1.5 Labour supply, taxes and benefits Flashcards
In order to look at Labour supply and the idea of households/individuals maximising their utility, we need to learn some notation, what does T n T-n mean?
T= endowment of time e.g. one year = 365 x 24 = 8760 hours ( this is fixed ) n = hours not in paid employment ( leisure T-n = hours in paid employment ( work)
What are the notations c p W and w?
c = consumption of a composite good
p = price of a composite good
W - hourly wage rate ( earnings per hour - nominal wage)
w - W/p - real wage rate ( wages per unit of consumption)
What are the 2 labour decisions we have?
To work or leisure, The whole point of working is that you generate enough income to buy goods. ( money doesn’t fall from the sky)
Usually utility is a function of good 1 and 2 U(x1,x2) but now we are going to adjust it for labour supply so now what is utility a function of? and is it realistic
Utility is a function of U(c,n) - i.e. it depends on consumption c and ‘leisure’ time n. It is not realistic to think people choose their labour supply ( how many hours to work( uber driver do lol)
Is non-saitation satisfied here and is there a an assumption that we don’t like doing something?
Utility is increasing for both consumption and ‘leisure’ - this must mean you are working less, as there is a fixed amount of time, meaning our model assumes you hate working.
Do completeness, transitivity, continuity and convexity hold and if they do what does it allow us to do?
Yes they all do. They allow us to write down u (c,n) and draw indifference curves that are downward sloping and convex to the origin.
We are now going to conduct and make the budget constraint similar to one of consumer theory. As we know that that income is a function of number of hours worked wage, what is the price of leisure and what is the price of consumption?
Price of consumption = P
Price of leisure = W ( this is because wage is the opportunity cost of time ( for every hour you don’t work, you sacrifice per hour wage.
Derive the budget constraint, what do we know about what spending on consumption must be equal or less than?, once we got that rearrange it to look more like the consumer theory budget constraint?
What do we do from here?
So now what is the price of consumption and leisure, what does non-satiation imply that the budget constraint?
Rearrange the budget line where c is a function of real wage rate, endowment of time and leisure? • Where does the budget line meet the axes? • What is the gradient of the budget line? • What happens to the budget line when wage increases?
Why is the horizontal axis also fixed.
Time is fixed, so the horizontal intercept can never be changed so pivoting can only happen not shifting.
What is the labour supply decision, how are preferences showed here, allocate bundle and show the number or works worked when choosing that bundle A, what are the y intercepts and gradient?
Individual/household pickes consumption bundle that maximises his utlitiy subject to prices and income.
In our model is it possible to see a corner solution?
It is rear to have corner solutions because it implies preferences that someone chooses not to work, but in our model it is difficult as it will mean no consumption.
Show an effect of an increase in the real wage and explain it?
Remember the horizontal axis is fixed it doesn’t shift.
Decompose the subsitution and income effect of a rise in real wage and explain it carefully. ( what are type of goods are c and n)
What is the implications of the income and subsitution effect.
We draw another budget line to reflect the impact on demand of a change in wage rate, keeping utlitiy constant at original indifference curve. The movement from A to D is the subsitution effect and the movement from d to b is the income effect.
There is an ambigious effect of labour supply.
We know for sure consumption will increase as the IE AND SE effect reinforce each other for C
If B is on the left of A then the subsitution effect > income effect which it does in this diagram. Meaning overall you work more and enjoy consumption.
If B is on the right of A then IE> SE, meaning overall you work less and enjoy leisure.
So we have discovered a rise in real wage can raise or lower labour supply, depending on the size of the income effect relative to the subsitution effect. .
Now recall the slutsky equation, the income effect on demand for a good is small when what?
income elasticity ( how elastic is demand for a good in respect to income). budget share p1x1/m is small.
What is the budget share of leisure and is it small, what about decsicions on consumption)
Leisure is a big component of our time ( income effect may be important, income effects play a big part in labour decisions but not on our decisions on consumption)
What does the backward bending labour supply show?
The substitution effect states that a higher wage makes work more attractive than leisure. herefore, in response to higher wages, supply increases because work gives greater remuneration. SE > IE
An increase in real wage beyond a certain point, people will work less and substitute to leisure because higher wage means workers can achieve a target income by working fewer hours. so here the IE>SE
Lets analyse this?
1) What does the mean uncompensated labour supply elasticity mean for husbands and wives, and explain why women is higher.
2) what does the mean compensated labour supply elasticity?
1) women are more responsive to changes in wages than men. ( so if there is a 10% increase in wages, there would be a 8% increase in labour supply) - WHY this might be due to starting wage, women have lower starting wage so more responsive to price changes.
