So what would an NDP government do?

[cybercoma]

Next time I'm in the city if I remember I'm going to take pics at the welfare offices.

What are pics going to show? You can't tell many people's disabilities from looking at them and you are not a doctor. But I'm not going to hold my breath on you spying on people in public. We're still waiting for those Toronto Price pics you promised many years ago. Or was that someone else? I can't remember. Your posts are so similar.

[Canada_First]

What are pics going to show? You can't tell many people's disabilities from looking at them and you are not a doctor. But I'm not going to hold my breath on you spying on people in public. We're still waiting for those Toronto Price pics you promised many years ago. Or was that someone else? I can't remember. Your posts are so similar.

being sad sometimes us not a disability. Those ppl need to get back to work and free up the money for those that need it. The pics will show whatever in the office at the time. I'll blur out faces before posting here if I need to. I don't wish to break ant rules. I'll go on a random day and show what I see.

[-1=e^ipi]

Hindering upwards mobility is exactly what people have been saying for at least a generation about flat tax schemes.

People have also being saying for generations that leprechauns exist. Just because people have been saying something, doesn't mean it is true.

It disproportionately burdens lower income earners, which is exactly what I said at the beginning of this discussion, but Euler wanted mathematical proof.

You haven't properly defined disproportionate and I doubt your definition is in agreement with mine.

Evidently he can't even see the proof in his own math.

Looks to me that you are in denial about the lack of justification for your claim about it 'disproportionately' affecting the poor.

You're going to eventually come back to a system of credits and exceptions that will leads you to where we are today again, effectively making the "flat" tax meaningless.

It doesn't have to be overly complicated. And actually under log utility, it's difficult to justify much variation in guaranteed income.

[-1=e^ipi]

do you really think businesses are going to pay people anything at all when you abolish minimum wage?

Yes Mr. Economically illiterate, they will. Wage will be determined via supply and demand. If a business is unwilling to pay workers anything, then they will have difficulty finding workers. Also, low income workers will be arguably less desperate to find work if they have a guaranteed income, which increases their reservation wage.

They may even pay them $0/hour and just offer benefits as an incentive to go work.

If an employer wants to pay someone benefits and an employee wants to work for those benefits then that is a Pareto improvement compared to the employer not having a worker and an employee not working.Two people are better off, no one is worse off. Do you reject the Pareto principle?

We can already see in many places how Walmart relies on government subsidies to keep their employees afloat.

Please define what you mean by 'afloat' are provide proof for your claim.

The government giving everyone a guaranteed allowance would give Walmart even less of a reason to pay their employees.

1. Lower wage isn't necessarily bad if more people are working.

2. Guaranteed income increases people's reservation wage for working, so a priori you can't say that wages will decline.

[-1=e^ipi]

A cutoff age of 16 may be more appopriate, a 10 year old would not need 25k a year, on top of their parents 25k. Even if that parent is a single parent.

A 10 year old does not contribute enough to society to recieve 25k of tax payers money, sorry but that's just how I see it. It's their parents job to feed, cloth and shelter them.

You look at this differently from me. For me it's a matter of maximization of expected utility for a person in society, so there is some reason to justify guaranteed income for children.

being sad sometimes us not a disability.

Depression? Mental health issues? Not sure I agree with this.

[cybercoma]

being sad sometimes us not a disability.

What makes you think I'm talking about being "sad" sometimes? There are plenty of conditions that you wouldn't seen at a glance, but would prevent someone from putting in a full day of work.

Moreover, if you're referring to depression and other mental illnesses, calling it "being sad sometimes" is incredibly ignorant of just what kind of effect that condition has on people. Depression is far from "feeling sad." The opposite of "depressed" is not "happy."

[cybercoma]

.

Edited June 26, 2015 by cybercoma

[cybercoma]

Yes Mr. Economically illiterate,

Your argument is so good you have to resort to insults. Good job.

