ECO204Y1 Chapter Notes - Chapter 2: Convex Preferences, Utility, Weighted Arithmetic Mean

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14 Oct 2017

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Infinite utility functions for one preference due to pmts. Use mrs to gauge intensity of preferences at arbitrary bundle. Sum of utility from good 1and good 2. Mu1 = , mu2 = : mrs = - / . Convex preferences: any bundle c that is weighted average of bundles a and b where a ~ b is indifferent. =1: good 1 is equally preferred to good 2. >1: good 1 is preferred more than good 2. <1: good 2 preferred more than good 1. Q2: mu1 = q1, mrs = - q2 / q1, q2 = (u/q1, when both goods good: Interior bundles are preferred: both goods are necessary, strictly convex preferences, mrs not constant. U = (q1: u = lnq1 + lnq2. Complements (leontief) utility function: u = min( q1, q2, good as long as , > 0, q1 = q2.

Related Questions

PROBLEM 1 (This one is designed to give you lots of math practice!)

Consider the following individual utility functions:

Adam Smith: U(x,y) = xy

Jeremy Bentham: U(x,y) = x^ay^b (Cobb-Douglas utility function)

Alfred Marshall: U(x,y) = ln x + ln y

John M. Keynes: U(x,y) = x + y^b (Quasi-linear utility function)

Joan Robinson: U(x,y) = aX + bY (linear; perfect substitutes)

a. Solve for the MRS(XY) for each of the gentlemen (i.e., all but Joan Robinson). Show that Mr. Smith, Mr. Bentham, and Mr. Marshall all have diminishing MRS(XY), but that if a=b=2, their preferences exhibit constant, increasing, and decreasing marginal utility respectively. (Fortunately, diminishing MRS is the condition that guarantees convex indifference curves, so we can solve for their optimal consumption!)

b. What is the general formula for Mrs. Robinsons (constant) MRS?

c. Find demand functions for x and y for Smith, Bentham, Marshall, and Keynes. Assume that each man has income I and that the prices of x and y are p_x and p_y respectively.

d. Now assume that a=1/2, b = 1/4, I = 600, p_x = 10 and p_y = 20. Use these values to find the utility-maximizing quantities of x and y consumed by each man.

e. Find an expression for the share of income Mr. Bentham and Mr. Marshall each devote to the consumption of good x, where this share, s_x, is defined as

(p_x x)/I.

f. Find expressions for the income elasticity of demand for good x for Mr. Bentham and Mr. Marshall. Find expressions for the (own) price elasticity of demand for good y for these two men.

aquamarinescorpion997

True or False. If False explain why it's false.

1. If all prices are doubled an money income is the left the same, the budget set does not change because the relative prices do not change.

2. A decrease in income pivots the budget line around the bundle initially consumed.

3. If preferences are well-behaved, then for any commodity bundle (x1,x2), the set of commodity bundles that are worse than (x1,x2) is a convex set.

4. The marginal rate of substitution measures the distance betwee one indifference curve and the next one.

5. If someone has the utility function U= 2min(x, y), then x and y are perfect complements for that person.

6. At the boundary optimum, a consumers indifference curve must be tangent to her budget line.

7. If two goods are substitutes, then an increase in the price of one of them will increase the demand for the other.

8. An Engel curve is a demand curve with the vertical and horizontal axes reversed.

9. An inferior good is less durable than a normal good.

10. When other variables are held fixed, the demand for a Giffen good rises when income increase.

ECO204Y1 Chapter Notes - Chapter 2: Convex Preferences, Utility, Weighted Arithmetic Mean

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