1    Review questions

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

1. The optimal linear tax rate is never above the revenue maximizing tax rate.  True. It is Pareto-improving to lower a tax rate above the revenue maximizing tax rate.

2. Standard labor supply theory suggests that the EITC should increase labor sup- ply.  Uncertain.  There are different predictions between decisions on the the inten- sive/extensive margin, as well as within the intensive margin.

3. Taxation of utes at a lower rate than sports cars can be justified on efficiency grounds.  True, assuming that sports cars are more complimentary with leisure than utes.

4. A tax that is not salient to consumers is equivalent to a lump-sum tax.  Uncertain. Yes if there are no income effects, no if there are income effects.

2    Problems

2.1    Consumption Taxation

Consider the utility function u(x1 ,x2 ,l) = aln(x1 )+bln(x2 )+cln(l), where x1  and x2  are goods, l < 1 is leisure, which cannot be taxed, and a + b + c = 1.

Total hours of work is normalised to one so the budget constraint is w = p1 x1 +p2 x2 +wl . The government needs to raise revenue R and can tax goods x1  and x2  but not leisure.  Producer prices are equal to one so prices for good i are pi  = 1 + τi .

(a) Write down the consumer’s utility maximisation problem. Find the first-order con- ditions for x1 , x2  and l . Then find the demand curves for x1 , x2  and l .

utility maximization problem:

L = aln(x1 ) + bln(x2 ) + cln(l) + α (w − p1 x1 − p2 x2 − wl)

FOCs:

a 

x1  =

x2

l =

Demand curves:

p1

p2

l = c

(b) Write down the consumer’s expenditure minimisation problem and find the demand functions for x1   and x2 .   Note that the demand functions from the expenditure minimization problem are compensated (Hicksian) demand curves. [Hint: see notes from Lecture 1 .]

Consumer’s expenditure minimisation:

L = p1 x1 + p2 x2 + wl + γ ( − aln(x1 ) − bln(x2 ) − cln(l))

FOCs:

aγ

x1  =

p2

cγ

l =

Compensated (Hicksian) demand curves:

x1(c)  = eu  · ab+c  · b −b  · c −c  · p  p2(b)  · wc

x2(c)  = eu  · a−a  · ba+c  · c −c  · p1(a)  · p wc

lc  = eu  · a−a  · b −b  · ca+b  · p1(a)  · p2(b)  · w − (a+b)

(c) Using your results from (b), find the compensated elasticities of demand for x1  and x2 .[Hint:  there are 9 elasticities of demand in this problem.]

ε11(c)  ≡  ε12(c)  ≡  ε1(c)l  ≡  ε21(c)  ≡  ε22(c)  ≡  ε2(c)l  ≡  εl(c)1  ≡  εl(c)2  ≡  εll(c)  ≡ 

·   = −(b + c)

x1

w

x1

p1

x2

2

2

w

x2

1

l

2

l

l

Notice that εi1 + εi2 + εil  = 0 for each i ∈ {1, 2,l}.

You can also verify symmetry of compensated demand curves:  ∂xi             ∂xj

(d) Does the Ramsey rule recommend the same tax rates on x1  and x2 ?

Ramsey rule: −  =对 εij(c) .

Plug in elasticities from part (c).   Re-arrange and use the fact a + b + c  =  1 to get  =  :   ⇒ τ1  = τ2  at an optimum.

(e) Calculate the optimal tax rates required to raise revenue of R.  [Hint:  recall that pi  = 1 + τi  and the demand functions derived in part (a) .]

τ =

2.2    Income Taxation

Joe starts a new job as a barista at Zachary’s Cafery. The job is unusually flexible: Joe can pick to work any number of hours between 0 and 4,000 in a given year at a wage rate of $20 per hour. His utility is given by:

2

After Joe starts his job, the government announces a new progressive income tax schedule:

• Income up to $10,000: MTR = 0

• Income between $10,000 and $40,000: MTR = 20%

• Income above $40,000: MTR = 30%

(a) Draw a graph with pre-tax earnings on the x-axis and consumption on the y-axis.

Draw Joe’s consumption function with and without the new tax system.

(b) For each tax bracket, what is the sign of the income effect, substitution effect, and

total effect on labor supply of the new tax system (compared to the baseline with no taxes)?

(c) Solve for Joe’s optimal choice of labor supply without the progressive income tax.

l ∗ = 2,000 hours (note that this is feasible because it is less than 4,000) c∗ = $40, 000

(d) Solve for Joe’s optimal choice of labor supply after the introduction of the progressive income tax. [Hint:  optimize on each portion of the budget constraint separately, as if that budget constraint applied everywhere.]

l ∗ = 1,600 hours (note that this is feasible because it is between 500 and 2,000, that is, pre-tax earnings are between $10,000 and $40,000)

c∗ = $27, 600

(e) What is the impact of the progressive income tax on Joe’s optimal choice of labor supply?

The progressive income tax results in Joe decreasing his labor supply by 400 hours (i.e., it discourages labor supply, implying the substitution effect dominates).

(f) What is Joe’s marginal tax rate? What is his average tax rate?

In this tax bracket, Joe’s marginal tax rate is 20%. His average tax rate is 13.75%.

(g) Now assume the cut-off point for the top tax bracket is $30,000 instead of $40,000.

What is Joe’s optimal choice of labor supply now?

Joe will“bunch”at the kink in the marginal tax rate schedule and choose to work exactly $30,000/$20 = 1,500 hours.