2. A consumer has the utility function U(x, y) = x0.6 0.4. The price of x is $6 and the price of y is $10. The consumer's income is $140. Use the Lagrangian method of constrained optimization to calculate how much of each good the consumer will demand
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- Ayana is pitching an idea for a startup company that makes and sells solar-powered phonechargers (C). Her market research has found that consumer demand for this product can beexpressed as a function of the price of the charger itself (PC), the price of phones (PF), andthe consmer’s income (I). Consumer demand can be described by the function C(PC, PF, I) =(i−10PC)/ (PF) Suppose her chargers come in all different capacities to meet any quantity demanded, so youdon’t need to worry about restricting C to whole numbers for this problem. (a) Does this product satisfy the law of demand?Explain.Ayana is pitching an idea for a startup company that makes and sells solar-powered phonechargers (C). Her market research has found that consumer demand for this product can beexpressed as a function of the price of the charger itself (PC), the price of phones (PF), andthe consmer’s income (I). Consumer demand can be described by the function C(PC, PF, I) =(i−10PC)/ (PF) Suppose her chargers come in all different capacities to meet any quantity demanded, so youdon’t need to worry about restricting C to whole numbers for this problem. (a) Does this product satisfy the law of demand?Explain. (b) Calculate consumers’ income elasticity as a function of the parameters. (c) Anaya is targeting a market whose income is expected to double over the next five years.Explain whether she should expect the demand for chargers to increase or decrease, andby how much, using the language of normal, inferior, luxury, and necessity goods. (d) Calculate the cross-price elasticity ofCwith respect toPF. (e)…Explain the correlation between marginal utility and consumer's demand curve
- The utility of a consumer derives from consuming the two goods X and Y can be assumed to be determined by the utility functionU(X,Y)= 4X03Y05. If good X costs 10 birr a unit and good Y costs 15 birr a unit and the consumer's income is 120 birr. What combination of X and Y that will maximize utility?Harry consumes 2 goods, X and Y and he spends N$60 per month. The price of good X is N$4 and the price of good Y is N$10. Harry's utility function is U(X, Y) = XY. (i) What is Harry's marginal rate of substitution? (5) (ii) What is the expression for Harry's budget constraint? (3) (iii) What is the slope of Harry's budget constraint? (2) (iv)Find the values of good X and Y maximizes Harry's utility. Show all your work. (6) (v) Illustrate Harry's utility maximizing combination on a clearly labelled graph. (4) (vi) Suppose the price of good X increased to N$6. Illustrate Harry's income and substitution effects on a well labelled graph. Show your work. (10) (vii) Suppose Genie is consuming two goods, coffee and beer. She spends all her income on a combination of coffee and beer where MUcoffee/Peoffee is 5 and MUbeer/Pbeer is 3. Explain why this combination is not maximizing her satisfaction. What should she do to maximize her utility? (5)A consumer with I dollars budget has the utility u(x,y) = x(y+1) over amounts of cake (x) and ice cream (y) she consumes. The prices are px , py respectively. (a) Derive her demand for cake (x), as a function of prices px , py and her budget I. (b) Looking at the demand function in (a), is cake a normal good or an inferior good? Are cake and ice cream complements or substitutes? (c) Calculate the (i) (own) price, (ii) income, (iii) cross- price elasticity of demand for cake at the point where I = 80, px =10, py = 20.
- Raymond consumes meatloaves and pineapples. He has decided that hismonthly budget will be $1500. Suppose that one meatloaf costs $375, while one pineapple costs $150. What is the expression for Raymond’s budget constraint? Draw a graph of Raymond’s budget line (meatloaf on the y axis).Would you expect marginal utility to rise or fall with additional consumption of a good? Why? b. Raymond consumes meatloaves and pineapples. He has decided that hismonthly budget will be $1500. Suppose that one meatloaf costs $375, while one pineapple costs $150.i. What is the expression for Raymond’s budget constraint? ii. Draw a graph of Raymond’s budget line (meatloaf on the y axis). iii. Show how Raymond’s budget line changes if the price of pineapplesincreases to $370. iv. Suppose Raymond decides to cut his monthly budget in half.Coincidentally, the next time he shops, he learns that meatloaves andpineapples are on sale for half price. Show what happens to Raymond’sbudget line? c. Shannon consumes dresses and lemonade. The price of a dresses is $10 and the price of a lemonade is $30. She is spending all her income on the two goods and her marginal rate of substitution of lemonade for dresses is 2. Is she at an optimum? ExplainAnatoly's marginal utility for DVDS (D) is MU X and his marginal utility for Xbox games (X) is MUx D. Anatoly's income is $120, the price of a DVD is $20.00, and the price of an Xbox game is $10.00. How many DVDS and Xbox games does Anatoly consume to maximize his satisfaction? Anatoly consumesDVDs andXbox games. (Enter your responses using integers.)
- Consider a consumer that consumes 2 teaspoons of sugar with each cup of coffee. For each cup of coffee with sugar the consumer gains 10 utils. a)Write down the utility function that gives the total utility if the consumer consumes S teaspoons of sugar and C cups of coffee. The consumer has assigned £7 per week to be spent on drinking coffee with sugar. The current price of coffee is £0.50 per cup and each spoon of sugar costs £0.10. b)Calculate the optimal weekly consumption bundle for this consumer. c)Does the consumer view C and S as complements or substitutes?To help with his busy schedule, Joe drinks a lot of cappuccinos. In particular, his weekly demand for cappuccinos is: cap = 0.1 M-8 Pcap Where Xcap is the number of cappuccinos drunk per week, Mis Joe's weekly income and Pcap is the price of a cappuccino. (Note that Xcop is an average demand and need not be a whole number). (a) Suppose that income M = 240 euro, and the price is Pcap = 1.5 euro. How many Cappuccino, on average, will Joe drink per week? Suppose that the price increases to Pcap = 1.6 euro. (b) What is Joe's new demand for cappuccinos? What is his total change in demand due to the price change? (c) Given the new price (Pcap = 1.6), how much extra income would Joe need to allow him to continue buying the number of Cappuccinos calculated in part (a), without needing to reduce his demand other goods? (In other words, how much would Joe's expenditure on cappuccinos have gone up if he had continued to buy the number of cappuccinos calculated in part (a)?) (d) Compute the change…1 Alice has a monthly lunch budget of 360 dollars which she spends on either burgers (B) or pizza (Z). The price of each pizza is 16. dollars and the price of each burger is 8 dollars. Alice's utility is given by: u(B, Z) = B'/2 +4Z!/2 + 47!/2 a) Find Alice's budget constraint. Set up the Lagrangian and find her optimal consumption. What is her new consumption if the price of burgers becomes 16 dollars?
