# Chapter 4 Social Efficiency and Welfare

We need to develop a framework for discussing the benefits gained by society through open trading in a free market. This idea is formally known as the social efficiency or surplus. We’ll develop this idea in more detail in this chapter with particular function on how we measure social surplus.

Social surplus alone is not the end of our discussion, we ideally want to relate that to social welfare. Through a social welfare function, we can consider how a surplus is distributed and whether that distribution is “just.” The social welfare function may suggest that while total “raw” surplus is maximized by the free market, this is not the optimal outcome from a social welfare perspective.

## 4.1 Social Efficiency or Surplus

Surplus is the idea that we derive additional benefits for certain trades and smaller (but still positive) benefits from other trades. We break social surplus into two parts: consumer surplus and producer surplus.

### 4.1.1 Consumer Surplus

Suppose you just finished a long hike and are extremely hungry. If I asked you how much would you be willing to pay for a pizza slice, you might say $12 - you are that hungry. You walk into the store and the buy a pizza slice for$1.50.18 Even though you were willing to pay $12, you only paid$1.50. The difference between these amounts is extra value that you realized. The $10.50 is your surplus on that purchase. We could repeat this process for each additional slice of pizza. Suppose your willingness to pay per slice of pizza is given by the table: Pizza Slice Price You’d Pay Price Charged Surplus 1 12 1.50 10.50 2 8 1.50 6.50 3 4 1.50 2.50 4 2 1.50 0.50 5 1.50 1.50 0.00 We’d see your total surplus is $\text{surplus} = 10.50 + 6.50 + 2.50 + 0.50 + 0 = 20$ In other words, for those 5 slices of pizza you would have been willing to pay$27.50 but were only charged $7.50. The$20 of surplus was additional value you realized above and beyond the amount you paid.

This idea - that trades where the willingness to pay exceeds the price are associated with additional value to the consumer - is the definition of the consumer surplus.

You might realize that embedded in the earlier table is also the demand schedule for pizza for our post-hike lunch.

Price Total Pizza Bought
12 1
8 2
4 3
2 4
1.50 5

The consumer surplus is then given by the difference between the demand curve and the market price. Graphically, the area shaded in blue in Figure 4.1. The area above the price that clears the market and below the demand line is part of the consumer surplus.

Note that the consumer surplus is represented as an area - it is the entire surplus summed over each of the possible number of units consumed.

### 4.1.2 Producer Surplus

The government deems this outcome to be unacceptable and instead restricts the number of trades allowed to 75. This would be similar to the rental cap passed by Iowa City. This change is shown as the dashed red line in the panel of Figure 4.4 labeled “With Quantity Restriction.” With the cap in place, only 75 trades can be made and the equilibrium must shift. The new equilibrium (denoted QR for Quantity Restricted in Figure 4.4) still takes place at the intersection of the supply and demand curves. Since demand is unaffected, the price that clears the market will be higher than at the competitive equilibrium ($2,401.67). Because the market price shifted, we also need to redraw the consumer and producer surplus. It is still given by the area between the demand line and the price (consumer surplus) or the supply line and the price (producer surplus). You’ll note that some of the region that was previously consumer surplus is now producer surplus because of the increased price. The percentage of the total surplus that goes to producers has increased at the expense of the share going to consumers. Meanwhile, the total social surplus has decreased. The new supply line reflects an imposed constraint on the market and prevents parties from making mutually beneficial trades. Before the restriction, suppliers were happy to sell the 25 additional goods at lower prices; however, they are prevented from doing so. As a result, society has to forgo these trades. We label the lost value of these trades as the dead weight loss of the policy. The dead weight loss is the area between the supply and demand lines, before the policy, for quantity above $$Q_\text{QR}$$. #### 4.2.1.2 Rent Control A policy commonly mentioned as a way to control housing costs is rent control. Under a rent control program, the rent charged for a housing unit is restricted to some maximum level. This has the effect of transforming the supply line into a horizontal line once the price equals the threshold price defined by the policy. No matter how many units are offered, suppliers cannot charge a rent above the cap. Figure 4.5 shows such a market. Before a rent control policy, a total of 100 units were rented out at a market price of$1,500. This is shown in the panel labeled “Before Rent Control.” A rent control policy was enacted, limiting the maximum rent landlords could charge to $1,050. The new equilibrium shifts to the point labeled RC (rent control). Since the marginal cost for landlords to rent out more than 75 units is greater than the price they can charge in rent, the supply curve becomes flat. Landlords are only willing to provide 75 units. Compared to the point CE, at RC, the rent is reduced by$450 but there are 25 fewer units open to be rented.

As with the quantity restriction, there are changes in the consumer and producer surplus. The consumer surplus, as a share of the total surplus, increases to reflect the new lower price. However, the total social surplus is reduced by the dead weight loss (the green area in Figure 4.5).

#### 4.2.1.3 Taxes

Rent control and quantity restrictions are brute force interventions. The government also has the option of taxing a good they wish to discourage. A tax would reduce supply since it would increase the cost of production above the bare marginal cost.

