Lexicographic Preferences: Suppose that a consumer has lexicographic preferences over bundles of non-negative amounts of each...

Lexicographic Preferences: Suppose that a consumer has lexicographic preferences over bundles of non-negative amounts of each of two commodities. The consumer’s consumption set is R2+. The consumer weakly prefers bundle a = (a1,a2) over bundle b = (b1,b2) if either (i) a1 > b1, or (ii) both a1 = b1 and a2 ? b2. In any other circumstance, the consumer does not weakly prefer bundle a to bundle b. (Note that these preferences are not continuous. Furthermore, they cannot be represented by a utility function.)

1. Under what circumstances will bundle c be strictly preferred to bundle d?

2. Under what circumstances will bundle c be indifferent to bundle d?

3. Are these preferences weakly complete? Explain why.

4. Are these preferences reflexive? Explain why.

5. Are these preferences strongly complete? Explain why.

6. Are these preferences transitive? Explain why.

7. Are these preferences rational? Explain why.

8. Show that these preferences are strongly monotone. (Note that this means that they are also monotone and locally non-satiated.)

9. Show that these preferences are strictly convex. (Note that this means that they are also convex.)

10. More challenging and non-examinable: Show that these prefer- ences are not continuous.

Solutions

Expert Solution

Answer (1): Bundle C will be strictly preferred to bundle D if c1>d1 and c2>d2.

Answer (2): Bundle C will be indifferent to bundle D if c1>d1 and c2<d2.

Answer (3): These preferences are weakly incomplete, as these do satisfy some other situations such as what will happen when c1=d1 or c2=d2, or c1<d1 and c2>d2.

Answer (4): The preference are reflexive, as they reflects the nature of c1 and d1. These preferences do not interfere.

Answer (5): No, these preferences are not strongly complete, as they do inform as to what will happen when c1=d1 or c2=d2, or c1<d1 and c2>d2.

Answer (6) : These preferences are transitive, as they safely keep distance from each other and do not get intermixed.

Answer (7): The preferences are rational, as they explain logical and true nature of c and d or a and b

Answer (8): Since, these preferences give us the information of one type, i.e. who is greater than who, or who is lesser than who, therefore these information are all of one type. Hence they are monotonous in nature.

Answer (9): Convex nature implies that the preferences are bounded in a certain group or collection of ideas. Since all these preferences are of one type and monotonous, they can be grouped under one collection, i.e. who is greater than who or who is lesser than whom. Hence these are also convex in nature.

Answer (10): Continuous preferences are always of one nature or one type such as a>b, b>c, c>d etc. Here, these preferences are not continuous because they also show pattern where b>c or c>d. Therefore these preferences are not continuous.

Rahul Sunny answered 1 year ago

Next > < Previous

Related Solutions

Consumer Theory. A consumer has preferences over goods x and y that can be represented by...

Consumer Theory. A consumer has preferences over goods x and y that can be represented by the utility function ?(?,?) = ?+??(?) where ln is the (natural) log function. The consumer has income I (all to be spent on x and y) and the price of x and y are px and py respectively. (You may assume the “at least as good as x” set B(x) is a convex set, so the solution to the consumer’s problem will be a...

Lyuba has preferences over peonies (p) and tulips (t). The following list shows all feasible bundles...

Lyuba has preferences over peonies (p) and tulips (t). The following list shows all feasible bundles that Lyuba can consume in a month: A = (pA,tA) = (1,3), B = (pB,tB) = (3,1), C = (pB,tB) = (2,2). If Lyubas preferences are given by: A is indifferent to B, C is strictly preferred to A and C is strictly preferred to B, then: (a) A function that represents Lyuba’s preferences is U(p,t)=3p+t. (b) A function that represents Lyuba’s preferences is...

(a) Suppose our consumer has two possible consumption bundles, one with 1 unit of clothes and...