[cybercoma]

Euler, I'm not going to do first hand proofs myself because I don't have the time nor inclination to waste that energy on this forum for someone who resorts to petty insults and isn't persuaded by any manner of argument (as can be seen by your past "debates" with Michael Hardner as well). However, if you were an actual scientist, you would look at research that contradicts your opinions instead of just mentally masturbating on the forum here. A flat tax was empirically shown to exacerbate inequality by concentrating income and wealth at the top (source). It disproportionately benefits the already wealthy (source). There's also the issue of how expensive and complicated it will be to implement such a system, as well as how easy it is to avoid paying the tax (source). At the end of the day, a flat tax disproportionately benefits the top income earners and places more of a burden on the rest of society, even when you place a personal exemption in there to create some for of "progressiveness" (source). So you can sit here and call names all you want and throw all the mathematical proofs around that you want, but this issue has been debated amongst economists for the last 20 years. The majority of the research shows that a flat tax disproportionately benefits the rich and actually increases inequality. Then we can get into a discussion of how inequality harms societies because that's exactly the problem your proposal creates. And that's not according to me. That's according to the last 2 decades of academic research on the topic.

Edited June 26, 2015 by cybercoma

[-1=e^ipi]

A flat tax was empirically shown to exacerbate inequality by concentrating income and wealth at the top (source). It disproportionately benefits the already wealthy (source).

Congratulations on not understanding the difference between comparative statements and absolute statements. Way to take comparative statements from various studies and somehow pretend they support your absolute statements.

There's also the issue of how expensive and complicated it will be to implement such a system

Yes, cause it's not like our current more complicated tax system costs more to implement.

At the end of the day, a flat tax disproportionately benefits the top income earners and places more of a burden on the rest of society, even when you place a personal exemption in there to create some for of "progressiveness" (source).

Again, there is a difference between a comparative statement and an absolute statement, and none of these studies discuss the combination of guaranteed income and flat tax.

Edited July 7, 2015 by -1=e^ipi

[-1=e^ipi]

Cool, so apparently I want to both harm the poor (Cybercoma) and give them a free ride (Canada_First).

[cybercoma]

none of these studies discuss the combination of guaranteed income and flat tax.

Apparently you didn't even read the study that was linked in the quote directly above this comment.

[cybercoma]

Cool, so apparently I want to both harm the poor (Cybercoma) and give them a free ride (Canada_First).

Harm the poor?

I said create greater inequality. I didn't say harm the poor. I guess if I'm economically illiterate, you're just plain old illiterate.

[-1=e^ipi]

Since you guys insist, I'll try to give a rough estimate of the magnitude of flat tax and income tax that I think would make sense using a crude and simplistic model. I'll start by trying to estimate the social welfare function.

First, let's assume that people have a logarithmic utility function (explained some justification of this earlier) of consumption and leisure (i.e. U = ln( c) + ln(l), where c is consumption and l is leisure). The social welfare function will simply be the sum of the individual utilities.

Now I need to estimate the tradeoff between additional taxes and hours worked. For simplicity, let's consider a country with a representative consumer. The consumer will consume (1-s) times their income, where s is the savings rate. The consumer's income is (1-t)*(w*L + r*K) + T, where t is the tax rate, w is the consumer's wage, L is their labour in hours worked, r is the interest rate, K is their Savings and T is transfers.

For government expenditure, let's suppose that the expenditure consists of essential services and transfers. Let's look at federal (http://www.statcan.gc.ca/tables-tableaux/sum-som/l01/cst01/govt49b-eng.htm) and provincial (http://www.statcan.gc.ca/tables-tableaux/sum-som/l01/cst01/govt56b-eng.htm) expenditures in 2009. Let's assume that general government services, protection of persons and property, transportation and communication, health, education and environment are essential services and everything else is transfers (you can agree or disagree with this assumption, but I need to make assumptions in order to quantify things). This works out to 53.7% of government expenditure being essential and everything else being transfers. Roughly 32.2% of Canada's GDP is tax revenue (https://en.wikipedia.org/wiki/List_of_countries_by_tax_revenue_as_percentage_of_GDP), so this works out to ~17.3% of GDP being essential government services.

If essential government services is 17.3% of GDP, then everything else is the 82.7%. Let's assume that rK is 1/3 of the GDP (i.e. physical capital's share of income is 1/3). Then we get that rK = ((1-t)*(w*L + r*K) + T)/3/0.827
=> (1.481+t)*rK = (1-t)*wL + T => rK = ((1-t)*wL + T)/(1.481+t).
Substituting this back into consumer income gives (1-t)*(wL + ((1-t)*wL + T)/(1.481+t)) + T
= 2.481*((1-t)*wL + T)/(1.481+t).