One common excise or sin tax is a tax on tobacco products. The government wants to discourage smoking and uses higher taxes to do so. Suppose the government applies a 25% tax to all tobacco products. The market may look something like Figure 4.6. The resulting figure is somewhat more complicated.

The equilibrium shifts from CE to TE (tax equilibrium) with an increase in price and decrease in the quantity traded. The consumer surplus is the region above the new market price ($$P_\text{TE}$$) and below the demand line. The producer surplus is the region below the value of the original supply curve at $$Q_\text{TE}$$. This reduction is the result of the tax with the value of the tax shown as the region in gold between the producer and consumer surplus. The dead weight loss is highlighted in green and shows the loss of surplus due to forgoing those trades.

## 4.3 Social Welfare

The competitive equilibrium maximizes social surplus but social surplus is not the be-all, end-all. We might find that the particular distribution of surplus and goods at competitive equilibrium does not match what we want in a just society. We might want to place our “thumb on the scale” to favor a different outcome that is more “just” or “fair” than that offered by just the free market.

This highlights an limitation of the first fundamental theorem of welfare economics. It is possible for the Pareto efficient, competitive equilibrium to be undesirably from a social point of view. For instance, nothing in the first fundamental theorem would have issue with one person owning everything and everyone else having nothing so long as that outcome was Pareto efficient.

The Second Fundamental Theorem of Welfare Economics says that we can reach a socially more desirable outcome by redistributing resources and then allowing the market to guide the outcome. In practice, this redistribution of resources is hard politically and also will almost certainly create some degree of inefficiency.

We generally “put our thumb on the scale” using different approaches then pure redistribution. However, when we do this, we almost always arrive at an equilibrium point that is less efficient than the free market competitive equilibrium. We have a trade-off to consider between the social surplus and the overall fairness of our outcome.

We describe this trade-off between efficiency and equity using a social welfare function (SWF). The social welfare function will guide us when determining when and to what extent we want to put our thumb on the scale. We’ll consider two common social welfare functions in this class: utilitarian and Rawlsian social welfare functions.

### 4.3.1 Utilitarian SWF

The utilitarian social welfare function stems from the philosophy of utilitarianism developed by the philosopher Jeremy Bentham20 Utilitarianism as a philosophy, simply put, is the idea that we should make the choices that leave the affected parties the most happiest and to maximize well-being.

As a result of this, the utilitarian social welfare function says that we should prefer changes that increase the total utility in society. In other words, we should not focus on how individual people are affected by a policy but rather how the whole of society is changed.

As a general case, we would say the utilitarian social welfare function is, in a society with $$k$$ people,

$\text{SWF}_u = U_1 + U_2 + U_3 + \ldots + U_k$

where $$U_i$$ is the utility of the $$i^\text{th}$$ person. Alternatively, we could express this using summation notation as

$\text{SWF}_u = \sum_{i = 1}^k U_i$

For instance, suppose our society has three people: Bill, Ted, and Rufus. We have some current state of affairs will Bill has 100 utility units, Ted has 50, and Rufus has 10. Our total level of utility across all society is then $$100 + 50 + 10 = 160$$.

To determine if a new outcome is more or less just, we consider how the total utility over all members of society changes with the new policy. Suppose we pass a tax that transfers income from the highest earners to the lowest earning members in our society (e.g., from Bill to Rufus). Because of the property of diminishing marginal utility, it is likely that the harm to Bill’s utility will be smaller than the benefit to Rufus’s. After the tax and transfer, we have Bill having 95 utility units, Ted having 50, and Rufus having 20. The total utility in society is $$95 + 50 + 20 = 165$$, or 5 more than before the tax and transfer. Therefore, a utilitarian social welfare function would prefer this state.

Conversely, suppose we had a tax and transfer scheme the other way: Rufus paid the tax and Bill benefited.21 Let’s say that Rufus lost 5 utility units of income to the tax and Bill got 2.5 units of extra utility. We then have a situation where total utility is $$102.5 + 50 + 5 = 157.5$$, or 2.5 units lower than the baseline. Based on the utilitarian social welfare function, we would not prefer this outcome.

Note that the utilitarian social welfare function is not sensitive to the distribution of utility. Suppose we took all of the utility away from Rufus and Ted and transferred it all to Bill. We now have Bill having 160 utility points while Ted and Rufus have none. Based on the social welfare function, this outcome is just as good as the one where Bill had 100, Bill had 50, and Rufus had 10.

### 4.3.2 Rawlsian SWF

While the utilitarian social welfare function often leads to the right decision for a society, it does have drawbacks due to its lack of distributional sensitivity. The Rawlsian social welfare function, derived from the arguments made by the American philosopher John Rawls, addresses that shortcoming.

Rawls argued that we should be chiefly concerned with how the least well off member of society is affected by a policy. He argued for using a social welfare function that has as its value the lowest utility of all members on society. Formally,

$\text{SWF}_R = \min(U_1, U_2, U_3, \ldots, U_k)$

This is attractive because we are often most concerned with how a policy might impact the least well off and so having a social welfare function that centers on that idea is useful. This is different from the utilitarian social welfare function in that the total utility in society can decrease but that new situation may still be preferred if the lowest utility was increased.

