonnegative quantities (so the consumption set is R2+). A consumer has preferences over consumption bundles that are represented by the following utility function: u(c, m) = min{c, m} + c + m, where c is the quantity of coffee (in grams) and m is the quantity of mineral water (in liters). The consumer has wealth in Dirhams of w > 0. The price of coffee is p > 0 (in grams/Dirham) and the price of mineral water q > 0 (in liters/Dirham). Now assume that w = 100, p = 10 and q = 10. In addition to the monetarybudget constraint, the consumer has a time constraint. The consumer hasonly 100 minutes available for the consumption of both goods. It takes5 minutes/liter to consume water and and 20 minutes/gram to consumecoffee. (a) In an appropriate diagram, illustrate the consumer’s constraint set. Your diagram should illustrate the monetary budget constraint, the time constraint, and indicate the consumer’s overall…
- There are two goods, coffee and mineral water, available in arbitrary nonnegative quantities (so the consumption set is R2+). A consumer has preferences over consumption bundles that are represented by the following utility function:u(c, m) = min{c, m} + c + m, where c is the quantity of coffee (in grams) and m is the quantity of mineral water (in liters). The consumer has wealth in Dirhams of w > 0. The price of coffee is p > 0 (in grams/Dirham) and the price of mineral water q > 0 (in liters/Dirham). Now assume that w = 100, p = 10 and q = 10. In addition to the monetarybudget constraint, the consumer has a time constraint. The consumer hasonly 100 minutes available for the consumption of both goods. It takes5 minutes/liter to consume water and and 20 minutes/gram to consumecoffee. The government wants the consumer to consume an equal quantity of coffee and water. To achieve this policy objective, the government implements a tax of τ > 0 Dirhams/gram for coffee. This tax…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…Summer break is approaching! Suppose you derive utility from days spent traveling on vacation domestically, D, and days spent traveling on vacation in a foreign country, F. Your utility function over these two “goods” is: U(D, F) : 4D0.25F 0.75 Let your budget constraint be I(D, F) = pDD + pF F; where I is your income, pD is the price of domestic travel per day, and pF is the price of foreign travel per day. a. Determine the demand functions for domestic travel and foreign travel. Make sure you show your work – show the steps used. b. Suppose that you’ve saved $800 for your summer travel, the price of domestic travel per day is $25 and the price of foreign travel per day is $100. How many days of each type of travel will you embark on? c. Illustrate the indifference curve, budget constraint, and the utility maximizing bundle associated with (b). Make sure you show the level of utility, the budget constraint intercepts, and the optimizing equilibrium.
- If the utility function of an individual takes the form: U = U ( x 1, x2) = (x1 + 2) 2 (x2 + 3) 3 Where U is total utility, and x1 and x2 are the quantities of two commoditiies consumed: (a) Find the marginal-utility function of each of the two commodities (b) Find the value of the marginal utility of the first commodity when 3 units of each commodity are consumed.Khan lives in a world with two consumption goods x and y. Her utility function is U (x, y) = √x² + y². a. If px = $3,py = $4, and her income, I, is equal to $50, what will be the quantities of x and y that Maya should buy to maximize her utility? Make sure that you write out the Lagrangian and the first-order conditions. (Hint: It may be easier to maximize U² than U). Have you found a true maximum? Explain your answer.8. What is the optimal consumption bundle of the individual, whose utility function and income, along with market prices of two commodities, are given below? √√x1x2 ; 1 = U (x1, x2) = (x₁ = 30, x₂ = 15) (x₁ = 40, x2 = 15) 100; P₁ = 1; P₂ = 4 (x₁ = 50, x2 = 12.5) (x₁= 50, x2 = 20)