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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
*What does v mean in this question? Also, could you solve this problem without using Lagrange multipliers? Thank you.
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- The utility maximization problem is given by 0-1 Ө-1\0-1 max uf = Cit,C2t,St 0 + a, (c2t) subject to Cit + St = Wt + e C2t = (1+ rt+1)s, By solving the maximization problem, characterize the saving function depending on the value of 0, i.e., there are three cases.When Amanda wakes up in the morning, she will either purchase Timbits (T) or a bagel (B). This is represented by the utility function U(T, B) = 15T + 10B. %3D In addition, the price of a bagel is $2 and the price of Timbits is $0.50 apiece. Amanda has a daily budget of $5 to spend on breakfast. She is planning to purchase 1 Bagel (B) and 6 Timbits (T). Is this allocation of goods utility-maximizing? a) Yes, since MUt/Pt = MUb/Pb b) No, Amanda should only consume Timbits c) Yes, since this bundle lies on Amanda's budget line d) No, Amanda should only consume bagels1.2) Suppose that a consumer’s utility function is U=10 lnx+20 lny. A) Find the marginal utility of x, MU_x, and the marginal utility of y, MU_y.B) Suppose that the consumer has R3600 to spend on x and y, while p_x=200 and p_y=400. C) Use the Lagrange multiplier (LM) method to find the levels of x and y that will maximise the consumer’s utility, subject to her budget constraint. D) Find and interpret the value of the Lagrange multiplier. Show that MRCS=p_x/p_y at the utility maximising levels of x and y.
- Solve the following utility maximization problem, max u (x1, x2) subject to Pix1 + p2x2 = I meaning find x (P1, P2, I) and x (p1, p2, I) and u* (p1, p2, I) = V * . In solving this optimization problem, assume the following functional form for the direct utility function, и (х1, X2) —а: · In(x1) + b· In(x2) where a and b are just parameters (i.e., representations of single numerical values). Assume that the Second-Order Conditions for a constrained maximum hold. Write clearly and show your work!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…Zeynep is at the supermarket buying her bi-weekly groceries. As she arrived at the store right before closing, not much is left on the shelves; so her purchases are limited to beef, B, and an assortment of organic vegetables, V. Beef costs £50/kg and vegetables Ł10/kg. Her utility function is expressed as U(B,V) = B0-6v0.3. She has ±150 to spend. Please answer the following questions using this information: A) Given the vegetables cost less than beef, explain why Zeynep would not only buy vegetables. B) Write Zeynep's constrained maximization problem (objective function subject to her budget constraint). C) Using the Lagrangian technique, find how much beef and vegetables would Zeynep buy? D) Show that for the preferences represented as above, demand for beef is a function of only its own price, pg, and income, m, but not the price of vegetables, py. Note: Rather than using the given prices and income, use PB, Py and m. Find and comment on the comparative statics, i.e., how do changes…
- The utility derived by a consumer from the consumption of two commodities is given by the function U (A, B) = 0.5In (A) + 0.5 In (B) where A are the number of units of the first commodity consumed and B are the number of units of the second commodity consumed each month. A unit of the first commodity costs $8 and a unit of the second commodity costs $ 4 using the Lagrange multiplier method determine the optimal quantity of each of the commodities consumed each month given that consumer has $32 to spend on both commodities each monthSolve the following utility maximization problem, и (х1, х2) max subject to P1x1 + p2x2 = I meaning find x" (p1, p2, I) and x (p1, p2, I) and u* (p1, p2, I) = V* . In solving this optimization problem, assume the following functional form for the direct utility function, u (x1, x2) = a · In(x1) + b· In(x2)2. Consider a consumer whose preference relation over the consumption set X = R can be repre- sented by the utility function: u (11, 12, 13) = A (x1 – #1)° (x2 – #2)® (r3 – 3)" where A, a, B and y are all strictly positive. (a) Why can you assume A = 1 and a + 3 + y = 1 without loss of generality? Do so for the %3D rest of the question. (b) Solve the UMP and derive the consumer's Walrasian demand. (c) Find the indirect utility function, the minimum expenditure function, and the Hicksian demand function.
- of 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 consumeProblem 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).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.
