Essays on Government: The Framework of a Social Insurance
“So that our free will may not be extinguished, I think it may be true that Fortune is the arbiter of half of our actions, but that she still leaves the other half of them, more or less, to be governed by us.”
— Niccolò Machiavelli
In the previous essay, I defined the social justice to be the outcome after trade under equal initial endowments. My series of essays would have ended there were it not for the fact that nature often plays a significant role in determining whether or not a just solution to welfare problems exist. This essay formalises the rule of nature and its far-reaching consequences. I argue that the only means by which nature could be reconciled with is through progress in technology. However, it is only with the correct transfer parameters conducted by an arm of the government, the so-called ‘welfare state’, that a social justice suffices to occur. These transfer parameters we analyse in part of our formal construction of the welfare tax regime. Towards the end, we pick between socially just transfer parameters that differ in its ability to insure the welfare of individuals under the Rawlsian veil of ignorance.
Nature’s Rule
Recall the idea of an allocation space as represented, in the two-by-two case, by an Edgeworth box. The government initialises the economy by picking a point of initial endowment and allow the rule of trade to establish an equilibrium outcome. If the economy is initialised in the centre of the Edgeworth box, then the mechanism is Rawlsian and Paretian, and so, by definition, socially just. But the centre of the Edgeworth box is not necessarily available as a point of initial endowment. More generally, the government could be restricted to only some proper subspace of the allocation space, as is true in the case when the goods of concern are non-redistributable human traits, such as talent or the number of limbs one is born with. To illustrate, imagine an economy of two individuals, one of whom is born with one leg and the other with two. The total number of legs in the economy is three, and the economy is modelled by the following Edgeworth box of legs against some good A:
Obviously, the government cannot initialise the economy such that each individual has a leg and a half, and expect the same dynamics of justice to play out, assuming that half a leg is a non-contributing factor to utility. There is also no transferring of one’s talent to another: our human government simply does not know how to do this. In such scenarios there is an additional restriction to the allocation space. Assuming that the good A is a redistributable resource, the allocation space becomes the proper subset of the Edgeworth box as represented by the thick black line:
The government cannot horizontally deviate from the black line to change the endowment that nature has, as unfairly as we think it to be the case, nevertheless proposed. In specifying an initial endowment, the government could only change each individual’s property along the vertical. This extra rule that both the government and its population are subject to we call a ‘nature’s rule’.
Notice here a rather foreboding insight: though a Pareto-efficient outcome is available under any nature’s rule (since the contract curve starts and ends at the upward sloping diagonal corners of the Edgeworth box), a social justice is available seemingly only if nature’s rule is coincidental with the social justice itself:
Moreover, if markets are necessary for the delivery of justice, then the government cannot even know whether or not nature’s rule in fact coincides with the socially just outcome. This is true if we recall the conditions that make markets a necessary means: asymmetry in information about preferences between the state and the private sector. But of course the rational mind is not always unpredictable. I now turn our attention to the analysis of preference structures in relation to types of goods, which will bring us to a rather powerful theorem, the necessary and sufficient conditions for the existence of a social justice.
The Levels of Perfect Substitutability
The marginal rate of substitution measures the instantaneous substitutability of one good for another so that an individual remains indifferent. Imagine that you have 50 slices of pizza and a small glass of water over a certain time frame, say, a day. Then, you might be willing to trade 10 slices of pizza for another glass of water in order to remain equally as satisfied. In this scenario, the marginal rate of substitution of pizzas for a glass of water is 10. Imagine now that you have made this trade (and are equally as satisfied), then, you might be willing to trade less pizzas for another glass of water because you have just gotten a little bit poorer in pizzas and richer in water after the initial trade. You are more willing to hold on to pizzas. These two goods are said to be imperfect substitutes: your valuation of each good in terms of the other changes as the configuration of your wealth changes. In my previous essay, the reason for why Pareto-efficient outcomes lie on a contract curve is due to our assumption of imperfect substitutability between the two goods in concern, because indifference curves of imperfect substitutes are mutually tangential to the Rawlsian rule at a unique point:
Now imagine that you are trading Coke against Pepsi. You may like Coke more but it is likely that you do so in a linear fashion: for any wealth configuration, two cans of Pepsi, say, will always give you the same satisfaction as do one can of Coke. This implies that your indifference curve has a constant slope that is not -1. In fact, it is -2 if Coke is represented on the x-axis. Goods that induce such a preference structure from individuals we call a ‘weak perfect substitute’. If the other individual has the same preference structure, then the aforementioned contract curve is rather a ‘contract space’ that spans the entire Edgeworth box since the mutual tangent to the indifference lines for any initial endowment is the indifference line itself. As opposed to ‘touching’ at a unique point in the Edgeworth box, the indifference lines touch at every point on itself. I illustrate this below:
Each red line are the two individuals’ indifference curves situated on each other. Therefore, each of it is the Paretian rule for a given initial endowment, and so by extension, the union of all the red lines, which is equivalent to the entirety of the Edgeworth box itself, is the contract space. From this insight we establish a proposition:
Proposition 1: If goods are weak perfect substitutes and preferences are cross-individual symmetric, then every allocation is Paretian.
