Which one of the statements below is false for Cobb-Douglas preferences over bundles of x1 and X2 with prices p1 and p2, respectively? a) The optimal bundle is always interior for a positive income, m. b) The Marshallian demand function for x; depends both on p; and pj, j i. c) These preferences are strictly convex and strictly monotone. d) The Engel curves for x, and x2 are positively sloped.
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- Further, provide arguments for oragainst the following statement: “The optimal bundle chosen by a consumer with welldefined preferences (rational, continuous, strictly monotone and strictly convex) alwayscontains luxury goods.”Max’s preferences can be represented by the utility function u(x1, x2) = 2√ x1 + x2. Take good 2 to be the numeraire (p2 = 1). a) For any price p1 and income m, write down and solve his utility maximization problem, i.e. find the demand x1(p1, m) and x2(p1, m) using the Lagrangian method (assuming an interior solution). Illustrate why this method is equivalent to using the two conditions (1) |MRS| =price ratio and (2) Budget constraint. b) In doing so, also solve for λ ∗ . c) Given any p1 and income m, find the utility of Max corresponding to his demand. Denote this by v(p1, m). That means, find v(p1, m) = u(x1(p1, m), x2(p1, m)). d) To understand how v(p1, m) changes with m, i.e. compute ∂v(p1,m) ∂m . Compare with your answer to b).Q4. Explain with the aid of well labelled diagram(s) whether this statement is True, False or Uncertain, given the information provided below:“Two goods, A and B are perfect substitutes with prices $PA and $PB respectively. The income of an individual is given by $Y. Initially, $PA = $PB.It is claimed that in a particular situation where $I is assumed to remain unchanged and if due to some exogenous factor, $PB < $PA, the budget line will pivot, become flatter and optimal bundle will be a corner solution (0,B*) and no other possibility would exist.
- Suppose Coke and Pepsi are perfect substitutes for me, and right and left shoes are perfect complements. A. Suppose my income allocated to Coke/Pepsi consumption is $100 per month, and my income allocated to right/left shoe consumption is similarly $100 per month. a. Suppose Coke currently costs $0.50 per can and Pepsi costs $0.75 per can. Then the price of Coke goes up to $1 per can. Illustrate my original and my new optimal bundle with "Coke" on the horizontal and "Pepsi" on the vertical axis. b. Suppose right and left shoes are sold separately. If right and left shoes are originally both priced at $1, illustrate (on a graph with "right shoes" on the horizontal and "left shoes" on the vertical) my original and my new optimal bundle when the price of left shoes increases to $2. c. True or False: Perfect complements represent a unique special case of homothetic tastes in the following sense: Whether income goes up or whether the price of one of the goods falls, the optimal bundle…Suppose that Monica has income of $120, and faces prices Px = $2.9 and PY = $2.5, where x = number of cups of Starbuck coffee and Y = number of cups of MacCafe. Her goal is to maximize her utility, described by the function U=2x+ Y. a) Represent graphically the problem Monica faces. Make sure you identify your axes as well as your budget constraint and indifference curves. Also identify the optimal bundle. b) Calculate the optimal bundle. Make sure you show all your calculations.Amy chooses between two goods, x and y, with prices px and py, respectively. She has an income I and her preferences are represented by the utility function U(x, y) = √x + y. 1. Assuming that an interior solution exists to the constrained utility maximization problem, derive Amy ’s Marshallian demand function for each of the two goods. Are both goodsnormal? Explain 2. Find the indirect utility function, V (px, py, I). 3. Derive Amy’s Hicksian demand function for each of the two goods and the expenditure function. Compare the Marshallian demand for good x and the Hicksian demand for good x. Arethese different functions? If so, why? If not, why not? 4. Suppose that I = 100, px = 1 and py = 2. How much of good x and good y will Amy optimally choose? 5. Now the price of good x rises to px = 2, while income (I = 100) and the price of good y, py = 1, remain unchanged. What quantities does Amy buy and what is her resulting utility?Illustrate graphically 6. Find the income and substitution…
- The preferences of a typical Californian can be represented by the following utility function: U (x1 , x2 ) = α ln(x1) + (1 − α) ln(x2) Here, x1 and x2 are the quantities of electricity and gasoline, respectively. The consumer faces prices given by p1 and p2 and has income m. Currently, the government has decided to impose a consumption restriction so that any person in the state is allowed to consume at most 50 units of electricity (x1 ≤ 50). Call this restriction a rationing constraint. (a) If α=0.25, m=100,and p1 =p2 =1, find the optimal consumption bundle of gasoline and electricity. Is the electricity rationing constraint binding (meaning does x1∗ = 50)? (b) Suppose that α = 0.75, but the other parameters are the same. What is the optimal consumption bundle? Is the rationing constraint on electricity consumption binding? (c) Now, assume that there is no rationing constraint. Assume m = 100 and p1 = p2 = 1, but α remains as a generic parameter. Solve for the optimal quantity…Suppose that your tastes over coke(X1) and burgers(X2) can be summarized by the utility function: u(X1, X2) = (X12X2)1/3. (a) Calculate the optimal quantity of coke and burgers' consumption as a function of P1, P2and I. (b) Illustrate the optimal bundle A when P1 = 2, P2 = 10 and weekly income I = 180. What numerical label does this utility function assign to the indifference curve that contains bundle A? (c) Using your answer, show that both coke and burgers are normal goods when your tastes can be summarized by this utility function. (d) Suppose the price of coke goes up to $4. Illustrate your new optimal bundle and label it, C.Suppose that the utility function for two commodities is: U(q1, q2) = q1α q2(1-α) Let the prices of the two commodities be p1 and p2 and let the consumer’s income be M. (1) Check the properties of marginal utilities. In particular, check whether it satisfies diminishing marginal utilities. (2) Assuming all income is spent on these two commodities, derive the demand curves for the two commodities. (3) What happens if U(q1, q2) = q1α q2β?
- 1. Consider a consumer who chooses bundles consisting of two commodities, x and y. Suppose that, regardless of the prices px and py (which are always positive), the consumer chooses to consume x and y in a ratio of 2:1. This behavior is consistent with: a. A consumer having Cobb-Douglas utility function. b. The consumer’s utility function exhibiting perfect complements. c. The consumer’s utility function exhibiting perfect substitutes. d. Any of the above could be correct.Suppose that a fast-food junkie derives utility from three goods-soft drinks (x), hamburgers (y), and ice cream sundaes (z)− according to the Cobb-Douglas utility function U(x,y,z)=x0.5y0.5(1+z)0.5. Suppose also that the prices for these goods are given by px=1,py=4, and pz=8 and that this consumer's income is given by I=8.a. Show that, for z=0, maximization of utility results in the same optimal choices as in Example 4.1 . Show also that any choice that results in z>0 (even for a fractionalz ) reduces utility from this optimum.b. How do you explain the fact that z=0 is optimal here?c. How high would this individual's income have to be for any z to be purchased?Consider a consumer whose utility function is U(q1,q2) = q1*q2. This consumer has $100 and prices of two goods are p1 = $1 and p2 = $2.a) Draw indifference curves corresponding to utility levels 1 and 4.b) Now consider utility function U(q1,q2) = (q1*q2)2. Check that it generates same indifference curve map as initial utility function and JUSTIFY this result.(c) Calculate marginal utilities of both goods and COMMENT on their properties.d) Calculate marginal ratio of substitution (MRS) at points (1,1) ; (1/2,2) ; (2,1/2) ; (2,2) ; (3,1) and (1,3) and INTERPRET results. THANKS.