6. (a) Draw upon Sen's analysis to demonstrate that attainment of Pareto optimality is not possible in a society marked by interdependencies in utility functions and a commitment to the basic cardinals of “liberal values".
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- 18. Declining MRSXY implies that:(a) The total utility is decreasing along an indifference curve(b) The consumer’s preferences do not satisfy the more-is-better principle(c) The consumer is willing to give up more and more X for additional Y as the consumption of X increases along an indifference curve(d) The consumer is willing to give up less and less X for additional Y as the consumptionof X increases along an indifference curve15. The assumption that preferences are complete requires the consumer:(a) To rank any two bundles(b) To say bundle C is preferred to bundle A if bundle B is preferred to bundle A andbundle C is preferred to bundle B(c) To rank a bundle with more units of all goods higher than a bundle with fewer unitsof all goods(d) To have a diminishing marginal rate of substitution(c) Now suppose that the relevant budget constraint is I2. Would the individual choose B? Why? 3. Suppose that bundle A in the following figure is chosen by a rational individual (who respects the revealed preferences axiom) when the relevant budget constraint is Ij. (a) Is A revealed preferred to B? (b) Suppose that the relevant budget constraint is I3. Would the individual choose B? Why?
- 1. Multiple choice: Choose only one correct answere a) If a person's indifference curves can be represented as downward-sloping straight lines, the person views the goods as substitutes but not perfect perfect substitutes perfect complementse complements but not perfecte economic badse4. Consider a society that has three individuals - 1,2 and 3 - and three alternatives - x, y and z. The society has a preference aggregation rule F. In year 2007, the profile of individual preferences were as follows: 1 2 3 X y Z y Z V It is known that the rule F satisfies Completeness, Transitivity, Anonymity, Unanimity and Independence of Irrelevant Alternatives. Also, in year 2007, F ranked x strictly above y. Now, we are in year 2008. The individual preferences have changed to: 1 2 3 Z y y X Z Z y X x (a) How does the society rank between y and x in 2008? Why? (b) How does the society rank between x and z in 2008? Why? (c) How does the society rank between y and z in 2008? Why? (d) What were the social preferences over all three alternatives in 2007?3. [In class, we noted that the consumer will spend all their money if they have in- creasing preferences. Here, we give a calculus-based condition for the consumer to spend all their money, that doesn't assume more is always better. The con- dition is a stronger version of what is sometimes called "local non-satiation".] Let E X, and let MU, denote the marginal utility for good i € {1,2} at r. (a) Suppose some good i has MU, #0 (it could be positive or negative), and suppose the consumer is not spending all her money at bundle x (that is pr 0 if MU; <0. Argue that the consumer would not optimally choose r. x (b) Suppose MU₁(x) = MU₂(x) = 0, the good is affordable (p ≤ m), and u is concave. Explain why r is optimal.
- 2. The standard model of consumer behavior assumes that income is exogenous. Of course, a person's income usually depends upon the number of hours the individual works. Suppose that an individual has T hours each day which can be allocated toward working time, H, or leisure time, L-that is, T = H + L. The individual earns w dollars for each hour worked. Then the individual's income is M + wH where M denotes any non-labor income. The individual has preferences over leisure time, L, and a consumption good, X, which can be represented by a quasi-concave utility function, U(X, L). The consumption good, X, can be purchased at the price Px. The individual seeks to maximize the utility subject to the constraints on time and money. (1) Formulate the individual's problem as an optimization problem with two constraints a time constraint and a money constraint. Derive the first-order conditions. Give an economic interpretation of the Lagrange multipliers. (2) The two-constraints problem can be…3. Consider the following utility function: X2 if: 1 2 u(x1, x2) X1 if: 1< x2 < 2 X2 + x1 otherwise Answer the following questions. (a) What is the Walrasian demand for this utility function?/
- - Assume an individual has quasilinear preferences of the form U(x1,x2) = 3x1¹2 + 2x2. If the price of Good 1 is $1 and the price of Good 2 is $2, how many additional units will the individual consume of Good 2 if her income increases by $1, assuming we're at an interior optimum? A. B. C. D. The individual's consumption of Good 2 will increase by 1/2 unit. The individual's consumption of Good 2 will increase by 1 unit. The individual's consumption of Good 2 will decrease. The individual's consumption of Good 2 will not change.2(b)There are three people and two goods in an economy. The utility functions and the initial bundles are given below: U1 = X11X 12 U = 2x 21+ X 22 U3 = X 31 X 3z+X 32 W; = (0,8) W;= (2,2) Wy= (8,0) Suppose it is suggested that the three traders move to the allocation x given by x;= (1,2), x2= (8,4), X3= (1,4) b. Show that x makes no one worse off than the original allocation.2. Suppose there are n identical consumers, each with a utility function u(x, y): = ax = x²+y. Each consumer has different disposable income w. The price of good x is p dollars, and the price of composite good y is equal to 1 dollar. Assume a > 0 and b > 0. (Hint: Question 1) (a) Calculate the marginal utility of good x and good y. Then, calculate the marginal rate of substitution between good x and good y. (b) Calculate the individual demand of consumer 1 for good x. (c) Calculate the aggregate demand for good x. (d) Draw a graph showing the market demand curve for good x. (e) Discuss whether an increase in disposable income for each consumer lead to an increase in the consumption of good x? This increase in disposable income can result from various factors, such as a decrease in taxes, government stipends, or other forms of income support. (f) Discuss the effect on demand for good x if there is an increase in the price of good x. This increase in the price of good x can result from…