Let us ignore all Paretian configurations except for the one that passes through the centre of the Edgeworth box:
Recall that the tangent to indifference curves that passes through the centre is Rawlsian. By the assumed preference structures, they are in this case also Paretian. Therefore, interestingly, the social justice is any point on this line. Weak perfect substitutability not only produces a continuum of solutions but also a slanted continuum, creating a space within which nature’s rule could be situated and yet a social justice could occur:
It is important to note too the following proposition.
Proposition 2: The likelihood of the existence of a social justice decreases in the proportional rate at which one good is favoured over the other.
This proposition has a natural explanation: the more proportionally imperfect a substitute is for some good, the smaller the social justice ‘window’ becomes. From this we see that if the goods are one-for-one substitutable at any wealth configuration for all individuals and the economy has equal stock of each resource, or that the goods are weak perfect substitutes under a proportional rate such that the entire stock of one resource yields equal utility as the entire stock of the other resource for all individuals, then a social justice exists under every possible nature’s rule because the Paretian rule that is also Rawlsian starts and ends at the downward sloping diagonal corners of the Edgeworth box. Goods that are one-for-one substitutable, that is, have a constant marginal rate of substitution of -1, we call a ‘strict perfect substitute’. Whatever nature specifies its rule to be it nevertheless intersects with a point in the Edgeworth box that is Paretian and Rawlsian:
Arriving at an economy configured as such allows us to ignore the brutish effects of nature in making or breaking the possibility of a socially just outcome. The arrival at a social justice, therefore, would depend wholly on the mechanisms that the state institution constructs. I don’t hesitate to say that this is the most important result in this essay if not the entire series, and so we capture this as a fundamental theorem, powerful by virtue of the fact that it speaks of a logical equivalence between existence and the conditions for it.
Fundamental Theorem: A social justice exists for any nature’s rule if and only if there exists a strict perfect substitute for the non-redistributable goods without excess of the total stock of the non-redistributable goods, or there exists a weak perfect substitute for the non-redistributable goods such that the stock of both resources reflect the proportional rate at which the former is a weak perfect substitute.
Of course, we could have simply said that the condition for the existence of a social justice are perfect substitutes such that there is an indifference line that touches both the North-West and South-East corners of the Edgeworth box. But our theorem is captured as above because I want to fine-grain our insights by showing that there are two mutually exclusive and collectively exhaustive cases under which that condition holds. Moreover, it is hard to translate into real life what ‘touching both the North-West and South-East corners’ entails.
What we should observe by now is the position of technological progress in rendering obsolete the brutish restrictions that nature’s rules can often be. For the voiceless there needs to be an invention, a machine of sorts that is a perfect substitute for the vocal cord. If nature unfairly redistributes legs, then prosthetics need to exist that could perfectly substitute for, although not necessarily in a one-to-one manner, the natural limbs with which others have been endowed. These inventions, evident already in the modern economy as they have been spurred on by the human engineering and creativity, are the carriers of hope in the context of a fair and just distribution of welfare. Still, they are not invented without cost. So a question arises: who is to pay? The state provides an answer to this question by asking the ‘unborn child’.
Defining the Welfare State
We start by defining two primitive objects that I will soon demonstrate come together very naturally to form the welfare state. These primitive objects are the population index set and the private property.
The ‘population index set’ of an economy is a lower bound of the set of natural numbers such that its greatest element is the total number of inhabitants in the economy. This is a somewhat simple object. For an economy with a population of 5, the population index set I is:
I = {0, 1, 2, 3, 4, 5}.
The element ‘0’ denotes the state. The ‘private property’ of an individual is a portfolio set of utility-inducing goods owned by that individual. Each element of the set comes in the form of a constant with a superscript and a subscript. The constant denotes the good that it refers to, the subscript denotes which particular item the element refers to and the superscript denotes which individual this item belongs to. The following is an example of a private property, P₁, for individual 1:
P₁ = {A₁¹, A₂¹, A₃¹, O₁¹, O₂¹}.
In this example, individual 1 has 3 apples (A) and 2 oranges (O).
From this definition of private property let us define the ‘national income’ Y as the union of private properties across all individuals:
Y = ∪ [i∈ I] Pᵢ .
Then, the ‘welfare state’ W is a triple consisting of the national income and the population index set equipped with a function f which we will call the redistribution function:
W = (Y , I, f).
