lives on an island where she produces two goods, apples (x) and bananas ccording to the production possibility frontier 200 = x² + y2, and she consumes all the herself. Her utility function is U (x, y) = xy³. Find her utility maximizing x and y as = the value of 2 (Lagrange multiplier).
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- Ruby has the following utility function: U(X, Y) = X^3/4 , Y^1/4, where X is her consumption of food items, with a price of $10, and Y is her consumption of clothing items, with a price of $30. She plans to spend $360 on food and clothing over the next week. Using the Lagrange multiplier technique, determine the number of food and clothing items that will maximize Ruby's utility.A consumer consumes two agricultural products: Red Meat, and Tomatoes according to the following utility function: U = RT That is, the total utility is the multiplication of the quantity consumed of the two products. Given that consumer's income is 210, price of R is 10, and the price of Tis 2, a)Write down the budget constraint (budget line equation) for this consumer. b)Determine the quantities that the consumer should consume of each of the two products c)Calculate the value of the lagrangian multiplier and derive the demand function for Red Meat and for tomatoes.Mr Banda faces the following utility function from consuming nshima [X] and Rice [Y] U = 3XY 1 3 Mr Banda’s budget allocation for nshima and rice is K240. The price of nshima is K60 per Kg while that of Rice is K20 per Kg. A. How much nshima and rice should Mr. Banda consume to maximize his utility and What is the total utility at the optimum? b. What is the Lagrange multiplier? c. What will be the increase in utility when the budget allocation is increased to K241?
- Michael does not like to mix peanut butter and jelly in the same sandwich. However, he will consume them separately; for him, a sandwich with 1 spoon of peanut butter is exactly the same as a sandwich with 2 spoons of jelly. Michael has an income of m = 50, and the prices per spoonful of peanut butter and jelly are pPB=5 and pJ=11. Please write down Michael’s utility function over peanut butter (PB) and jelly (J). 2. Please determine Michael’s Marshallian demands PB*m and J*m3. Please determine Michael’s new Marshallian demands PB*m and J*m, when the price of peanut butter falls to pPB = 1.4. What are the (Hicks) SE and IE? Draw a diagram to show your analysis, with peanut butter on thehorizontal axis, and jelly on the vertical axis. 5. Recall that there are two different types of substitution effects. For example, in Q2 we have used theHicks SE. Does your answer to the last part change if we use the Slutsky SE? Justify. 6. Nam likes his peanut butter and jelly sandwiches with exactly…1.2) 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.You are given the following utility function: ? = ?? The budget is K100 and the price of X is K2 while the price of Y is K5. a) Derive the demand for X and Y by the Lagrange multiplier method. b) What will be the demand when all the prices are doubled and the income is doubled? c) What is the utility when the budget is increased by K1?
- A consumer is maximising her utility function: U(x, y) = (x¹/³+y¹/³)³, subject to the budget constraint x + 3y = 100. (a) Set up the Lagrangian function of this utility maximisation problem and derive the first-order conditions. (b) What are the utility maximizing amounts of x and y? Also, calculate the Lagrange multiplier. (c) What are the utility maximising amounts of x and y if the budget constraint changes to x + 3y = 50? Also, calculate the Lagrange multiplier.I. A)MRS=1/3, B)MRS=1/4, C)MRS=4/3, D)MRS=12 II. A)Rhea should buy more bread, less milk, B)Rhea should buy more milk, less bread, C) Rhea maximizes her utility at bundle A, so she should not change her consumptions of either good, D) Rhea cannot afford bundle A, so she should buy less of both goods III. A)Rhea's MRS at bundle A is greater than the price ratio (Pb/Pm), B)Changing her bundle moves Rhea to a higher indifference curve, C)The slope of the indifference curve (where bundle A is found) is not equal to the slope of Rhea's budget constraint, D)All of the above statements explain your answer to part II.You have £20 per week to spend, and two possible uses for this money: telephoning friends back home, and drinking coffee. Each hour of phoning costs £2, and each cup of coffee costs £1. Your utility function is U(X,Y) = XY, where X is the hours of phoning you do, and Y the number of cups of coffee you drink. What are your optimal choices? What is the resulting utility level? You can use the standard result on the constrained maximization of such a function, but must state it clearly. Now suppose the price of telephone calls drops to £1 per hour. What are your optimal choices? What is the resulting utility level? How much income per week will enable you to achieve the same quantities at the new prices as the ones you chose before? What income will enable you to attain the same utility as you did before? Comment on your answer in the context of equivalent variation and compensating variation.
- 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 monthConsider an overtime rule that requires that workers get paid double for any weekly hours over 40. Draw a picture that shows how a worker decides how much to work. Label everything in your picture and explain what is happening. Consider an investment that costs $100 and pays back $10 each year as long as the person making the investment is alive. Construct an equation for the net present value of the investment. An individual has a utility function, U = AX1 X2 where X1 and X2 are consumption of goods 1 and 2. The individual also faces a budget constraint. Show mathematically how an increase in Aa§ects the individualís decisions about consumption of each good.Options: 1. A) Yes, you can still afford the bundle at your previous consumer equilibrium B) No, you now cannot afford the bundle at your previous consumer equilibrium C) It is uncertain whether or not you can still afford the bundle at your previous consumer equilibrium 2. A) For each peck of apples, you will buy 1/3 pounds of oranges B)For each peck of apples, you will buy 1 pounds of oranges C)For each peck of apples, you will buy 3 pounds of oranges D) For each peck of apples, you will buy 6 pounds of oranges