Question

Consider my tastes for consumption and leisure. A. Begin by assuming that my tastes over consumption and leisure satisfy our five basic assumptions.

a. On a graph with leisure hours per week on the horizontal axis and consumption dollars per week on the vertical, give an example of 3 indifference curves (with associated utility numbers) from an indifference map that satisfies our assumptions.

b. Now redefine the good on the horizontal axis as “labor hours” rather than “leisure hours.” How would the same tastes look in this graph?

c. How would both of your graphs change if tastes over leisure and consumption were non-convex, that is, if averages were worse than extremes?

B. Suppose your tastes over consumption and leisure could be described by the utility function u(c,l)=c^2\1*l^2\1.

a. Do these tastes satisfy our five basic assumptions?

b. Can you find a utility function that would describe the same tastes when the second good is defined as labor hours instead of leisure hours? ( Hint: Suppose your weekly endowment of leisure time is 60 hours. How does that relate to the sign of the slopes of indifference curves you graphed in part A(b)?)

c. What is the marginal rate of substitution for the function you just derived? How does that relate to your graph from part A(b)?

d. Do the tastes represented by the utility function in part (b) satisfy our five basic assumptions?

Answer 1

So I'll take the five basic assumptions as:

1) Completeness: Given any two commodity bundles X1 and X2, either X1 is weakly preferred to X2, X2 is weakly preferred to X1 or both.

2) Transitivity: For three bundles X1, X2 and X3; if X1 is weakly preferred to X2 and X2 is weakly preferred to X3 then it holds that X1 is weakly preferred to X3.The transitivity assumption implies that indifference sets have no intersection.

3) Non-satiation: More is better, or a consumption bundle X1 will be preferred to X2 if X1 contains more of at least one good and no less of any other.

4) Continuity: The graph of an indifference set is a continuous surface

5) Convexity. Given any consumption bundle X1, its better set is strictly convex. This is akin to Mix Is Better i.e. averages are preferred to extremes.

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a)

Shown above are the three types of indifference curves - the first is the regular case, the second is the case of perfect substitutes and third of perfect complements.

Let's quickly set up a mathematical structure before proceeding.

(Labor hours) + (Leisure hours) = (Total time available) => [ Labor Hours = {(Total time available) - (Leisure hours)} ]

Income = (Wage Rate) * (Labor Hours)

implies Income = {(Wage Rate) * { (Total time available) minus (Leisure hours)} }

So assuming Income = Consumption (i.e. no saving)

there's a negative relationship between Consumption and Leisure. The Consumption-Leisure graph will come out to be negatively sloped.

Shown below are the diagrams for the three cases:

b) If the variable plotted on the X axis is Labor Hours, then the graph is upward sloping.

c) If averages were worse than extremes, then the

i) The consumption-leisure indifference curves would become negatively sloped concave.

ii) The consumption-labor graph becomes upward sloping and concave.

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