Week 10: Welfare Economics and Policy Evaluation

DSAN 5450: Data Ethics and Policy
Spring 2025, Georgetown University

Jeff Jacobs

jj1088@georgetown.edu

Wednesday, March 26, 2025

Foreshadowing Next Week / HW4

In the stu (RStudio), cooken up some 🔥 for yall. Bringing data (.csv files) back into all this econ game theory nonsense

Descriptive Ethics: Foreshadowing W14

\[ \DeclareMathOperator*{\argmax}{argmax} \DeclareMathOperator*{\argmin}{argmin} \newcommand{\bigexp}[1]{\exp\mkern-4mu\left[ #1 \right]} \newcommand{\bigexpect}[1]{\mathbb{E}\mkern-4mu \left[ #1 \right]} \newcommand{\definedas}{\overset{\small\text{def}}{=}} \newcommand{\definedalign}{\overset{\phantom{\text{defn}}}{=}} \newcommand{\eqeventual}{\overset{\text{eventually}}{=}} \newcommand{\Err}{\text{Err}} \newcommand{\expect}[1]{\mathbb{E}[#1]} \newcommand{\expectsq}[1]{\mathbb{E}^2[#1]} \newcommand{\fw}[1]{\texttt{#1}} \newcommand{\given}{\mid} \newcommand{\green}[1]{\color{green}{#1}} \newcommand{\heads}{\outcome{heads}} \newcommand{\iid}{\overset{\text{\small{iid}}}{\sim}} \newcommand{\lik}{\mathcal{L}} \newcommand{\loglik}{\ell} \DeclareMathOperator*{\maximize}{maximize} \DeclareMathOperator*{\minimize}{minimize} \newcommand{\mle}{\textsf{ML}} \newcommand{\nimplies}{\;\not\!\!\!\!\implies} \newcommand{\orange}[1]{\color{orange}{#1}} \newcommand{\outcome}[1]{\textsf{#1}} \newcommand{\param}[1]{{\color{purple} #1}} \newcommand{\pgsamplespace}{\{\green{1},\green{2},\green{3},\purp{4},\purp{5},\purp{6}\}} \newcommand{\prob}[1]{P\left( #1 \right)} \newcommand{\purp}[1]{\color{purple}{#1}} \newcommand{\sign}{\text{Sign}} \newcommand{\spacecap}{\; \cap \;} \newcommand{\spacewedge}{\; \wedge \;} \newcommand{\tails}{\outcome{tails}} \newcommand{\Var}[1]{\text{Var}[#1]} \newcommand{\bigVar}[1]{\text{Var}\mkern-4mu \left[ #1 \right]} \]

Thucydides and the Kindly Slavemaster

[What is] right, as the world goes, is only in question between equals in power; otherwise, the strong do as they please and the weak suffer what they must. (Thucydides 2013 c. 411 BC) (Think of necessary vs. sufficient conditions!)

	liberalism	republicanism
Definition of Injustice	Strong do bad things (Berlin 1959)	Strong can do bad things (Skinner 1998; Pettit 1997; Lovett 2022)
Thucydides Question	Strong do as they please \(\overset{?}{\Rightarrow}\) Strong do bad things	Strong do as they please \(\overset{?}{\Rightarrow}\) Strong can do bad things
Answer	No, not necessarily!	Yes, necessarily!
Frederick Douglass	My feelings [towards slave masters] were not the result of any marked cruelty in the treatment I received…	…they sprung from the consideration of my being a slave in the first place. It was slavery—not its mere incidents—that I despised. (Douglass 1855)
A Doll’s House	Our home is nothing but a playroom. I have been your doll-wife, just as at home I was papa’s doll-child; and here the children have been my dolls. (Ibsen 1879)

(Plz notice the lowercase “l”, lowercase “r”!)

Relevance for This Week (Where We Left Off)

• Can we develop policy interventions that equalize power, so that world looks like normative ethics from W03-W08 (“what is right”)?
• (Hidden antecedent: non-Nietzschean ethical framework e.g. Utilitarianism/Kant)
• Point of prev slide: From now til W14, keep in mind how definition of power (and hence “effectiveness” of policy intervention) depends on antecedent
  • liberal definition \(\Rightarrow\) focus on equilibria (no injustice if bad thing doesn’t happen in equilibrium)
  • republican definition \(\Rightarrow\) also take off-equilibrium possibilities into account (no injustice if bad thing doesn’t happen in equilibrium and doesn’t happen if one player deviates “on a whim”) (Jacobs and Naidu 2025 😉)
• Prisoners’ Dilemma 😫 \(\prec\) Assurance Game 🤨 \(\prec\) Invisible Hand Game 🥳

