1    Review questions

Indicate whether each statement is true, false, or uncertain, and give a brief explanation.

1.  Suppose that,  as a population policy, government taxes families for each car in excess of two in the family.  Consequently no family has more than two cars and the government collects no revenue from the policy. Does this mean that the policy has no deadweight loss?  Uncertain.  It could be that the tax is not bindingânobody would have had more than two cars in the absence of the taxâin which case there is no change in behavior and no deadweight loss. However, it could be that people’s behavior is so responsive to taxes that nobody has more than two cars as a result of the tax. In this case the tax has distorted behavior and created a deadweight loss.

2. The theory of optimal commodity taxation argues that tax rates should be set equal across all commodities. False/uncertain:  The efficiency costs of commodity taxation depend on the size of the elasticities of supply and demand for each good. Hence, it is more efficient to have higher tax rates on inelastic goods. Equal tax rates across all commodities is desirable only under certain conditions, e.g. if elasticities are the same across goods.

3. Taxing dividends distorts investment decisions for a firm.  Uncertain. It depends on whether the investment is being financed by cash that is already held inside the firm, or cash that is outside the firm. If the investment is done out of retained earnings, dividend taxation does not distort the firm’s investment decision because the money will be subject to the tax either way.  If the investment is done with cash is held outside the firm, investing in the firm subjects the investment to the dividend tax and therefore the tax distorts the investment decision.

4. Debt financing is cheaper than equity financing because it is tax deductible and is therefore always the preferred source of financing for firms.   Uncertain.   Debt financing is cheaper than equity financing if q (1 − τp ) > q (1 − τc )(1 − τe ) . If there is no corporate tax (or it is in practice eliminated because of imputation) and the tax rate on interest income and equity returns is the same (τe  = τp ), then debt financing has the same return as equity financing. In general, whether debt or equity financing is cheaper depends on the relative sizes of τp , τc   and τe .  Even if debt financing is cheaper than equity financing, debt is risky and a firm would not want to be 100 per cent debt financed.

2    Problems

2.1    Corporate Taxation

This question considers the effect of the corporate income tax on wages for a small open economy.

The value of the firm is given by

V = (1 − τc ) (F(K,L) − wL) − rK

where τc is the corporate income tax rate, = F(K,L) is output produced by the firm using labor L and capital K , w is the wage rate and r is world interest rate.  The production function has the form F(K,L) = Kα L1−α , with 0 < α < 1.  The firm’s problem is to choose L and K to maximize the value of the firm taking τc , w and r as given.

(a)  Show that the firm’s first-order condition for its choice of labor  L is    =  (1 −

τc )(FL  − w) = 0.  Does the corporate tax distort the firm’s choice over how much labor to employ?

The first-order condition for L simplifies to FL  = w .  The corporate tax does not distort the firm’s labor demand decision because labor is tax deductible.

(b)  Show that the firm’s first-order condition for its choice of capital K is   = (1 −

τc )FK  − r  = 0.   Does the corporate tax distort the firm’s choice over how much capital to employ?

The first-order condition for K simplifies to FK  =  .  The corporate tax reduces demand for capital.

(c) Note that F(K,L) can be written as F(K,L) = F (  , 1) L = f(k)L where k =  is the capital to labor ratio and f(k) = F(k,1). Show that the first-order condition for labor can be re-expressed as w = f(k) − kf\ (k) and that the first-order condition for capital can be expressed as r = (1 − τc )f\ (k).

FL (K,L) =                  =

= f(k) + f\ (k)L

Hence, w = FL  = f(k) − kf\ (k).

FK (K,L) =                  =

∂K

Hence, r = (1 − τc )FK  = (1 − τc )f\ (k).

(d)  Show that the total derivative with respect to k of the first-order condition for L is dw = −kf\\ (k)dk and that the total derivative with respect to k of the first-order condition for K is dr = −dτc f\ (k) + (1 − τc )f\\ (k)dk .

w = f(k) − kf\ (k)

dw = f\ (k)dk − f\ (k)dk − kf\\ (k)dk

= −kf\\ (k)dk

r = (1 − τc )f\ (k)

= f\ (k) − τc f\ (k)

dr = f\\ (k)dk − dτc f\ (k) − τc f\\ (k)dk

= −dτc f\ (k) + (1 − τc )f\\ (k)dk

(e) In a small open economy, r is unaffected by changed in the home corporate tax rate so dr = 0.  Using dr = 0, combine the two expressions in part (d) to show that dw/dτc  = −kf\ (k) ·  .

dr = 0 implies dk = dτ .  Substitute this into the expression for dw from part (d) and re-arrange to get the desired result.

(f) Using the result in part (e) and the functional form of the production function, show that elasticity of wages with respect to 1 − τc  is εw,(1−τc)  = kf\ (k)/(f(k) − kf\ (k)) =

α  

1−α .

dw      1 − τc

εw,(1−τc)  =

f(k) − kf\ (k)

Plug in the functional form of the production function, f(k) = k α  to achieve the final result.

(g) The parameter α is the elasticity of output with respect to capital. A typical value

is α =  . Plug in this value to the result from (f) and interpret the result. A one percent increase in the net-of-tax rate raises wages by half a per cent.

2.2    Tax Evasion

Consider the Allingham-Sandmo model of tax evasion.  Suppose the probability of audit is p, there is a tax rate τ per dollar of income earned, and the penalty on detected evasion is equal to π per dollar of income underreported e. Utility is given by u(c) = ln(c).

(a) For what values of p will a risk-averse taxpayer engage in at least some amount of evasion?

The taxpayer will evade if p <  .

Yitzhaki (1974) argued that having the penalty be a function of the amount of tax evaded wasn’t realistic and proposed an alternative form of the penalty function.

(b) Assume the loss from being caught evading is πτe instead πe. Now for what values of p will a risk-averse taxpayer engage in at least some amount of evasion?

The taxpayer will evade if p <  .

(c) Write down the maximization problem for the taxpayer’s decision over how much income to evade using Yitzhaki’s form of the penalty function and assuming the amount of income earned is fixed. Solve for the taxpayer’s optimal evasion choice.

Taxpayer’s problem:

max  (1 − p)u (y (1 − τ) + τe)+ pu (y (1 − τ) − πτe)

Optimal evasion choice:

y (1 − τ) (1 − p(1 + π) )

πτ

(d) How does an increase in the tax rate τ affect the taxpayer’s optimal evasion choice? How does this differ from the result discussed in lecture using the classic Allingham- Sandmo penalty function?

An increase in the tax rate τ lowers the amount of evasion.  This is because there is no substitution effect when the penalty rate is proportional the amount of tax understatement and there is a negative income effect (higher tax rate −→ less real wealth).  This differs from the result using the classic Allingham-Sandmo penalty function, which has ambiguous theoretical predictions for the impact of an increase in the tax rate on the optimal evasion choice.

Note, this prediction is inconsistent with the bunching evidence in Kleven et al.  (2011), who concludes minimal impact of the marginal tax rate on evasion.