The ‘redistribution function’ is a mapping f : Y → I under a certain function rule, the analysis of which will span the remaining part of this essay. The mapping represents the redistribution of owner-indexed goods to a particular individual, or else the state, in the economy.
A Trivial Welfare State in Action: An Example
Consider the special case where the function f is defined by:
f(xⱼⁱ) = i, for all i ∈ I.
In words, the redistribution function f maps every item owned by individual i into the individual i himself, and every state property to the state itself. Since this is true for every individual, the redistribution function has no effect on the initial distribution of resources. Welfare states whose redistribution function is as described above we call a ‘null welfare state’.
The Mean-Pivot Welfare State
Imagine an economy of 5 individuals, two of whom have only one leg. Then, the total number of legs in the economy is 8. Then, given that a prosthetic leg is a perfect substitute for a real leg at least up to the second leg, then the availability of two prosthetics is sufficient for the existence of a social justice. Suppose that each prosthetic is providable at a cost c. Then, it is clear that the government needs to collect 2c. The average number of real legs in the economy is 8/5 which is 1.6. Let us construct a mechanism that transfers (2–1.6)c = 0.4c from every individual with two legs to the state and (1–1.6)c = -0.6c from every individual with one leg to the state. Under this mechanism, the state receives 1.2c from individuals with two legs and transfers 1.2c to individuals with one leg. Further, every individual pays according to the same scheme, a pivot at the mean number of real legs whose value may be, due to perfect substitutability, attached to the cost of production of a prosthetic leg. Such a welfare state we call a ‘mean-pivot welfare state’.
This welfare state operates under the following redistribution function:
f[Pxⁱ | Pxⁱ = 0.4c] = 0 for all i such that i has two legs.
f[Px⁰ | Px⁰ = 0.6c] = j for all j such that j has one leg.
As a point of definition, a restriction of the redistribution function that maps private property to the government is called a ‘welfare tax’ (as in the function restricted to the property of individuals with two legs) and one that maps state property to a private individual is called a ‘welfare subsidy’ (as in the function restricted to the property of individuals with only one leg).
Because everyone plays by the same rule, the mechanism is fair. But notice very importantly that the state is required to produce property to the equivalent of 2c. For the fact of state appropriation, this welfare state is inefficient. If the welfare state does not appropriate property and is null, then it is efficient but unfair. Equivalently stated, both the lucky and unlucky group in the population nevertheless bear a brunt, an after-effect of sorts, of guaranteeing the existence of a social justice under an unfair nature’s rule. This is our second encounter of an imperfection: though we have rendered obsolete the effects of nature, we have done so at a non-zero cost. In fact, the only way out of this mess is for society to invent at zero cost, a task that is impossible given the fundamental problem of scarcity. This appropriation of property from the state we shall refer to as ‘nature’s rent’, a deadweight loss of welfare due in the first place to the very imperfections that come with living.
But which configuration of the welfare state shall we settle with? Let us investigate the expected utility associated with the null and the mean-pivot welfare states. In our world, there is a 2/5 chance that an individual is born with one leg and a 3/5 chance that an individual is born with two legs. For the null welfare state, if an individual is born with one leg, then he receives a payment to the equivalent of 0. If the individual is born with two legs, then he pays the government 0. The expected utility of the null welfare state is:
2/5(c)+3/5(2c) = 0.4c+1.2c = 1.6c.
For the mean-pivot welfare state, if an individual is born with one leg, then he receives a payment to the equivalent of 0.6c (implying that he pays 0.4c on top of his state allowance for his prosthetic). If the individual is born with two legs, then he pays the government the same amount of 0.4c. The expected utility of the mean-pivot welfare state is:
2/5(0.6c + c)+3/5(-0.4c + 2c) = 1.6c–1.6c = 0.
In the first case, the expected utility is better than in the second because c > 0. However, there is a real risk of being born and left to one’s own device under the circumstance of being naturally endowed with only one limb as opposed to two. There is now an ambiguity: an individual that is mildly risk-averse may still choose the risky option because the risky option yields strictly higher expected utility than the certain option. Which type of welfare state is best depends wholly on a moral philosophy on the acceptability of risk-loving behaviour.