If a person works an additional hour, then get one less hour of leisure, so l = H - L, where H is some constant (note that it isn't really known a priori because for example it is unclear if sleeping counts as leisure). So the person's utility becomes U = ln((1-s)*2.481*((1-t)*wL + T)/(1.481+t)) + ln(H-L)
= ln(1-s) + ln(2.481) - ln(1.481+t) + ln((1-t)*wL + T) + ln(H-L).

The consumer will try to maximize their utility. Their utility is maximized when the derivative of U with respect to L is zero. This gives 0 = w*(1-t)/((1-t)*wL + T) - 1/(H-L)
=> H - L = L + T/w/(1-t).

If the government is revenue neutral then the tax revenue (t*(wL + rK) = t*(2.481*wL + T)/(1.481+t)) must equal the government expenditures (T + 17.3% of GDP = T + 0.173*((1-t)*(wL + rK) + T)/0.827 = T + 0.519*((1-t)*wL + T)/(1.481+t)).

This gives t*(2.481*wL + T)/(1.481+t) = T + 0.519*((1-t)*wL + T)/(1.481+t)

=> T*(1 + (0.519 - t)/(1.481+t)) = (2.481*t - 0.519*(1-t))*wL/(1.481+t)

=> T*2/(1.481+t) = (3*t - 0.519)*wL/(1.481+t)

=> T = (1.5*t - 0.2595)*wL

Combining the condition of revenue neutrality with utility maximization gives
H - L = L + (1.5*t - 0.2595)*L/(1-t) => H = L*(1.7405-0.5*t)/(1-t) => L = H*(1-t)/(1.7405-0.5*t).
This describes the relationship between labour in hours worked and the tax rate. A higher tax rate results in less hours worked.

If I take the average annual hours worked per worker for developed OECD countries (http://stats.oecd.org/index.aspx?DataSetCode=ANHRS) and tax revenue as a percentage of GDP (https://en.wikipedia.org/wiki/List_of_countries_by_tax_revenue_as_percentage_of_GDP) then I find that the calculated H values are roughly constant at ~ 4000 hours. In Canada's case I get 3974.

However, using a 32.2% tax rate for Canada isn't really accurate for the above calculation since Canada's tax system is 'progressive'. So I should try to take this into account. I can get the distribution of income for Canadians here (http://www.statcan.gc.ca/tables-tableaux/sum-som/l01/cst01/famil105a-eng.htm). I think it only includes working age Canadians, so for the sake of simplicity I'll exclude everyone else. If we treat everyone in each income category as having the middle income of the category (except the last category which I will assume earn $381000 http://www.cbc.ca/news/canada/who-are-canada-s-top-1-1.1703321) then I can calculate the marginal tax rate, the effective tax rate, the percentage of the population and the percentage of total income for each group. For simplicity, I'll pretend everyone lives in Ontario and look at income taxes. I'll also assume that everyone that earns less than $10,000 doesn't pay tax.

Income Range: Approximate Income: % of Population: % of Total Income: Marginal Tax Rate: Effective Tax Rate:

5000- 2500 7.5% 0.4% 0% 0%
5000-10000 7500 6.7% 1.1% 0% 0%
10000-15000 12500 9.0% 2.5% 20.05% 20.05%
15000-20000 17500 9.6% 3.8% 20.05% 20.05%
20000-25000 22500 8.1% 4.1% 20.05% 20.05%
25000-35000 30000 12.7% 8.5% 20.05% 20.05%
35000-50000 42500 15.9% 15.1% 24.15% 20.20%
50000-75000 62500 15.4% 21.5% 31.15% 23.46%
75000-100000 87500 7.7% 15.0% 33.16% 25.78%
100000-150000 125000 4.9% 13.7% 37.16% 29.14%
150000-200000 175000 1.3% 5.1% 41.16% 32.19%
200000-250000 225000 0.5% 2.5% 42.16% 34.21%
250000+ 381000 0.8% 6.7% 42.16% 35.49%