With our examples above, the Rawlsian social welfare function is supportive of a policy that transfers income from Bill to Rufus as Rufus’s utility is increased. This would remain true even if the decrease in Bill’s utility was greater than the increase in Rufus’s.

When we consider the extreme example of a policy that changes utility to 160 for Bill and zero for both Ted and Rufus, the Rawlsian social welfare function would reject this outcome as being equally “good” as the baseline. The lowest utility has changed from 10 (Rufus) to 0 (Ted or Rufus) and so the policy would not be preferred.

A Rawlsian social welfare function is not immune to pathology. In particular, society can be clearly worse off but if the least member is improved, then that policy is preferred. For instance, suppose Bill earns $100,000 while Rufus earns$5,000. If we take $50,000 from Bill and give$1 to Rufus and burn the $49,999 that remain, a Rawlsian social welfare function would prefer this alternative. This outcome is clearly not desirable and wastes a considerable amount of utility by burning$49,999 but because Rufus is $1 richer, it is better. ### 4.3.3 Applying a SWF In practice, you would rarely want to decide to only use a Rawlsian or a utilitarian social welfare function. Both have strengths but also significant weaknesses. The lack of attention to distribution of the utility in a utilitarian setting is a major weakness - distribution matters - but reducing that measure to only how the worst off is doing likely distorts decision making by an equal amount. You would likely want to consider and weigh the decision using both social welfare functions, adding a slight bias towards one or another depending on the question. Consideration of the question is critical. There are some questions where we clearly would be very interested in how the least well off person is doing. For instance, with health care, which we feel is important to health and a human right, is a case where we might prefer the decisions closer to those made with the Rawlsian social welfare function. Since we are concerned with how all people are doing, we are particularly concerned with how the worst off is doing. But other questions are less compelling. If we were considering access to LASIK eye surgery as opposed to regular health care and access to adequate vision care. We would want everyone to be able to see and might use a Rawlsian social welfare function when considering access to adequate vision care, but we don’t feel as though using glasses or contacts is so burdensome that we really care that everyone can get LASIK. We would likely use a utilitarian framework to understand policies related to access or reducing the barriers to access to optical care like LASIK. As a general matter, the Rawlsian SWF will prefer more redistribution of resources while the utilitarian SWF will prefer somewhat less. ## 4.4 Equity Criterion Governments and society do not only use the desire to obtain social surplus or the results of the social welfare function to identify their goals. They also use an equity criterion when determining whether to intervene. Two equity criteria are in common use. The first, called commodity egalitarianism, states that after baseline needs are met, income and utility inequality are not relevant. For instance, once the government has ensured that each citizen reaches a minimally acceptable standard of living, how much people earn does not matter. Suppose that it costs$50,000 to live in a city and the city has three residents - Jane who earns $1,000,000, Elizabeth who earns$100,000, and Mary who earns $50,000. The incomes are highly unequal but all three have enough to live comfortable and intervention or redistribution is not indicated. However, if Mary earned$40,000 then intervention to redistribute resources would be indicated.

The second commonly used criterion is equality of opportunity. Under this standard, the government should ensure all citizens have an equal change of success but has no role in determining who succeeds or who fails. Intervention to alter outcomes after ensuring equality of opportunity is beyond the proper scope of the government.

1. The firm in unaware of the price that you are willing to pay and can only charge one price for all customers in this example. This is fairly common overall: there is an information asymmetric between consumers and producers on how much they value a particular good. This is not always the case though and some firms engage in price discrimination, or charging different people different prices based on their expected willingness to pay. Airlines are famous for doing this, largely by attempting to determine who is traveling for fun (and most price sensitive) and those traveling for business or last minute (and the least price sensitivity). Crudely, you could charge lower rates that include weekend stays or that are booked further in advance and more for trips that lack weekend stays, are on weekdays, or booked on shorter notice.
The word “discrimination” sounds bad because generally discrimination is bad (e.g., sex or race discrimination). However, this practice is actually beneficial to consumers. If an airline was unable to charge different prices for different consumer types and instead a constant fare, consumers would pay more and the company would have lower profit.↩︎

2. Policies like this were banned by the State of Iowa in spring 2019 in response to Iowa City and other locations passing these ordinances. Detractors of the policies made “free market” arguments.↩︎

3. Interesting tidbit: in his will he directed for his body to dissected and them converted into an “auto-icon” to be publicly displayed. Today, his body is on public view at University College London. You can see an picture of his current display location here.↩︎

4. This is actually very common. For instance, until recently, people who owned property could deduct their local property tax from their income. Meanwhile, renters, who did not own the property but did pay the property tax indirectly through their rent, could not deduct that value. This ended up reducing the tax bill of home owners, who tend to be wealthier, at the expense of renters, who tend to be less wealthy. Normally, taxes are design to be progressive, meaning that the burden of the tax increases with additional income. Since marginal utility of income decreases, this makes the utility lost to tax more equal. However, through various carve-outs, many taxes become increasing regressive, or falling more heavily on lower income households.↩︎