A "Conservative" Economic Justification of the Welfare State

My thoughts on the welfare state - atleast from an economic perspective. I will mark the sections without math so if this gets annoying, you can just skip to those. I hope they are accessible.

I am not going to make any Keynesian macroeconomic assumptions here for a few reasons, first being that I think it beocmes too easy to defend the welfare state and second that in the long run I think they are not necessarily right. My goal is also to demonstrate that "conservative economics," i.e. macroeconomics under New Classical assumptions and neo-classical production functions, is not incompatible with welfare structures. In fact, I argue, it encourages the existence of welfare states - especially in the regions I care about dearly (the developing world).

Ok. I will do any math in discrete time, and try to skip over anything that isn't really important to show. I just want to show the general results, because they are pretty interesting.



[General Argument (no math)]
Basically, I want to show the following: whether or not I assume the government to be productive, it is productive. And thus we have a justification for the welfare state.

I will show this by, first, assuming that the govt is 100% productive and second assuming that it is 0% directly productive, and show that we end up with basically similar results.

The first model allows us to deal with taxes. Say the govt infrastructure (roads, hospitals) was 100% effective and not wasteful at all. And say that it directly enters each firm's production function positively. I will show that ultimately, we end up with a growth function that has the following property: when government revenue = 0 (because of tax rate = 0), we have negative growth and when govt revenue = "max" since they steal everything when tax rate = 1 (i.e. 100%), we will also experience negative growth. Rolle's theorem tells us that if f(a)=f(b), then there exists some c within a,b such that f'(c)=0. i.e. if a function takes on the same negative growth rate because of 100% tax and 0% tax, it takes on some maximum growth rate with some intermediary tax. (ok sorry for the rolle's theorem bit, but i mean that is fairly intuitive right?)

The second model allows us to deal with the welfare state. We will assume that the government is not at all productive. 0% productive. All it does is redistribute from the rich to the poor. We will assume that inequalities decline as there is more distribution, and since distribution means taxes, inequalities decline as taxes rise. We will also assume ( rather fairly, I think) that too much inequality leads to instability. I mean, at the extreme, there will be revolution - we see this intuitively. So as taxes rise, instability goes down. Finally, we look at firms and direct impact to growth rates. There is a probability that a firm retains its own product. As we get more unstable, the firm is more likely to have people loot it and steal it. So basically, this probability is higher when there are really high inequalities, and lowere when there are low inequalities. The end result of this model will be, we will see, the same thing as above. That if taxes are too low or two high we end up with negative growth. So at some point in between, at some optimal tax rate, there will be an optimal welfare state contributing to growth and the government expenditure is effectively productive, though - by assumption - it wasn't!

Thus we can conclude that welfare structures are essential to strong economic growth under these assumptions. Notice I never assumed that the welfare was being used for anything other than appeasing the disgruntled masses. I could have leaned on things like externality benefits of education - I didn't.



[Feel free to ignore this part - I justify my growth eqn (Math)]
Assume: 1. infinite horizon of love. Basically parents care about their kids, and they about their kids, etc. So they take this into consideration - but they prefer themselves. 2. Also, say they prefer to smooth consumption over time.

To make it straightforward to write down, let's deal with it in discrete time. We will end up with Growth=1/J(r-p) where J is the curvature parameter on how they want to smooth consumption, r = rate or return on saving, and p = the preference of themselves.

Now say I assume a neo-classical production function (constant returns to scale, diminishing returns to capital, and these things called Inada conditions that we don't really need to go into ...). The basic deal there is ... if I double my inputs I will double my outputs. But if I only buy machines, and hold workers constant, then hell - I can't make a whole lot more Tickle-Me-Elmo dolls because it is hard for the same number of workers to operate more machines.

What we get is, r = f'(k)-d where f'(k) is marginal product of capital and d = depreciation rate of our machines. So we end up with Growth=1/J(MPK-p-d)



[The First Model: Governments 100% Productive (Some Math)]
A firm's output is given by y=f(A,k,g) where A = state of tech, k = capital, g = govt spending. Now let us assume that a government finances g with a tax at rate t. Assume it is flat for now. The govt can borrow and lend too, but in the long run, they cancel out (Ricardian Equivalence Hypothesis) so we just look at G = tY is the Govt tax revenue. So in per capita terms, g = ty. If we substitute this into a neoclassical production function, we end up with Growth=1/J((MPK of y-ty) - p - d) since at the end of the day, f'(k) for a firm now is f'(k)-taxed amount.

So we have Growth=1/J((1-t)MPK-p-d). Now remember that MPK is y'=f'(A,k,g) here. But A = state of technology is exogenous, so we can make y'=Af'(k,g). But recall that g=ty. Then MPK = Af'(k,ty))=Af'(k,t*f(k,ty)).

