In the real world, government provision of public goods usually entails the use of distortionary taxes to raise the required revenues. Consider the pure public good “national defense”, a good provided exclusively by the government (with no private contributions).
A: Consider varying degrees of inefficiency in the nation’s tax system.
(a) In our development of the concept of deadweight loss from taxation, we found that the deadweight loss from taxes tends to increase at a rate K2 for a k-fold increase in the tax rate. Define the “social marginal cost of funds” SMCF as the marginal cost society incurs from each additional dollar spent by the government. What is the shape of the SMCF curve?
(b) True or False: If the public good is defined as “spending on national defense”, then the marginal cost of providing $1 of increased funding for the public good is $1 under an efficient tax system.
(c) How does the marginal cost of providing this public good change as the tax system becomes more inefficient?
(d) Use your answer to (c) to explain the following statement: “As the inefficiency of the tax system increases, the optimal level of national defense spending by the government falls.”
(e) What do you think of the following statement: “Nation’s that have devised more efficient tax systems are more likely to win wars than nations with inefficient tax systems.”
B: Suppose we approximate the demand side for goods by assuming a representative consumer with utility function u(x, y) = x1/2y1/2 and income I, where x is private consumption (in dollars) and y is national defense spending (in dollars).
(a) If the government can use lump sum taxes to raise revenues, what is the efficient level of national defense spending?
(b) Next, suppose that the government only has access to inefficient taxes that give rise to deadweight losses. Specifically, suppose that it employs a tax rate t on income I, with tax revenue equal to TR = t I/ (1+βt) 2. How does this capture the idea of deadweight loss? What would β be if the tax were efficient?
(c) Given that it has to use this tax to fund national defense, derive the efficient tax rate and level of national defense. (It is easiest to do this by setting up an optimization problem in which t is the only choice variable, with the utility function converted to logs.) How does it compare to your answer to (a)?
(d) Suppose I = 2,000. What is national defense spending and the tax rate t when β = 0? How does it change when β = 0.25? What if β= 1? β= 4? β = 9?
(e) Suppose next that the government provides two pure public goods — spending on national defense y1 and spending on the alleviation of poverty y2 (where the latter is a public good in the ways developed in exercise 27.8). Suppose that the representative consumer’s tastes can be described by u(x, y1, y2) = x0.5y γ1y2 (0.5−γ) 2. Modify the optimization problem in (c) to one appropriate for this setting—with the government now choosing both t and the fraction k of tax revenues spent on national defense (versus the fraction (1−k) spent on poverty alleviation.)
(f) Does the optimal tax rate differ from what you derived before? What fraction of tax revenues will be spent on national defense?