LECTURE 7 Flashcards
How does commodity tax distort consumer choice?
But affecting relative prices and causing demand to switch from one good to another.
Also affect total consumption/saving.
How do taxes affect consumer and producer prices?
drive a wedge between the 2
What type of tax is commodity tax usually? What does this mean?
Proportional, not lump sum = DISTORTIONARY
How does commodity tax affect IC/BC diagram?
BC PIVOTS INWARDS for good the tax is levied on.
Change in slope of BC.
EQUIVALENT VARIATION asks…
How much money would the consumer be WTP in advance to avoid the tax and be at constant original (pre-tax) prices?
i.e. How much does income need to change before the tax such that utility is equal to the level if the tax is implemented?
EV what hypothetical BL do we draw?
Draw BC parallel to OLD BC and tangent to NEW IC.
- At original prices
- Utility the same level as under the tax.
- Consumers WTP this amount of income shift that brings utility no lower than that under the tax.
How do we calculate EV? What does it tell us?
EV = difference in income the original and hypothetical BLs represent. It tells us the $ value of the welfare loss of the price change induced by the tax.
Whats IE and SE with commodity tax on good 1?
IE: decreased disposable income = reduce consumption of both goods.
SE: good 1 relatively more expensive = substitute 2 for 1.
How do we show a lump sum tax on the same diagram that generates the SAME revenue as the commodity tax?
Draw BL parallel to original BC (no change in prices) that goes through the final commodity tax bundle (ensures generates same revenue)
How can we compare welfare effects of commodity tax and equivalent lump sum?
Lump sum = reaches higher IC
EV higher for commodity tax than lump sum
How do we define DWL of a commodity tax?
The welfare loss (measured in monetary terms) created by the commodity tax over and above the tax revenue generated.
DWL calculation in monetary units
DWL = EV(commodity) - EV(lump sum) DWL = EV(commodity) - tax revenue
How else can we calculate DWL apart from using EV?
Simple demand and supply diagram: DWL = change in CS + PS + government revenue
2 other words for DWL
excess burden / deadweight burden
Uniform tax rates for all goods are NOT optimal if…
consumer has different PED for the goods.
Ramsey optimal tax rule
Optimal tax rates where marginal DWL for the last $1 of tax collected is the SAME across all goods.
Does the Ramsey rule consider both efficiency and equity?
only efficiency - assume just 1 consumer so abstract from redistribution.
How does inverse elasticity rule relate to Ramsey rule? What assumption does it make?
It’s a simplified version of ramsey rule.
Assumes NO CROSS PRICE EFFECTS: demand for a good depends on own price only.
after tax price =
qi = pi + ti
IER BC for 2 goods
q1x1 + q2x2 = wL = L for w=1
What do we assume about individuals? Implication
Assume all individuals are homogenous = we can optimise for one representative individual/
Lagrangian for consumer optimisation
L = U(x1, x2, L) + alpha(L - q1x1 - q2x2)
What Lagrange Multiplier do we use for consumer optimisation?
alpha
3 FOCs for IER consumer optimisation
MU of consumption = lagrange multiplier * after-tax price for both goods.
MU of labour hours = - lagrange multiplier
For consumer IER optimisation, what does alpha represent?
alpha= shadow price = MU of income i.e. change in utility if income rises by $1
Gov revenue constraint in terms of p and q
R = t1x1 + t2x2
ti = qi - pi
R + p1x1 + p2x2 = q1x1 + q2x2
How does the gov obtain t*?
Maximise representative consumer’s utility while satisfying their revenue constraint
What are the government’s choice variables? Why?
x1 and x2
cannot choose L - this is endogenously determined from individual BC once x1 and x2 chosen.
What Lagrange Multiplier do we use for gov optimisation? What does it measure?
lambda = MU of income for the government
How are lambda and alpha related???
Distortionary taxes imply lambda > alpha
MU of Income for gov > MU of income for consumer
Gov valuation of rev > ind valuation of income
What’s the inverse demand function?
qi=qi(xi)
demand only depends own price.
How does the gov use consumer optimisation FOCs in their own?
They embed the consumer’s 3 FOCs into their own FOCs
PED formula
PED = dxi/dqi * qi/xi
Use after-tax prices.
INVERSE ELASTICITY RULE FORMULA
ti/(pi + ti) = - [(lambda - alpha)/lambda] * 1/PED
What is the sign of IER tax and why?
Assume lamba > alpha as distortionary
So +VE proportional tax
IER Relationship between tax and PED
Higher PED = more elastic = lower tax
Lower PED = more inelastic = higher tax
Inversely related!!
How does the constant of proportionality vary across goods?
It is CONSTANT across goods - the taxes only vary across goods if PED different.
IER implication for necessities and luxuries.
Necessities = inelastic = higher tax Luxuries = elastic -= lower tax
Is IER equitable?
NO - higher tax on necessities = lower income individuals bear relatively more of the tax
What does the ramsey rule introduce over IER?