(a) Suppose our consumer has two possible consumption bundles, one with 1 unit of clothes and 5 units of food, the second with 3 units of clothes and 4 units of food. For which is the MRS of clothes for food the highest? (b) For the example in (a), what property of indifference curves tells us which will have the highest MRS of clothes for food? (c) Suppose your income is $100. The price of food is $15, and the...

Randy consumes two goods: X and Y. Randy’s preferences over consumption bundles (X,Y) are summarized by the...

Randy consumes two goods: X and Y. Randy’s preferences over consumption bundles (X,Y) are summarized by the utility function: u (X,Y) = X3Y. Write algebraic expressions for Randy’s demand functions for goods X and Y to be If PX= 1, PY= 1, and m = 100, what would be Randy’s optimal consumption of goods X and Y? Suppose now that the price of X rises to 3, while the price of Y and income remain unchanged. What is Randy’s new optimal consumption...

Consider a one period economy in which the representative consumer has preferences over leisure (l) and...

Consider a one period economy in which the representative consumer has preferences over leisure (l) and consumption (c) described by the utility function u(c, l) = c0.5l0.5 This consumer has 1 unit of time h to spend between leisure, l, and labor supply, Ns. The representative firm’s production function is Y = zNd where Nd is labor demand and z = 2 is total factor productivity. The government buys one unit of consumption good, meaning that, G = 1. (a)...

Ethel has preferences over amounts of goods 1 and 2 represented by the utility function u(x1,...

Ethel has preferences over amounts of goods 1 and 2 represented by the utility function u(x1, x2) = (x1)^2 + x2, where x1 denotes how much of good 1 she has and x2 denotes how much of good 2 she has. Write an expression for Ethel’s marginal utility for good 1. Does she like good 1? Explain your answer. Write an expression for Ethel’s marginal rate of substitution at any point. Do Ethel’s preferences exhibit diminishing marginal rate of substitution?...

Suppose a consumer has preferences represented by the utility function U(X,Y) = X2Y Suppose PY =...

Suppose a consumer has preferences represented by the utility function U(X,Y) = X2Y Suppose PY = 1, and the consumer has $300 to spend. Draw the Price-Consumption Curve for this consumer for income values PX = 1, PX = 2, and PX = 5. Your graph should accurately draw the budget constraints for each income level and specifically label the bundles that the consumer chooses for each income level. Also, for each bundle that the consumer chooses, draw the indifference...

Consider the case of Timmy, a consumer with preferences over oranges (O) and Sauerkraut (S) that...

Consider the case of Timmy, a consumer with preferences over oranges (O) and Sauerkraut (S) that give him a utility function U=ln(O) + S. A. Given that Timmy has an allowance of I and faces prices of Po and Ps, use the Lagrangian Multiplier method to find his optimal consumption of O and S. B. Demonstrate whether Sauerkraut is a normal or inferior good for Timmy. C. Find Timmy’s indirect utility function. In a world where I=20, Po = 1...

"Suppose a consumer has preferences represented by the utility function U(X,Y) = X(^2)Y Suppose Py =...

"Suppose a consumer has preferences represented by the utility function U(X,Y) = X(^2)Y Suppose Py = 1, and the consumer has $360 to spend. Draw the Price-Consumption Curve for this consumer for income values Px =1, Px = 2, and Px = 5. Your graph should accurately draw the budget constraints for each income level and specifically label the bundles that the consumer chooses for each income level. Also for each bundle that the consumer chooses, draw the indifference curve...

Suppose a consumer has preferences given by U(X,Y) = MIN[2X,Y]. Suppose PX = 1 and PY...

Suppose a consumer has preferences given by U(X,Y) = MIN[2X,Y]. Suppose PX = 1 and PY = 2. Draw the Income Consumption Curve for this consumer for income values • M = 100 • M = 200 • M = 300 To do this, carefully draw the budget constraints associated with each of the prices for good X, and indicate the bundle that the consumer chooses in each case. Also, be sure to label your graph accurately.

Subjects