2) both positive if wage goes up they supply more labour ( there is a negative subsitution effect for both, hence both positive for labour supply ( wage goes up less leisure) WHY IS IT HIGHER FOR WOMEN THAN MEN? - if you are working full time, there is less time to devote to working.
Now lets add some income tax to our model? what are we going to say about income tax, use a 20% example? and show it on our budget constraint?
Income tax is proportional to total earnings ( the more you earn the more you are taxed.
Show income tax on the diagram
1) show the budget constraint with income tax and show original budget constraint, draw the pivot inwards with income tax, show the gradient and they new optimal bundle.
When there is a tax, the budget line pivots inwards and gradient is less.
What is EV again and show it on the diagram and how much tax is paid, is it enough?
The amount of extra money taken away from the consumer without changing prices that has the same effect on utility as the price change. Draw a parralel line downwards until its tangent to the new indifference curve we get a new bundle D.
The amount of tax that is paid. I draw a line with the same gradient going through b, and the difference between the original budget line and this new line is the tax paid.
The tax raised is less than EV, this means it is not enough revenue raised, there is excess burden of tax.
What can we say about proportional tax in summary?
It gives rise to an excess burden.
What is a more realistic model of income tax?
Having multiple tax brackets is the norm; the marginal tax rate for each bracket, where the higher the bracket the higher the marginal tax rate.
Explain this?
Total income tax = 0 (bracket 1 income) + 0.20 (bracket 2 income) + 0.40 (bracket 3 income) + 0.45 (bracket 4 income).
Find the amount of total income tax and a post tax income for someone earning £100000. using these marginal tax rates.
How can we translate late this into a diagram where there is post tax income on the y axis and pre tax income on the x axis ? with the maximum pre tax income of being 100000?
The slope is 1-t t being the marginal tax rate
The first 12500 the marginal tax rate is 0 so the slope is diagonal up and post and pre tax income the same.
up to 50000 the marginal tax rate is 0.2, meaning slope 1- 0.2 = 0.8. you pay tax of 7500 meaning post tax income is 42500.
Up to 100000, the marginal tax rate is 0.4, meaning slope 1-0.4 = 0.6 ( less steep, more elastic), i pay tax of 20000, as it is an accumulation of tax, the post tax income is 72500.
How would i find the average tax rate?
say £100000, the amount of tax corresponding ( 100000,72500) and connect to orgin the slope is 1-average tax rate 27500/100000 = 0.275, so 1- 0.275 = 0.725.
What do we notice here?
As you start moving through different tax brackets, you start getting taxed more heavily, and there appears kinks.
We want to convert this model into the one where we have Post tax income on the y axis and pre tax income on the x axis, to one where we have consumption and leisure.
But first we want to convert the x axis into number of hours worked, how do we go about doing this and use w = 50.
We can convert the pre tax income into number of hours worked T-N, by dividing pre tax income by wage, this tells us how many hours worked, so in this case as its 50 we divide by 50 and we also times slope by 50 as slope is now (1-t)W.
Now how do i convert the y axis into consumption? e.g. lets say price is =1?
You divide by price if price = 1 then the y axis don’t change
Change y axis to CONSUMPTION
Imagine the Y axis says Consumption, what do we now want to do?
The next thing to do is i want to convert hours worked to leisure, by doing total hours - hours worked = leisure Total hours worked = 365 days x 24 hours = 8760
Now summarise all the steps to get this diagram as a function of consumption and leisure? Remembering prices = 1 and w = 50. Draw the diagram including 150000.
Divide pre tax income by 50, to find hours worked, then take this away from total hours to find leisure. To turn Post tax income into consumption, as price = 1, we divide post tax income by price to find quantity of consumption. So on the x axis total left to right 8760 - 250 = 8510 8760-1000 = 7760 8760 - 2000 = 6760 8760 - 3000 (150000/50) = 5760
On y axis the extra 50000 x 0.45 = 22500+ 27500 =50000 ( total tax)
Post tax income = 100000 / price =1 = 100000 on y axis.
How many kinks are there?
( BTW the optimal decision is assumed here to be A)
What effects can we analyse?
There are 4 kinks for every tax bracket.
we can analyse effects of changes in tax rate e.g. if the 40% tax bracket changed to 35% , this would mean less shallow (as gradient changes 1 -t changes, and we can analyse substuition and income effects.
So show the tax brackets when income is > 100000 and when income is less than 100000.
We are going to analyse the effect of a reduction of personal allowance using un-tapered removal ( all at once)
Find out the difference in tax and post tax income, when someone who earns 100000 ( so they don’t pay tax on 100000) and someone who earns 100001?
What can we see?
If you earn £1 more, you pay £5000 more tax, this is unacceptable.