Now this gives an average (weighted by income) marginal rate of 30.17% and an average effective rate of 24.61%; this works out to ~288 billion dollars. However, Canada's tax revenue is ~32.2% of GDP and GDP was ~1894 billion dollars in 2013 (http://www.statcan.gc.ca/tables-tableaux/sum-som/l01/cst01/econ04-eng.htm), so I'm missing ~322 billion dollars. Obviously Canadian governments have other means of obtaining tax revenue such as consumption taxes, employment insurance and corporate tax. Since these taxes are difficult to determine the distribution and affect everyone (corporate taxes reduce jobs available to poor people) I'll add 27.5% tax to each income category so that the total tax revenue is in agreement with 32.2% of GDP. I am skeptical of how correct this approximation is, but I need to use something to perform a calculation; if anyone wants to provide me with a better distribution of tax burden I will be happy to use that. Anyway, I get:

Income Range: Approximate Income: % of Population: % of Total Income: Marginal Tax Rate: Effective Tax Rate:

5000- 2500 7.5% 0.4% 25.70% 25.70%
5000-10000 7500 6.7% 1.1% 25.70% 25.70%
10000-15000 12500 9.0% 2.5% 45.75% 45.75%
15000-20000 17500 9.6% 3.8% 45.75% 45.75%
20000-25000 22500 8.1% 4.1% 45.75% 45.75%
25000-35000 30000 12.7% 8.5% 45.75% 45.75%
35000-50000 42500 15.9% 15.1% 49.85% 45.90%
50000-75000 62500 15.4% 21.5% 56.85% 49.16%
75000-100000 87500 7.7% 15.0% 58.86% 51.48%
100000-150000 125000 4.9% 13.7% 62.86% 54.84%
150000-200000 175000 1.3% 5.1% 66.86% 57.89%
200000-250000 225000 0.5% 2.5% 67.86% 59.91%
250000+ 381000 0.8% 6.7% 67.86% 61.19%

This works out to an average effective tax rate of 50.31% and an average marginal tax rate of 55.87%. Again, this is a rough idea of the tax burden due to all taxes, although I suspect I am underestimating 'progressivity'. Anyway, if I go back to my earlier equation that I obtained for imposing the condition that the consumer tries to maximize utility (H - L = L + T/w/(1-t)). I should point out that the t in this equation should correspond roughly to the average marginal tax rate. However, the t in my equation for revenue maximization (T = (1.5*t - 0.2595)*wL) should correspond to the average effective tax rate.

So I have H = L*(2 + (1.5*t_e - 0.2595)/(1-t_m)), where t_e is the effective tax rate and t_m is the marginal tax rate. For Canada's average hours worked per year (1706) this gives H = 5326. So I'll use ln(1-s) + ln(2.481) - ln(1.481+t) + ln((1-t)*wL + T) + ln(5326-L) as my utility function. However, the second term is a constant, so I can drop it and simplify my utility function to:

U = ln(1-s) - ln(1.481+t) + ln((1-t)*wL + T) + ln(5326-L)

To be continued...

Edited June 27, 2015 by -1=e^ipi

[cybercoma]

You do know that paying 50% of your income when you make $250k a year is less of a burden than someone who makes $30k a year paying 50% of their income in taxes, right? Equal proportion does not mean equal burden.

[-1=e^ipi]

You do know that paying 50% of your income when you make $250k a year is less of a burden than someone who makes $30k a year paying 50% of their income in taxes, right?

No, I did not know that.

You do know that paying 50% of your income when you make $250k a year is less of a burden than someone who makes $30k a year paying 50% of their income in taxes, right? Equal proportion does not mean equal burden.

Cool, can you define burden for me? What % of the $250k person's income results in equal burden as 50% of the 30k person's income?

[cybercoma]

No, I did not know that.

Cool, can you define burden for me? What % of the $250k person's income results in equal burden as 50% of the 30k person's income?

The amount that makes it as difficult to survive as someone with $15,000 left to their name.

[PrimeNumber]

The amount that makes it as difficult to survive as someone with $15,000 left to their name.

With the $25,000 GI their income is now $40,000.

[cybercoma]

He keeps reminding us that he never said $25,000.

[PrimeNumber]

He keeps reminding us that he never said $25,000.

No, that was me.

