LECTURE 6 Flashcards
3 uses of income tax by government
- Fund public services.
- redistribute income
- Provide insurance
3 sources of taxable income
- earned income
- non-earned income e.g. property income
- taxable benefits e.g. state pension
The UK PIT system is approx UNITARY. What does this mean?
Income is aggregated and tax is calculated on the aggregate.
basic, higher and additional rates in UK.
20%, 40% and 45%
The UK PIT system consists of 2 things…
- Tax: income and NICs
2. Tax credits - form of benefits.
Average tax rate formula
total tax paid / income = T(Y)/Y = tax paid per unit of income
Marginal tax rate formula
change in total tax paid / change in income = change in tax paid for 1 unit increase in income.
MTR under progressive system
MTR increases with income so higher income individuals pay a greater proportion of their income in tax.
MTR under regressive system
MTR decreases with income so higher income individuals pay a smaller proportion of their income in tax. Same tax amount e.g. sin taxes.
MTR under proportional system
MTR same for all levels of income so all individuals pay same proportion of their income in tax. Pay same tax rate = flat tax.
Extensive margin of labour supply
whether to be in LF or not.
Intensive marginal of labour supply
How many hours to work conditional on being in LF.
Consumer utility function and BC in consumption leisure optimisation problem
Max U(c, l) s.t. C = (1-t)wL + R
What’s the opportunity cost of leisure?
Wage
Effective wage =
(1 - t)w = post-tax wage.
What’s another constraint for consumer consumption leisure optimisation besides BC?
TIME constraint: L + l = T
How do consumption, leisure and labour affect utility?
U’(c) > 0
U’(l) > 0
U’(L) < 0 - utility decreases in labour
Labour-consumption trade off
U’(c) > 0, U’(L)<0 but need to work to earn money to consume.
What’s R in consumer c-l optimisation?
R = non-labour income. c = R if do not work still have some money.
Slope of BC in consumption-leisure space
slope = –(1-t)w = opportunity cost of leisure in terms of post-tax wage and so consumption lost.
What’s the SE if tax rate decreases?
Lower t = higher effective wage = higher opportunity cost of leisure. BC pivots so steeper. Substitute leisure for labour as leisure becomes relatively more expensive.
What’s the IE if tax rate decreases?
Tax rate decreases = higher disposable income. Assume consumption and leisure both normal goods = consume more of both.
SO: how does a decrease in t affect labour incentives through SE and IE?
SE: lower t = greater incentive to work
IE: lower t = more leisure since normal good = disincentive to work.
SO: how does an increase in t affect labour incentives through SE and IE?
SE: higher t = less incentive to work
IE: higher t = less leisure since normal good = incentive to work.
Tax function T(Z) =
How does tax depend on income?
T(Z) = T bar + tz
T bar = autonomous part
tax linearly depends on income.
Tax function if zero earnings but transfer benefits.
- T(0) = T bar < 0
- ve taxation = transfer.
Another name for MTR
phasing out rate
Formula for how much of a $1 increase in income you get to keep as disposable income.
D = Z - T(Z) dD/dZ = 1 - T'(Z) = 1 - MTR
the MTR affects what type of labour supply response?
INTENSIVE
Participation tax rate formula and explanation.
Tp = T(Z) - T(0) / z
Difference between taxes paid if z>0 and transfer receive if z=0, as a proportion of income.
1 - participation rate shows
proportion of earnings retained when moving from out of LF to into LF and earning Z.
Disposable income formula relating to participation rate.
Z - T(Z) = -T(0) + z(1 - Tp)
Break even earnings point is where…
T(z) = 0, pre-tax income = post-tax income. Neither net subsidy receiver nor net tax payer.
A graph with consumption on Y axis and pre-tax income on X axis aims to show us…
What’s the 45 degree line?
How the tax and transfer system affects the relationship between pre-tax income and disposable income/consumption. It shows the budget constraint / consumption function. The 45 degree line is where consumption=pre-tax income so T(Z)-0
Slope of consumption function =
1 - T’(Z) = 1 - MTR
IE how much of $1 increase in income you keep as disposable income.
How do the slopes of the consumption function and 45 degree line compare and why?
Consumption function slope flatter and < 1 for MTR > 0. When income rises by $1, get to keep
What does it mean when pre-tax income is before the break even point?
Z < Z* means T(Z) < 0 i.e. net subsidy receiver.
consumption > pre-tax/transfer earnings
What does it mean when pre-tax income is after the break even point?
Z > Z* means T(Z) > 0 i.e. net tax payer.
consumption < pre-tax/transfer earnings
How do we find the participation tax rate on c-z graph?