So show tapered removal with 1 column marginal tax rate, tax paid with £100,001 and tax paid with £100000.
As for every £1 you earn, you are personal allowances reduces by 50p, so in the first marginal tax rate of 0% £12.499.50 doesn’t get taxed.
Then the next 37500 is taxed at 20% ( by law) the remaining is 50000.5 this taxed at 40%.
What is the mariginal tax rate, why is this the case and why do we charge charge the 50p in the 40% bracket?
What we are doing is just shifting the brackets, and reducing the amount of personal allowance. So that 0.5 pound that is shifted, you may count it in the 20% bracket. But due to that shift, there is a 0.5 extra in 40% bracket too due to a downward shift. So effectively on that 0.5 pound shift into the 40% tax bracket, but you are paying 20 plus 40% so effective rate comes around to 60% (because not just there is a shift in the 40 bracket but also in the 20 percent bracket).
Marginal tax rate is 60%
So we know that tapered removal leads to weird marginal tax rates? is this intentional from policy makers?
No, its not well thought out
We first draw the diagram with post tax income as a function of pre-tax income.
We have to convert the y axis into consumption, so divide by p. As p = 1, Post tax income = consumption.
2) We need to change x axis, so first we do pre tax income / W to find number of hours worked . Then as we are working trying to get leisure on x axis we take number of hours worked from T( total hours) to find leisure, here we just use letters.
The introduction of the high marginal tax tate does change the first segment of budget constraint, the budget line becomes flatter.
remember the slope on the diagram on the right is all negative, as opposed to the left.
Do b
Here we need to analyse the substitution and income effects, as it is a change in marginal tax rate. To do so show the line where no increase in marginal tax rate vs when there is.
1) Originally the consumer is at bundle A. An increase in new high marginal tax, pivots the original line inwards ( slope becomes shallower 1-0.5 = 0.5) compared to the line if there was no marginal increase ( remember read the question, it states new high marginal tax rate, so there wasn’t one before).We get to bundle c.
2) To decompose IE AND SE, we draw a fake line from new line, parallel up to when its tangent to the original indifference curve. to get to BUNDLE B.
3) The movement from A to B is the substation effect. (working more hours now leads to a lower increase in income. Intuitively, leisure has become “cheaper” as its opportunity cost is now lower. ( Negative in labour supply)
4) As they feel poorer they work more hours, thus consume more and less of leisure( Positive income effect of labour supply).
Overall the IE AND SE effects reinforce each other for consumption, so this consumption deffo decreases, but an ambiguous effect on hours worked and leisure.
4) If the subsitution effect > IE which this diagram shows then overall labour supply falls.
5) If Income effect > SE then overall labour supply increases
It says assume that for all pre tax incomes the marginal tax rate was 40% so we draw a line with slope 1-0.4 = 0.6. ( solid line and also show a fainted 45* line.
Now there has been a change. in tax laws
1) A marginal tax rate for pre tax incomes below x, so we show this on Post tax income as a function of pre tax income diagram,
2) A marginal tax rate of 60% for those with pre tax from £x and £y and where its 40% for y. So draw that.
Now its time to convert on the diagram of Consumption and leisure, so for the y axis we do post tax income / p =1 so consumption = post tax income.
Now for the horizontal axis we do pre tax income / w =1 = pre tax income then take this away from T( total hours) to get leisure ON X AXIS.
remember slope is negative.
i)
no effect. People earning below £x are unaffected by the tax change. If they were optimally earning below £x before the change they will also optimally earn below £x after the change.
ii)
As there is an increase in marginal tax rate between T-y and T-x to 60% there are subsitution and income effects. Originally be at bundle A, but due to an increase in marginal tax rate, the consumer reoptimises at bundle C
We can decompose the SE AND IE. We draw a parallel line from the new line between T-y and T-x, up towards where the line is tangent to the original indifference curve.
1) The movement from A to B is the substitution effect. People work less as working leads to lower levels of increase in income. So leisure becomes attractive ( opportunity cost of leisure as fallen. ) ( corresponds to a negative subsitution effect on labour supply.
2) The movement from B to C is the income effect, as i feel relatively poorer, i have to work more hours and reduce my leisure. ( this is a positive IE). For consumption the effects reinforce each other, ( Negative IE AND SE). For leisure and hours worked its is ambigiuous as the effects go in opposite directions.
IF SE > IE which it shows on the diagram overall, you will work less. ( negative labour supply.
If IE> SE then you work more ( Positive labour supply.
(iii)
only an income effect. People who were optimally earning above £y before the tax change will now experience an income effect, leading them to work more hours and spend less time in leisure. If we assume both goods are normal , both will reduce. ( Consumption and leisure)
a
b
C.
