1. LO 1 Using a diagram show that if the consumer (c) Given perfect substitutes, is more preferred to prefers more to less, then indifference curves can- less? Do preferences satisfy the diminishing not cross. marginal rate of substitution property? 2. LO 1, 3 In this chapter, we showed an example in which the consumer has preferences for con- 3. LO 1, 3, 4 Consider a situation in which lump- sumption with the perfect complements property. sum tax is higher than dividend income and the Suppose, alternatively, that leisure and consump- consumer prefers an equal proportion of con- tion goods are perfect substitutes. In this case, an sumption and leisure. indifference curve is described by the equation (a) Suppose that h = 16, w =0.5, # = 0.6, and 7 = 4. Use a diagram to show the consumer's i = al + bC. optimal choice of consumption and leisure. (b) If the consumer likes a fixed proportion of 4 where a and b are positive constants, and u is the units of consumption and 1 unit of leisure, level of utility. That is, a given indifference curve how will your answer to part (a) change? has a particular value for u, with higher indiffer- Use the diagram from your answer to part ence curves having higher values for u. (a) to show the changes. (a) Show what the consumer's indifference (c) Show the new optimal choice of consump- curves look like when consumption and lei- tion and leisure on the same diagram if w is sure are perfect substitutes, and determine now doubled. Explain this change in terms of graphically and algebraically what consump- income and substitution effects. tion bundle the consumer chooses. Show 4. LO + Suppose that the government imposes a that the consumption bundle the consumer proportional income tax on the representative chooses depends on the relationship between consumer's wage income. That is, the consumer's ab and w, and explain why. wage income is w(1 - ()(h - D) where t is the tax (b) Do you think it likely that any consumer rate. What effect does the income tax have on would treat consumption goods and leisure consumption and labor supply? Explain your re- as perfect substitutes? sults in terms of income and substitution effects.4. Suppose that the utility function is u(c, () = log(c) - e (a) Find the equation of an indifference curve given a level of utility I. (b) Find the Marginal Rate of Substitution. (c) Show the optimal consumption-labor choice using a plot with c in the y-axis and { in the x-axis. (d) Find the optimal amount of consumption and leisure assuming that h = 1 and T = T = 0