(1 - Tp)*Z = vertical distance between consumption at z=0 and consumption at a given z>0
How do utility functions compare in diagrams in consumption-leisure and consumption-pre-tax income space?
C-l: ICs slope downwards as usual and higher to NE
C-Z: ICs slope upwards as more pre-tax income i.e. more labour (z=wL) requires more consumption as compensation. Higher to NW when more C but less work.
How does wage compare into c-l and c-z diagrams?
c-l: wage comes into BC and (1-t)w = slope of BC
c-z: wage not in BC, but affects utility.
How to inputs of utility functions compare into c-l and c-z diagrams?
c-l: U(c, l)
c-z: U(c, L=z/w)
How do ICs compare into c-l and c-z diagrams?
They’re equivalent
c-l: C = (1 - t)wL + R
c-z: C = z - T(z) = (1 - t)z + T bar
2 aims of US EITC
- redistribute income to low wage earners.
2. Promote labour supply among low wage earners since they’re in work benefits.
efficiency equity trade-off with tax and transfer system.
Equity: redistribution to reduce inequality
Efficiency: taxes and transfers create disincentives to work.
Effect of T(z)<0 on labour supply and through which effect?
Initially on 45 degree line, then receive subsidy in net so c > z. Increase in disposable income = increase leisure and consumption via IE since both normal goods = decrease labour = z decreases.
Why does it make sense intuitively that transfers reduce incentive to work via IE?
Because with transfers, can now work less and earn the same disposable income.
Effect of T(z)>0 on labour supply and through which effect?
Initially on 45 degree line, then pay taxes in net so c < z. Decrease in disposable income = decrease leisure and consumption via IE since both normal goods = increase labour = z increases.
Why does it make sense intuitively that taxes increase incentive to work via IE?
Because with taxes, now have to work more to earn the same disposable income.
Effect of MTR i.e, T’(Z) > 0 on labour supply and through which effect?
SE: MTR > 0 = lower effective wage = lower opportunity cost of leisure = increase leisure, reduce labour and so Z through SE. Regardless of whether z>z* or z
Summarise IE and SE on incentives to work for net subsidy payer
IE = disincentive to work SE = disincentive to work
Summarise IE and SE on incentives to work for net tax payer
IE = incentive to work
SE = disincentive to work
= AMBIGUOUS overall effect.
How do taxes and transfers affect utility for zz*?
Zz*: taxes decrease utility = move to lower IC via IE
The optimal income tax rate balances what 2 incentives?
equity and efficiency
How are consumers similar and how are they different in Mirrlees?
identical preferences
Different skills and so wages.
Whats the wage in Mirrlees
wage = skill assuming competitive economy
Income = f() mirrlees
Income = f(skill, hours worked)
Does Mirrlees look at extensive/intensive margin?
Assumes everyone works = NO EXTENSIVE
ONLY INTENSIVE
What’s the 1st best tax policy? Why is this not feasible for the government?
1st best = lump sum tax on individual characteristics = skill - like an initial endowment. BUT: skill = private information and cannot be observed by the gov - they only know distribution of skills.
In Mirrlees model, government are trying to determine…
Optimal T(Z)
What’s the 2nd best tax policy when 1st best isn’t feasible?
2nd best = tax behaviour i.e income since this is observed. While income depends on skill, it also depends on hours worked –> distortionary.
2 constraints in government’s problem Mirrlees. Explain them.
- revenue requirement
- incentive compatibility - no 2 consumers want to swap allocations, they find the allocations given U maximising so no incentive to change/pretend to be someone else.
In Mirrlees, the government assigns an allocation of what to each consumer?
Consumption pre-tax income allocation
Formula for pre-tax income Mirrlees
Z = wL
w=skill, s
SO: z = sL
In Mirrless, what functions depend on skill?
z(s)
c(z(s)) = z(s) - T(z(s))
U=f() in Mirrless
U(c, L=z/s) = U(c, z, s)
How do ICs of low and high skill consumers compare? WHY?
At any (z, c), LOW SKILL IC = STEEPER than high skill. Because income = f(skill, hours worked) so to achieve a given level of income, low skilled must put in relatively more labour hours = require higher consumption as compensation
What’s the agent monotonicity condition?
This requires ICs of low skilled to be steeper than ICs go high skilled at any (c, z)
Diagrammatically, a consumer’s optimum (c, z) is where…
Consumption function is tangent to highest IC
Consequence of agent monotonicity for (c, z) allocations of high vs low skilled.
cH > cL
zH > zL
HS optimum will never be to the left of LS optimum as LS would also find this preferable.
