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- For the following utility function find the MRS at (x, y) = (2, 3) a) U(x, y) =2x +y b) U(x, y) =x^3 y^2 c) U(x, y) =min(x, 2y) Solve subparts (c )correctlywhat is the derivative of the utility function UU = log (100-2PPii) + log(PPjj+PPdd) with respect to PPjj. Budget constaint XXii = 100 - 2PPiiPlease use Kuhn Tucker 5. A consumer's utility function is: u(x) = x1+ /®2 (c) Verify the properties of the indirect utility function. (d) Calculate the expenditure function and verify its properties.
- Paige has the utility function U(x, y) = ln(x) + 4y Do not worry about corner solutions when answering the following questions, you can use the Lagrangian multiplier method. Paige's budget constraint is px x +Pyy = IConsider the following indirect utility function: v(p, w) = w2 /4p1p2 How can we recover the direct utility function (u(x))?A consumer is maximising her utility function: U(r. y) = (r05 +y0.5)2, subject to the budget constraint 4x + 2y = 108. (a) Set up the Lagrangian function of this utility maximisation problem and first-order conditions. (b) What are the utility maximizing amounts of x and y? Also, calculate the multiplier. (c) What are the utility maximising amounts of x and y if the budget constrain to x+ 2y = 36? How would change? Explain your reasoning. (Hint: Yc need to calculate A, rather comment on how it would change and why.)
- 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 consume1. Prove if the indirect utility function is quasiconvex The indirect utility function: V(p,w) = w[P1^(p/p-1) + P2^(p/p-1)]^((1-p)/p)Suppose you have the following indirect utility function: 72 V(Px, Py, I) = In (. PxPy
- Consider the following utility functions. G(x,y) = -1/[ min (6x, 2y)+1] H(x,y) = min (6x,6y) L(x,y) = min (6x,2y) - 10000000000 U(x,y) = min (3x, y) W(x,y) = min (6x, y) Z(x,y) = min (x,2y) Select the function or functions, if any, representing the same preferences as function U?A utility function is given by the equation U = 20xe¬0.1x, where x is the number of glasses of wine consumed. (a) Show that this utility function has a maximum value and calculate the maximum utility. (b) Describe how marginal utility changes for glasses of wine consumed after the maximum utility is reached. Do you consider this reasonable? Give an explanation.A consumer has the following utility function: U(X,Y)= 400X2/5Y3/5 Suppose that the price of a spa day (x) is £200 and the price of a city break (y) is £300 Find the number of spa days and city breaks that minimise expenditure if 12,800 units of utility are obtained. Also calculate the consumer's minimised expenditure