Question

We showed an example in which the consumer has preferences for consumption with the perfect complement property. Suppose, alternatively, that leisure and consumption goods are perfect substitutes. In this case, an indifference curve is described by the equation:

u=al+bC

where a and b are positive constants and u is the level of utility. That is, a given indifference curve has a particular value for u, with higher indifference curves having higher values for u.

a) Show what the consumer’s indifference curves look like when consumption and leisure are perfect substitutes, and determine graphically and algebraically what consumption bundle the consumer will choose. Show that the consumption bundle the consumer chooses depends on the relationship between a/b and w, and explain why.

b) Do you think it likely that any consumer would treat consumption goods and leisure as perfect substitutes?

c) Given perfect substitutes, is more preferred to less? Do preferences satisfy the diminishing-marginal-rate-of-substitution property?

Answer 1

*Answer:

Step-by-step solution

1. Step 1 of 6

   a.

   It is given that leisure and consumption are perfect substitutes and an indifference curve is given by the equation

   

   where a and b are positive constants, l stands for hours of leisure, Cstands for consumption, and u stands for utility.

   To find out how indifference curves look, calculate the slope of an indifference curve.

   

   Since the slope of indifference curves is negative, we know that the indifference curves are downward sloping. Since the slope of the indifference curves is constant, we know that indifference curves are straight lines.

   

   The indifference curves are downward sloping straight lines, as shown in figure 1.

   Comment

2. Step 2 of 6

   The consumer’s budget constraint is given by the equation

   

   where h is the total number of hours available to be allotted between work and leisure, and w is the wage rate per hour.

   

   Whenever the indifference curves are downward sloping straight lines, the optimal consumption bundle depends on the magnitudes of the slope of the budget constraint and the slope of the indifference curves.

   Case I:

   

   When the magnitude of the slope of the indifference curves is higher than the magnitude of the slope of the budget constraint, the indifference curves are steeper than the budget constraint.

   

   As shown in figure 2, when indifference curves are steeper than the budget constraint, the optimal consumption bundle (that is, the consumption bundle that lies on the highest indifference curve) is given by the horizontal intercept of the budget constraint.

   Comment

3. Step 3 of 6

   Case II:

   

   When the magnitude of the slope of the indifference curves is less than the magnitude of the slope of the budget constraint, the indifference curves are flatter than the budget constraint.

   

   As shown in figure 3, when indifference curves are flatter than the budget constraint, the optimal consumption bundle (that is, the consumption bundle that lies on the highest indifference curve) is given by the vertical intercept of the budget constraint.

   Comment

4. Step 4 of 6

   Case III:

   

   When the magnitude of the slope of the indifference curves equals the magnitude of the slope of the budget constraint, the highest possible indifference curve coincides with the budget constraint.

   

   As shown in figure 4, when the highest possible indifference curve coincides with the budget constraint, all the bundles on the budget constraint are optimal consumption bundles. The consumer will randomly pick any of these bundles for consumption.

   Comment

5. Step 5 of 6

   b.

   It is highly unlikely that the consumer will see consumption and leisure as perfect substitutes. The marginal rate of substitute between any two goods is usually diminishing, but when two goods are perfect substitutes their marginal rate of substitution is constant, which is unlikely.

   Comment

6. Step 6 of 6

   c.

   Since the marginal utilities of both consumption and leisure are positive, we can say that more is preferred to less.

   As we know from part a of this solution, the marginal rate of substation is constant and therefore does not satisfy the property of diminishing marginal rate of substitution.

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