If we have a quasilinear utility function u(c) - z/s for Mirrlees, what’s the MRSz,c ? How does MRS vary with skill?
MRSz,c = MUz/MUc = -1/s/u’(c) = - 1/su’(c)
Higher skill = lower MRS = flatter IC = agent monotonicity satisfied.
Would a low skill ever mimic a high skill?
NO as it would be costly they’d have to undergo training.
Would a high skill ever mimic a low skill?
YES - could find it preferable to lower consumption and have more leisure instead by putting in less effort.
incentive compatibility constraint mirrlees
Only HS IC is binding.
u(cH) - zH/sH = u(cL) - zL/sH
sH regardless as high skilled.
How does a utilitarian government optimise for Mirrless? what are it’s choice variables?
MAX utility HS + utility LS
s.t. incentive compatibility constraint
s.t. aggregate resource constraint
4 choices = cH, cL, zH, zL
We simplify the government Mirrlees optimisation to make it unconstrained. Give the maximisation formula nw.
Max betaH u(cH) + betaL u(cL) - (sH + sL / 2sHsL) (cH + cL)
What’s beta h in Mirrlees?
Bh = sH + sL / 2sL
i.e. weight on HS utility function
What’s beta l in Mirrlees?
Bl = 3sL - sH / 2sL
i.e. weight on LS utility function
Range for sH in terms of sL. What does this ensure?
sL < sH < 3sL
Ensures + VE weight on LS utility.
How to weights on LS and HS utilities compare? What does this mean for outcomes?
beta H > beta L
HS always has more weight on utility function.
SO: cH > cL at the optimum.
Mirrlees FOC wrt cH + interpret
u’(cH) = 1/sH
MU of consumption of HS is inversely related to skill
Since u’‘(c) < 0, higher skill = lower MU of consumption = higher consumption. consumption proportional to skill.
At the optimum, what’s MRSz,c for HS ??
u’(cH) = 1/sH
MRS = 1/sHu’(cH)
So MRS = 1 at the optimum.
At the optimum, what’s MTR for HS? Explain.
We found MRS=1 at the optimum.
Optimum where consumption function tangent to IC.
1 - T’(z) = 1 –> T’(Z) = MTR = ZERO
MTR = 0 for high skilled is called…
No distortion at the top result
Mirrlees FOC wrt cL + interpret
u’(cL) = (sH + sL) / sH(3sL - sH)
At the optimum, what’s MRSz,c for LS ??
MRS = sH(3sL - sH) / sL(sH + sL) < 1
At the optimum, what’s MTR for LS? Explain.
MRS < 1
1 - T’(Z) < 1 –> T’(Z) = MTR > 0
+VE MTR for LS
SO: what’s the Mirrless MTR range?
0 <= t* < 1
What does Mirrless imply about progressive tax schemes?
That they cannot be optimal since progressive require MTR to rise with income so MTR is highest for highest income, not zero.
Mirrlees no distortion at the top result is only valid for…
The HIGHEST skill consumer - no prediction for 2nd highest.
How do Diamond and Saez adapt Mirrless?
They use ELASTICITIES
How do applied optimal income tax models differ to results on Mirrless?
They find:
- top MTR not zero.
- MTR < 0 for low earners i.e. subsidise
Optimal linear income tax AKA
LAFFER CURVE
Laffer curve show relationship between…
Tax revenue and tax rate
In Laffer, what do we assume the government do with revenue?
R = t*average Z
This R goes back to individuals in the form of a transfer: C = (1 - t)Z = R
What’s the only channel from tax rate to pre-tax earnings?
Lower tax rate = higher net-of-tax rate = higher effective wage = higher opportunity cost of leisure = SE increases labour = higher pre-tax earnings.
Tax revenue formula Laffer
R(t) = t*Z(1 - t) where Z = f(1 - t), not multiplied.
What’s R when t=0 and t=1?
t=0 R=0
t=1 R=0 since no-one works.
Elasticity of AVERAGE pre-tax income wrt net of tax rate
e = dZ/d(1-t) * (1-t)/Z
Mechanical effect of higher t on R
Higher t = higher R holding Z constant
Behavioural effect of higher t on R
Higher t = lower Z = lower R
How do we find the revenue maximising t* i.e. the peak of Laffer curve? What is it?
dR/dt = 0 t* = 1 / 1 + e
Why is the revenue maximising t* not necessarily optimum t*?
Because government objective is NOT to maximise revenue - its to maximise SWF. They must account for behavioural effect of higher t on Z.