[-1=e^ipi]

Firstly, I want to resolve an issue. There is quite a difference between the total income earned by the individuals on that table (~1170 billion) and Canada's 2013 GDP (~1894 billion) there are numerous reasons for this such not everyone filing their taxes, not all income being taxed, error in the method of how I calculated total income, or accounting discrepancies. In my above model, I need GDP to equal the sum of rK + wL for all individuals otherwise there is too much internal inconsistency. So I'll assume that Canada's GDP is roughly 1170 billion. Note that the tax burden being ~32.2% of 1894 billion is still reasonable and is consistent with the government expenditure estimates on the government of Canada website for 2009 (so I'll leave it as is).

From here on, I'll use t_e as the effective tax rate of a specific individual, t_m as the marginal tax rate of a specific individual and t as the average effective tax rate of society (weighted by income). I'll also use the functions mean and sum to refer to the mean and sum over all individuals (in an attempt to make notation simpler). T will refer to transfers per capita.

Recall that we had rK = ((1-t)*(w*L + r*K) + T)/3/0.827 for the representative consumer. In the more general case, this condition (physical capital's share of income being 1/3 of GDP) becomes
r*sum(K) = ((1-t)*(sum(wL) + r*sum(K)) + sum(T))/2.481
=> (1.481+t)*r*sum(K) = (1-t)*sum(wL) + sum(T)

The revenue neutrality condition becomes:
t*(sum(wL) + r*sum(K)) = sum(T) + 0.173*((1-t)*(sum(wL) + r*sum(K)) + sum(T))/0.827
Substituting in r*sum(K) = ((1-t)*sum(wL) + sum(T))/(1.481+t) gives:
t*(2.481*sum(wL) + sum(T))/(1.481+t)) = sum(T) + 0.519*((1-t)*sum(wL) + sum(T))/(1.481+t)
=> sum(T) = (1.5*t - 0.2595)*sum(wL) (this is basically what we had with the representative consumer)

Combining the revenue neutrality condition with physical capital being 1/3 of GDP gives:
(1.481+t)*r*sum(K) = (1-t)*sum(wL) + (1.5*t - 0.2595)*sum(wL) = (0.7405 + 0.5*t)*sum(wL)

If for the sake of simplicity let's choose a model where people's savings are proportional to their labour income, then the above equation becomes rK = (0.7405 + 0.5*t)*wL/(1.481+t).
So a person's taxable income is rK + wL = wL*(1 + (0.7405 + 0.5*t)/(1.481+t))
= wL*(2.2215 + 1.5*t)/(1.481+t)
=> wL = i*(1.481+t)/(2.2215 + 1.5*t), where i is taxable income.

For the utility maximization condition of an individual, we get H - L = L + T/w/(1-t_m)
Substituting in the revenue neutrality condition gives:
H = 2L + (1.5*t - 0.2595)*mean(wL)/w/(1-t_m) => wH = 2wL + (1.5*t - 0.2595)*mean(wL)/(1-t_m)
=> w = (2*i + (1.5*t - 0.2595)*mean(i)/(1-t_m))*(1.481+t)/(2.2215 + 1.5*t)/H

Using this, I can calculate the effective hourly wage for each income group (remember that t = 50.31%). Once I have w I can get L as L = i*(1.481+t)/(2.2215 + 1.5*t)/w. I get:

Income Range: Approximate Income: % of Population: Marginal Tax Rate: Hourly Wage: Hours Worked:

5000- 2500 7.5% 25.70% $4.37 383
5000-10000 7500 6.7% 25.70% $5.61 892
10000-15000 12500 9.0% 45.75% $8.24 1012
15000-20000 17500 9.6% 45.75% $9.49 1229
20000-25000 22500 8.1% 45.75% $10.74 1396
25000-35000 30000 12.7% 45.75% $12.62 1585
35000-50000 42500 15.9% 49.85% $16.17 1753
50000-75000 62500 15.4% 56.85% $22.07 1888
75000-100000 87500 7.7% 58.86% $28.64 2037
100000-150000 125000 4.9% 62.86% $38.76 2150
150000-200000 175000 1.3% 66.86% $52.17 2236
200000-250000 225000 0.5% 67.86% $64.95 2309
250000+ 381000 0.8% 67.86% $104.00 2442

Now I will point out that this gives an average number of hours worked at 1502, which is slightly lower than the empirical 1706. The reason for this is that my estimate for H in the first part was a rough estimate. Rather than try to analytically rederive H, I'll just vary H until average hours worked is 1706. This gives H = 6051. This modifies the above table to:

Income Range: Approximate Income: % of Population: Marginal Tax Rate: Hourly Wage: Hours Worked:

5000- 2500 7.5% 25.70% $3.83 435
5000-10000 7500 6.7% 25.70% $4.94 1013
10000-15000 12500 9.0% 45.75% $7.25 1149
15000-20000 17500 9.6% 45.75% $8.35 1397
20000-25000 22500 8.1% 45.75% $9.45 1587
25000-35000 30000 12.7% 45.75% $11.11 1801
35000-50000 42500 15.9% 49.85% $14.23 1991
50000-75000 62500 15.4% 56.85% $19.43 2145
75000-100000 87500 7.7% 58.86% $25.21 2314
100000-150000 125000 4.9% 62.86% $34.11 2443
150000-200000 175000 1.3% 66.86% $45.92 2541
200000-250000 225000 0.5% 67.86% $57.17 2624
250000+ 381000 0.8% 67.86% $91.54 2775

I'll just reiterate that these are the values that are internally consistent with the model. You guys wanted a rough estimate, so I'm giving a ROUGH estimate. Most likely the assumption that someone's savings are proportional to their wage income is incorrect and this is causing an underestimating of the hours worked and hourly wages of poor people and overestimating these factors for rich people (so to correct for this I would have to greatly complicate the model to look at consumption smoothing behaviour; or alternatively find empirical values for average income by income group and then infer savings by income group to be internally consistent). Another issue is that policies like minimum wage would make this model give inaccurate values for poor people because it doesn't take into account the fact that poor people aren't as able to find as much work as they would like.

Anyway, I needed to be able to get values for w by income group and H in order to continue with my rough estimate of optimal flat tax and guaranteed income.

To be continued...

Edited June 27, 2015 by -1=e^ipi

[-1=e^ipi]

Okay, so now let's consider a flat tax so that t = t_e = t_m. The utility function becomes U = ln(1-s) - ln(1.481+t) + ln((1-t)*wL + T) + ln(6051-L). Using the revenue neutrality condition (sum(T) = (1.5*t - 0.2595)*sum(wL)) gives:
U = ln(1-s) - ln(1.481+t) + ln((1-t)*wL + (1.5*t - 0.2595)*mean(wL)) + ln(6051-L)

The social welfare function becomes:
sum(U) = N*ln(1-s) - N*ln(1.481+t) + sum(ln((1-t)*wL + (1.5*t - 0.2595)*mean(wL))) + sum(ln(6051-L))
One can divide by N (number of people) to get an equivalent social welfare function:
mean(U) = ln(1-s) - ln(1.481+t) + mean(ln((1-t)*wL + (1.5*t - 0.2595)*mean(wL))) + mean(ln(6051-L))

Now I need to take into account the fact that L depends on the tax rate t. From individual utility maximization I obtained: wH = 2wL + (1.5*t - 0.2595)*mean(wL)/(1-t). Taking the mean over all individuals gives:
H*mean(w) = 2*mean(wL) + (1.5*t - 0.2595)*mean(wL)/(1-t)
=> H*mean(w) = (1.7504 - 0.5*t)*mean(wL)/(1-t)
=> mean(wL) = H*mean(w)*(1-t)/(1.7504 - 0.5*t)
=> wH = 2wL + (1.5*t - 0.2595)*H*mean(w)*(1-t)/(1.7504 - 0.5*t)/(1-t)
=> L = 0.5*H*(1 + mean(w)/w*(1.5*t - 0.2595)/(1.7504 - 0.5*t))

Substituting this back into the 3rd term of the social welfare function gives:
mean(ln((1-t)*wL + (1.5*t - 0.2595)*mean(wL)))
= mean(ln((1-t)*0.5*H*(w + mean(w)*(1.5*t - 0.2595)/(1.7504 - 0.5*t)) + (1.5*t - 0.2595)*H*mean(w)*(1-t)/(1.7504 - 0.5*t)))
= ln(0.5H) + ln(1-t) + mean(ln(w + mean(w)*(1.5*t - 0.2595)/(1.7504 - 0.5*t) + (1.5*t - 0.2595)*mean(w)/(1.7504 - 0.5*t)))
= ln(0.5H) + ln(1-t) + mean(ln(w + 2*mean(w)*(1.5*t - 0.2595)/(1.7504 - 0.5*t)))
Because ln(0.5H) is just a constant, we can drop it from the social welfare function without loss of generality. So we have ln(mean(w)) + ln(1-t) + mean(ln(w/mean(w) + (3*t - 0.519)/(1.7504 - 0.5*t))) as the 3rd term.