Nevertheless, one may argue that the principle of fairness should, at this level, trump the expectation that a rational individual also maximises expected utility, that the forgone 1.6c should in fact be part of the deadweight loss which we have referred to as nature’s rent. This is a rather compelling argument if the welfare state is interpreted to be the arm of the government that induces, through pure distributional changes, what is called the social insurance. The insurance aspect of a human society is independent of expected utilities, but rather wholly determined by the way those expected utilities are formed. A welfare state that diminishes the risk of gamble at birth relative to nature’s rule is said to induce a ‘social insurance’. A policy that induces a society where an unborn child does not gamble with his circumstances at all we call a ‘perfect social insurance’. While the inefficient but fair mean-pivot welfare state described above forces everyone to ‘give away’ 0.24c to arrive at a social justice equilibrium, this welfare state in fact induces a perfect social insurance. From this it follows that a perfect social insurance policy results in a greater deadweight loss to society due to the necessity of forgoing a positive expected utility in favour of nothing on top of state appropriation that we have defined to be nature’s rent.
Further Investigations of the Mean-Pivot Welfare State
Let us derive the redistribution function for a mean-pivot welfare state where half of the population is born with 2 legs and the other half with 1 leg and the cost of a prosthetic leg is c. Let us assume that there are 6 individuals in total instead of 5 (so that it isn’t the case that 2.5 individuals are in each group). Then, the mean number of real legs is 1.5 and the number of prosthetics needed to ensure equal initial endowment is 3. Individuals with two legs pay the state 0.5c in the form of a welfare tax and individuals with one leg receive a welfare subsidy to the equivalent of 0.5c, and therefore is responsible for paying 0.5c to the cost of producing their prosthetic leg. The state receives 1.5c in taxes and dispenses 1.5c in subsidies: the welfare state is efficient. Not only that, but everyone is forced to pay 0.5c in order to arrive at a social justice equilibrium: the welfare state is also fair. From this it follows that the existence of a perfect welfare state is independent of a moral philosophy on the position of risk-aversion if nature’s rule is such that unfairness, and by extension, fairness, is cross-individual symmetric. Further, though I state here without proof and the reader may verify themselves, the same result holds true for the case where nature’s rule specifies an unfairness on the majority as opposed to the minority. We capture these results in a proposition.
Proposition 3: A welfare state is fair and efficient if and only if it is a mean-pivot welfare state.
Reflections
Two times in this essay I alluded to the existence of a ‘talent’ being one of the non-redistributable, utility-inducing resources. Though, in the example that overshadows this essay, I opt instead to use legs as the goods we are concerned. The reason for this is simple: while the counting of legs allows for the construction of an Edgeworth box, I do not know how to count talent in much the same way. And even if I did know how, the measurement of would nevertheless prove to be difficult. This implies that there exists two classes of non-redistributable goods: the observables and non-observables, the biological ‘phenotypes’ and ‘genotypes’ of individuals. While I do not want to bore the reader with statistical methods and tables, I would like to point out that the existence of the non-observable genotypes do not present a conceptual problem to the feasibility of a social justice: given that we could come up with reasonable econometric models in the approximation of genotypes, we have a whole host of methods to correct its natural distribution. In exchange economies, asymmetric investment in education could be a reasonably good welfare state remedy for differences in talent. In production economies, simple money transfers would suffice if it is assumed that the entire private benefit of talent and education lies in wealth generation. The verdict, though, is clear: the existence of a social justice could exist only if gene editing is an available option for genotypes without an alternative substitute.
I would like also to highlight the difference between existence and occurrence. Recall our Fundamental Theorem. This theorem captures the necessary and sufficient conditions for the existence of social justice, and not its occurrence. Perfect substitutes do not suffice to concoct a socially just outcome. For all we know, technology could simply be discarded, or gene editing used to advance the interests of only a select few. What society also needs is a good welfare state that harnesses the power of technology in a certain manner so as to induce socially just outcomes. This was the topic in the latter half of this essay. In my opinion, the existence theorem is a powerful one still: in the static void of a hectic world, I consider it still a victory for reason and justice, to which the theorem is a significant weight of evidence.
Lastly, I would like to highlight the messy end of this essay. Suffice to say, we arrived at a very strange conclusion. In Proposition 3, it seems as though something good arises only out of a situation that is anything but good, that a fair and efficient welfare state exists only if justice has in the first place been paid for by society, only if, that is, nature has extracted its rent. In the choosing of a welfare state, we need not consent to this deadweight loss of welfare. Yet, in consenting to it and opting for the mean-pivot welfare state, I nevertheless used the word ‘efficient’ to describe the outcome. How could efficiency be compatible with the presence of a deadweight loss of welfare? It is compatible only because I have implied a tweak in my definition of efficiency, or rather to put it more accurately, a tweak in the set of assumptions under which the conditions for efficiency have been derived. In the grand scheme of things, the most efficient institution is no institution at all because any government intervention results in a deadweight loss of welfare. Yet, to guarantee the existence of a social justice, society must pay a price to nature (which is precisely this foregone welfare associated with the positive expected utility of the null welfare state) and the mean pivot welfare state results in no further loss of welfare in addition to nature’s rent. In this sense it is an efficient institution.