Point being that we end up with: Growth = 1/J((1-t)Af'(k,ty))-p-d)
Notice that, taxation rate has 2 effects:
1) in the (1-t) term, it slows down growth. This is the "distortion effect" of taxes.
2) in the f'(f,ty) term, it increases growth. This is tax revenue given to firms by government (in forms of roads, etc).

Also notice:
1. if t=0 then f'(k,ty) becomes 0 so we grow negatively at 1/J(-p-d)
2. if t=1 then (1-t) becomes 0 so we grow negatively at 1/J(-p-d)

Then, apply Rolle's Theorem to see that there is a maximum in between. Hell, we can solve for this maximum quite easily with basic calculus. Now it really depends on the type of production function, so say we go with our neo-classical production function, and we find that the optimal taxation rate is the government share of contribution to the firm's output. i.e. if the government contributes 0.3 out of 1 to producing 1 good, they should tax at 30%.

In Growth-Tax space, our curve is basically parabolic in form (more or less) taking a maximum at 0<1.



[The Second Model: Government is 0% Productive (Some Math)]
So here, y=A*p*f(k). g=ty is purely redistributive. In(t) = Inequality as a function of tax is a decreasing function. As t increases, inequalities fall. p(In) = probability of business keeping their product as inequality changes is a decreasing function. As inequality rises, probability drops. But then, p(In(t))=is probability depending on taxes. Realize that the probability of keeping product increases as t increases (since inequalities are falling).

So our MPK of (y-ty) is (1-t)*A*p(In(t))*y.
Thus we have:
Growth=1/J((1-t)*A*p(In(t))*y-p-d)

Notice that t again axts in 2 places.
1. if t=0 then p(In(t)) becomes 0 - i.e. 100% inequality leads to revolt so firms get screwed and have a 0% chance of holding on to their goods so growth is negative at 1/J(-p-d).
2. if t=1 then (1-t) becomes 0 so we grow negatively at 1/J(-p-d).

These results are identical to the taxation scheme we produced above, meaning that, by applying Rolle's theorem we see an maximum growth level exists at some t in between 0 and 1. Thus, the welfare state should exist to maximize economic growth. It also shows that welfare spending can be productive even if it is assumed not to be productive - since the A*p(Int(t)) term acts in the same manner as government spending financed by t acted when g was productive!

Thus, in Growth-Tax space we basically have the same curve.



[Tax and Welfare Policies (no math)]
Prof. Aaron Director at the University of Chicago discovered that, generally speaking, rich countries tended to tax beyond the optimal and poor countries would tax under the optimal (for whatever reasons). Thus, most econometricians, who collect data in OECD (rich) countries and plot taxation vs. growth rate see a declining relationship - that as these countries have more taxes = are more welfare oriented = have less inequality, they grow slower. This leads people at the WB, IMF, and conservative ideologues to argue that we should encourage everyone (even developing countries) to cut taxes and welfare to grow. I.e., they should privatize everything to grow. But what this neglects is that, poor countries usually don't have good data, since well - they are poor. So there is a bias. We are only looking at the part of the parabola that slopes downards - the part beyond the optimal taxation level (at which there is max growth). Since this is where rich countries reside, the conclusion is obvious.

But poor countries, they reside in the realm of under taxation. It is hard for them to collect taxes, for one thing. And there are a host of other reasons. But, our growth equation says that they will grow more if they indeed provide more welfare programs and tax more. Government spending would be good for growth. However, conservative ideologues (and WB and IMF) seem to blanketly apply the rich-country lesson that they should tax less and remove welfare structures to grow. That they fail to understand that the same model that accounts for the fact that rich countries probably have too much welfare also accounts for the fact that poor countries do not have enough social services. So over-privitization and encouraging minimal states in developing nations is idiocy.

Development aside, we also see that arguments for "minimal states" in any state is crazy - at least on economic grounds (with a conservative production function!). Unless, of course, by minimal state we mean optimal level of taxation. But I, for one, would call that cheating. Of course, our model didn't even take into account things like external benefits to education or welfare policy or health policy etc. All we did was considered completley unproductive government welfare-spending and we found that, by and large, there is definitely a productive element to this notion of "useless welfare" alone. Then take a minute and consider what it means for "useful welfare". Yah, that's what I thought. So people who think economic arguments blanketly reject welfare are sorely mistaken. In fact, we just demonstrated, using a "conservative" economic framework, whatever conservative means, that completely useless welfare is useful.

So let's not forget that a lot of time, a smarter economic policy might not be the thing that your conservative ideologue might be supporting!


**Disclaimer** Now this welfare justification on sheer basis of political volatility has no relation to social security. It's not as if old people are going to revolt any time soon. And social security is a huge expenditure. So that is for a different day. I am just making the case that, welfare states to whatever degree need not be only justified by political philosophy or Keynesian economics. There are definite structures in New Classical economics that support welfare states.