Substituting L into the 4th term gives:
mean(ln(6051-L)) = mean(ln(6051-0.5*H*(1 + mean(w)/w*(1.5*t - 0.2595)/(1.7504 - 0.5*t))))
= ln(0.5H) + mean(ln(1 - mean(w)/w*(3*t - 0.519)/(1.7504 - 0.5*t)))
= ln(0.5H) + mean(ln(1 - mean(w)/w*(3*t - 0.519)/(1.7504 - 0.5*t)))
Again, because ln(0.5H) is a constant, we can drop it. Furthermore, mean(w)/w is a constant in this model for a given individual, so without loss of generality the 4th term becomes:
mean(ln(w/mean(w) - (3*t - 0.519)/(1.7504 - 0.5*t)))

This means that the social welfare function becomes:
ln(1-s) + ln(mean(w)) - ln(1.481+t) + ln(1-t) + mean(ln(w/mean(w) + (3*t - 0.519)/(1.7504 - 0.5*t))) + mean(ln(w/mean(w) - (3*t - 0.519)/(1.7504 - 0.5*t)))

To maximize social welfare, the derivative of social welfare with respect to t must be zero:
0 = -1/(1.481+t) - 1/(1-t) + mean((3*(1.7504 - 0.5*t) + 0.5*(3*t - 0.519))/(1.7504 - 0.5*t)^2/(w/mean(w) + (3*t - 0.519)/(1.7504 - 0.5*t))) - mean((3*(1.7504 - 0.5*t) + 0.5*(3*t - 0.519))/(1.7504 - 0.5*t)^2/(w/mean(w) - (3*t - 0.519)/(1.7504 - 0.5*t)))
= -1/(1.481+t) - 1/(1-t) + mean(4.9917/(1.7504 - 0.5*t)^2/(w/mean(w) + (3*t - 0.519)/(1.7504 - 0.5*t))) - mean(4.9917/(1.7504 - 0.5*t)^2/(w/mean(w) - (3*t - 0.519)/(1.7504 - 0.5*t)))
= -1/(1.481+t) - 1/(1-t) + 9.9834/(1.7504 - 0.5*t)^3*(3*t - 0.519)*mean(1/((w/mean(w))^2 - ((3*t - 0.519)/(1.7504 - 0.5*t))^2))

I'm not sure if I can solve this analytically due to things like Abel's impossibility theorem (and I'm tired), so instead I varied t to obtain 0. I obtain a maximum social welfare when t = 24.73%.

To find T, I have T = (1.5*t - 0.2595)*mean(wL) = (1.5*t - 0.2595)*H*mean(w)*(1-t)/(1.7504 - 0.5*t). Using H = 6051 and t = 0.2473 gives T = $4588. So this would suggest that a flat tax of 24.73% and a guaranteed income of $4588 would be socially optimal (hopefully I didn't make an error). Note that 17.3% of GDP for essential government services is also funded by this tax system.

Now I want to talk briefly about the consumption tax. Recall that from post #274, the simple Solow model gives y = A*(sA/d)^(a/(1-a)). To be consistent with the solow model, let's suppose that mean(w) depends on factors we can't control (call it Q) times s^(a/(1-a)) (i.e. more physical capital gives a higher wage). For a=1/3, a/(1-a) = 0.5. Then the first two terms in the social welfare function become:

ln(1-s) + 0.5*ln(s) + ln(Q)

To find the optimal s, set the derivative with respect to s to be zero. This gives:

0 = -1/(1-s) + 0.5/s => s = 0.5 - 0.5*s => s = 1/3. Which is the same result as the solow model.