Gov’s optimisation to find optimum t* for Laffer curve?
Assume all individuals have same U(c, L)
Sub in consumption function
d(SWF=sum Ui)/dt
What’s the optimum t* from Laffer method?
t* = (1 - g bar) / (1 - g bar + e)
In laffer, e represents… and how does t* depend on e and why?
EFFICIENCY parameter
Higher e = 1% change in net of tax rate has a greater effect on Z i.e. greater behavioural effect = high efficiency cost of taxation = lower t*
In laffer, g bar represents… and how does t* depend on g bar and why?
EQUITY parameter: How much social value gov places on increasing consumption of 1 person by 1 unit vs distributing 1 unit among everyone.
g bar = inversely related to redistribute tastes and so inequality.
Higher g bar = less redistribution = lower t*
Laffer MAX redistribution values of g bar and t*
Max redistribution when g bar = 0.
g bar =0, t* = 1 / 1 + e i.e. revenue maximising laffer.
Government have higher redistributive tastes when…
Higher inequality.
MU of income decreases fast with income.
Laffer MIN redistribution values of g bar and t*
Min redistribution when g bar = 1.
t* = 0 - zero tax rate, zero revenue, zero transfers.
Government have lower redistributive tastes when…
Little inequality.
When we say tax rate, what kind of tax rate do we mean?
always MARGINAL tax rate
Pertubation method for top income tax rate
Consider the effects of a small increase dt on SW by ONLY looking at effects on individuals with income > certain threshold.
Explain mechanical effect and its sign
Higher MTR = increase in revenue, holding behaviour (labour hours) constant = +VE effect
Explain behavioural effect and its sign
Higher MTR = decrease in labour hours through SE = lower average earnings = lower revenue = -VE effect
Explain utility/welfare effect and its sign
Higher MTR = decreased utility for high income individuals holding behaviour constant.
How do we find optimum for top income tax rate?
mechanical effect + behavioural effect = utility effect
Must sum to zero so tax reform has no first order SW effects.
To simplify top income tax analysis, what do we assume?
NO IE - only SE of higher tax rate on labour supply.
For individuals over threshold, how do MTR compare?
Assume CONSTANT linear tax rate, t, above the threshold z bar.
Do we look at intensive/extensive for top tax?
Intensive only - assume everyone works.
What’s the effect of a small increase dt on taxes paid by 1 individual above z bar?
(z - z bar)dt
What’s the mechanical effect of dt on taxes paid, assuming population above z bar = 1?
M = (zm - z bar)dt
where zm = average income above threshold
Formula for reduction in pre tax earnings of one individual due to dt (using elasticity)
dz = -ez dt/(1 - t)
Formula for reduction in revenue due to behavioural effect of one individual due to dt (using elasticity)
dT(z) = t*d(Z)
Behavioural effect for all individuals
B = -e bar Zm tdt/(1 - t)
How do we show small increase dt for top earners on c-z diagram?
Slope = 1 - T’(z) so Higher MTR = flatter slope beyond z bar.
How do we show mechanical effect on c-z diagram?
At same z, reduction in disposable income from old to new consumption function = increase in tax receipts
How do we show behavioural effect on c-z diagram?
Show z decreasing in response to the reform.
Overall effect of tax reform on revenue: M + B =
[Zm/z bar - 1 - t/(1-t) * e bar * zm/z bar]
If M > B, raises revenue.
What’s the welfare effect of the tax reform driven by?
Increased tax liability so its driven entirely by the mechanical effect.
Utility effect =
g bar M
- M = increased tax liability
- g bar = SWF weight on top income individuals relative to weight on government revenue
The definition of g bar implies…
The government is indifferent between $1 more consumed by top income individuals and $g bar more government revenue
When we set M + B = g bar M, we get the optimal top income tax rate =
t* = (1 - g bar)/(1 - g bar + a*ebar)
How does g bar affect top t*?
Higher g bar = more social value on top income individuals = less redistributive taste = lower t*
How does e affect top t*?
Higher e = greater behavioural effect & so efficiency loss of tax = lower t*
What is a in top t* formula?
a = thinness of tail of income distribution
Higher a = thinner tail = fewer people in top half.
How does a affect top t*? Why?
Higher a = thinner tail = fewer people in top = lower revenue potential = lower t*. Also when a higher, more worried about B > > M.
How does top t* relate to Mirrlees result?
When zm=z bar i.e. only 1 person in top, a–>infinity, t* = 0. The same no distortion at the top result.
a formula:
a = zm / zm - z bar a = [zm/zbar] / [zm/zbar - 1]