CROSS PRICE EFFECTS i.e. goods can be substitutes or complements.
Does Ramsey rule consider equity?
NO - abstract from redistribution by considering an economy with 1 consumer.
Demand function for goods for Ramsey
xi = xi(q1, q2)
Side check: what 2 taxes are equivalent??
Uniform commodity tax (2 goods taxed at same rate) & income tax
What market structure does Ramsey assume? Implication.
Perfect competition: P=MC
SO: pi = w
pre tax price since this is the price the producer charges.
When analysing the equivalence of uniform commodity tax and income tax, what kind of taxes do we assume for each?
Income tax = proportional
Commodity tax = proportional
Compare after-tax price formula for IER and Ramsey
IER: qi = pi + ti - additive
Ramsey: qi = (1 + ti)pi - proportional
Uniform commodity tax are income tax are equivalent when…
tow = t / (1 + t)
where tow = proportional income tax
t = uniform commodity tax
A commodty tax only differs from an income tax when…
It is NON-UNIFORM
How do uniform and non-uniform commodity taxes distort choice? Which is more distortionary?
Uniform = only distorts consumption-leisure choice.
Non-uniform = also distorts choice between x1 & x2
= more distortionary.
Is a uniform tax better than a non-uniform one?
NO not necessarily - non-uniform taxes can exploit different PED for more revenue and lower DWL.
For actual Ramsey rule, what do we assume about income tax?
NO INCOME TAX - this simplifies without loss of generality.
BC for Ramsey rule
(1 + t1)p1x1 + (1+t2)p2x2 = wL
assume w = 1 therefore p = 1
(1 + t1)x1 + (1+t2)x2 = L
What does the gov optimisation do for Ramsey?
MAX INDIRECT utility function of consumer s.t. revenue constraint.
How do w find indirect utility function?
v(1 + t1, 1 + t2)
Sub optimal demand functions from consumer optimisation into x1, x2 and L to get as function of post-tax price.
Sii shows
slutsky compensated demand.
first i = which good’s quantity changes.
second i = which good’s price changes
What is Slutsky compensated demand?
Shows quantity a consumer buys if he is income compensated for a price change i.e. eliminates IE = SE remains. SE holds utility fixed.
OWN PRICE SLUTSKY IDENTITY and interpretation
dXi/dqi = Sii - Xi*dXi/dM
Sii = SE - hold utility fixed
Last term = IE - hold prices fixed
CROSS PRICE SLUTSKY IDENTITY
dXi/dqj = Sij - Xj*dXi/dM
Approx of change in compensated demand due to tax =
When is it a good approx?
ti*Sii
Good approx if ti is small.
Formula for change in compensated demand of good 1 due to introduction of taxes, as a proportion of pre-tax demand.
(t1S11 + t2S12) / X1
Formula for change in compensated demand of good 2 due to introduction of taxes.
(t2S22 + t1S21) / X2
RAMSEY RULE + EXPLAIN
(t1S11 + t2S12) / X1 = (t2S22 + t1S21) / X2
changes in compensated demand due to the taxes as proportions of pre tax demand should be EQUAL for the 2 goods. This minimises distortions in quantities due to non-uniform taxes.
Implication of Ramsey Rule relating to elasticities
Good 1 = inelastic
Good 2 = elastic
If t1 changes, S11 < S21 by PED, so tax good 1 more and good 2 less.
ADD ROY’S IDENTITY WE GET…
(t1S11 + t2S12) / X1 = (t2S22 + t1S21) / X2 = - THETA
another name for ramsey rule
equal proportional reduction rule
We can calculate actual taxes if we have…
utility function
production parameters.
Ramsey rule minimises distortion in terms of what? Why?
QUANTITIES, NOT PRICES.
Because quantities are what determine utility - so prices only matter so far as they determine demands.
SO: what does Ramsey rule suggest about uniform taxes?
NOT OPTIMAL - we want to minimise quantity, not price distortion.
Ramsey equity implications. Why does this arise?
Inequitable - only considers efficiency.
Arises due to 1 household assumption.
Another intuition from Ramsey
If there exists an untaxed good e.g. leisure, the optimal tax formula can be expressed in terms of complementarity or substitutability with the untaxed good. SO: efficiency can be improved by taxing more heavily goods that are more complementary with leisure.
Optimal commodity tax implication for production
When optimal commodity taxes are employed, production must be efficient - optimum lies on PPF boundary.
Condition for productive efficiency to be achieved
MRTS between any 2 inputs the SAME for all firms. This is achieved without the tax under perfect competition since each firm sets MRTS = p1/p2 and prices are taken from the market.
Implication of productive efficiency result of commodity tax
Do NOT tax inputs or intermediate goods as it creates distortion and leads to a worse outcome than without the tax. So only tax final goods i.e. consumption.
Limitations: the principle of not taxing intermediate goods is heavily dependent on 3 things…
- No pure private sector profits.
- Perfect competition
- Possibility to tax all final goods.