As I mentioned earlier, you would probably want to raise consumption taxes to 25% to get the savings rate to 1/3. A 25% VAT consumption tax at a savings rate of 1/3 is equivalent to a (1-s)*(1 - 1/(1 + 0.25)) = 13.33% income tax. So a 25% VAT tax and an 11.4% income tax would be preferable. Increasing the savings rate from 21% to 1/3 would increase GDP by (1/3/0.21)^(1/3/(1-1/3)) - 1 = 25.99%. So the corresponding guaranteed income would be $5780. Of course if you factored in all the efficiency gains from simplifying the tax code, getting rid of employment insurance, minimum wage, etc. then the justified guaranteed income would be even higher.

Edited June 27, 2015 by -1=e^ipi

[-1=e^ipi]

Thought I would try to calculate labour elasticity for the model to see if the model is consistent with empirical estimates of Canadian labour elasticity. From the utility maximization of the representative consumer I had: L = 0.5*H - 0.5*T/w/(1-t)

Taking the derivative with respect to w gives:
dL/dw = 0.5*T/(1-t)/w^2

Thus the labour elasticity is:
dln(L)/dln(w) = w/L*dL/dw = 0.5*T/(1-t)/L/w

For the representative consumer, T = (1.5*t - 0.2595)*wL.
=> dln(L)/dln(w) = (0.75*t-0.12975)/(1-t)

For an average effective tax rate of 50.3%, this gives a labour elasticity of 0.498.

It seems plausible, although it might be a bit on the high end of empirical estimates:

Edit: the two links I provided earlier do not have good overall empirical estimates of labour supply. One of the links was about labour demand not labour supply (so I'm a bit embarrassed about that).

If my model overestimates the labour elasticity (which means I am overestimating the effect taxes have on output), or if I am underestimating relative risk aversion (i.e. log utility isn't risk averse enough), then my earlier estimates for optimal flat tax & guaranteed income should be larger.

Externalities may justify greater income redistribution to poorer people. For example, there is a tendency for poorer people to commit more crime and empirical evidence suggests that lower poverty leads to a lower crime rate. Also, poorer people have worse health outcomes and thus greater income redistribution can lead to lower rate of spread of disease or lower health care expenditures. The magnitudes of these externalities can theoretically be quantified using empirical data.

If you put all that together, then I could see a guaranteed income as high as ~10k being justified based on empirical data. Although if consumption & income taxes were the only revenue sources then you would probably need a VAT of 25% and an income tax of 17% to achieve that (assuming you only kept the essential government services of 17.3% of GDP). Also, it would take a while to reach the new Solow equilibrium from a higher savings rate so initially you would only be able to afford ~8k guaranteed income.

Edited June 28, 2015 by -1=e^ipi

[Evening Star]

OK, a $10K GAI with 25% federal sales tax and a flat income tax rate of 17% definitely does not sound like a society I would want to live in.

I estimated a $25k GAI assuming that you were not making adjustments for people with children, disabilities, etc. If you are, then I could see how you could get away with a base-level GAI between $12K and $15K or so. However, any version of your scheme ends up with marginal tax rates being totally flat, with variation in effective rates only. Relative to a system of progressive marginal rates, this does seem to me like it would hinder upward mobility at lower income levels. It seems clear enough to me that it would also shift more of the tax burden to lower income levels. I'm pretty sure that this is why we ended up with progressive marginal rates in the first place. If you were going to come up with a method of plotting marginal rates on an increasing curve, I might be able to get on board with that.

[-1=e^ipi]

OK, a $10K GAI with 25% federal sales tax and a flat income tax rate of 17% definitely does not sound like a society I would want to live in.

Why does that sound so unappealing? If you don't think it is 'progressive enough' then may I ask what is your evaluation of what the 'progressiveness' of a society based upon? Why would the relative risk aversion of humans not reflect how much 'progressiveness' a society should have?

Relative to a system of progressive marginal rates, this does seem to me like it would hinder upward mobility at lower income levels.

I'm not sure I agree with this. You are increasing jobs available to low income people and arguably the GAI is high enough to avoid poverty traps.

It seems clear enough to me that it would also shift more of the tax burden to lower income levels.

Really? People earning less than ~$33000 would be on the net receiving end of the system.

I'm pretty sure that this is why we ended up with progressive marginal rates in the first place.

I don't think there was much logic behind how we got to the system we have, it just happened (with the help of lots of logical fallacies by politicians).