In the chapter, we discussed the deadweight loss from taxes on consumption goods when tastes are quasilinear in the taxed good, and we treated deadweight loss when tastes are not quasilinear for the case of wage taxes. In this exercise, we will consider deadweight losses from taxation on consumption goods when tastes are not quasilinear.
A. Suppose that x is a normal good for consumers.
(a) Draw the market demand and supply graph for x and illustrate the impact on prices (for consumers and producers) and output levels when a per-unit tax t on x is introduced.
(b)Would your answer to (a) have been any different had we assumed that all consumers’ tastes were quasilinear in x?
(c) On a consumer diagram with x on the horizontal and “all other goods” (denominated in dollars) on the vertical axes, illustrate the impact of the tax on a consumer’s budget.
(d) In your graph from (c), illustrate the portion of deadweight loss that is due to this particular consumer.
(e) On a third graph, depict the demand curve for x for the consumer whose consumer diagram you graphed in (d). Then illustrate on this graph the same deadweight loss that you first illustrated in (d).
(f) Now return to your graph from (a). Illustrate where deadweight loss lies in this graph. How does it compare to the case where the original market demand curve arises from quasilinear tastes rather than the tastes we are analyzing in this exercise?
(g) True or False: We will overestimate the deadweight loss if we use market demand curves to measure changes in consumer surplus from taxation of normal goods.
B. Suppose that consumers all have Cobb-Douglas tastes that can be represented by the utility function u(x, y) = xαy(1−α) and each consumer has income I . Assume throughout that the price of y is normalized to 1.
(a) Derive the uncompensated demand for x by a consumer.
(b) Suppose income is expressed in thousands of dollars and each consumer has income I = 2.5 (i.e. income of $2,500). There are 1000 consumers in the market. What is the market demand function?
(c) Suppose market supply is given by xs = βp. Derive the market equilibrium price and output level.
(d) Suppose α = 0.4 and β = 10. Determine the equilibrium pd , ps and xt when t = 10. How do these compare to what we calculated for the quasilinear tastes in Section 19B.2.1 (where we assumed α = 1000 and β= 10) graphed in Graph 19.10?
(e) What is the before-tax and after-tax quantity transacted?
(f) If you used the market demand and supply curves to estimate deadweight loss, what would it be?
(g) Calculate the real deadweight loss in this case — and explain why it is different than in Section 19B.2.1 where market demand and supply curves were the same as here.