Prisoners’ Dilemma (Fishers’ Dilemma)

• Single, unique Nash equilibrium, and it’s Pareto inferior

The Game

		\(j\)
		Fish 6 Hours	Fish 8 Hours
\(i\)	Fish 6 Hours	\(1.0,1.0\)	\(0.0,\boxed{1.2}\)
	Fish 8 Hours	\(\boxed{1.2}, 0.0\)	\(\boxed{0.4},\boxed{0.4}\)

• Boxes = Best Responses:
• \(\text{BR}_i(6\textrm{ hr}) = 8\textrm{ hr}\), \(\text{BR}_i(8\textrm{ hr}) = 8\textrm{ hr}\)
• \(\text{BR}_j(6\textrm{ hr}) = 8\textrm{ hr}\), \(\text{BR}_j(8\textrm{ hr}) = 8\textrm{ hr}\)

Pareto Dominance

Operationalizing Power

• Equally good outside options \(\implies\) can contract to Pareto-optimal point \(o^P\)
• \(i\) has better outside options \(\implies\) can make take it or leave it offer to \(j\):
  • “You (\(j\)) fish 6 hrs all the time. I (\(i\)) fish 6 hrs 41% of time, 8 hrs otherwise”

• Ever so slightly better for \(j\) \(\implies\) \(j\) accepts (Behavioral econ: \(j\) accepts if 41% meets subjective fairness threshold; observed across many many cultures!)
• Later / next week: observe policy with outcome \(o^{C}_{i \rightarrow j} \iff\) policy values \(i\)’s welfare more than \(j\)’s welfare (inferred social welfare weights \(\omega_i > \omega_j\))

Policy Interventions: Fish Dilemmas \(\mapsto\) Assurance Games

• Notice: To “escape” prisoners’ dilemma, we had to literally change the rules of the game (permanent intervention)
• Fishers’ Dilemma:
  • No institutions: \(a_i, a_j \in \{6\text{ hr}, 8\text{ hr}\}\)
  • Institutions (courts or social norms): \(\{\text{Accept}, \text{Reject}\}\)
• Driving “game”:
  • No institutions: \(a_i, a_j \in \{\text{Stop}, \text{Drive}\}\)
  • Institutions (stoplights installed by govt or community agreement): \(a_i, a_j \in \{\text{Obey Light}, \text{Run Light}\}\)
• Within assurance games, only need to nudge (one-time intervention) \(\leadsto\) new equilibrium (self-enforcing by definition)

Assurance Game

• Multiple equilibria; the particular outcome we observe is a function of history (path dependency)
• Drive-on-left vs. drive-on-right: Assurance game where neither equilibrium Pareto-dominates other option
  • Swedish Dagen H: Nudge from \(a^*_{\textsf{L}} = (\textsf{L},\textsf{L})\) to \(a^*_{\textsf{R}} = (\textsf{R},\textsf{R})\)
  • Either eq is self-reinforcing! (Unless you want to crash out)

• QWERTY vs. DVORAK / Palanpur farmers: Assurance game where observed equilibrium Pareto inferior

		\(j\)
		Early	Late
\(i\)	Early	\(\boxed{4},\boxed{4}\)	\(0, \, 3\)
	Late	\(3, \, 0\)	\(\boxed{2},\boxed{2}\)

Invisible Hand Game

• Single, unique Nash equilibrium, and it’s Pareto efficient
• \(\Rightarrow\) Acting in self interest \(\leadsto\) best possible outcome

It is not from the benevolence of the butcher, the brewer, or the baker that we expect our meal, but from their regard to their own interest (Smith 1776)

		\(j\)
		Corn	Tomato
\(i\)	Corn	\(2, \, 4\)	\(4, \, 3\)
	Tomato	\(\boxed{5}, \boxed{5}\)	\(3, \, 2\)

• Wealth of Nations SPOILER: The wealth comes from division of labor
  and also dumbleydore dies. semperus snake too. and even poor ron the weasel, who never deserved such a fate

An economic transaction is a solved political problem. Economics has gained the title “Queen of the Social Sciences” by choosing solved political problems as its domain. (Lerner 1972)

First Fundamental Theorem of Welfare Economics

Thm: [Antecedents (Coase Conditions)] \(\Rightarrow\) markets produce Pareto-optimal outcomes

• Even Jeff finds proof (and corollaries) compelling / convincing / empirically-supported
  • (It’s a full-on proof, in the mathematical sense, so doesn’t rly matter what I think; I just mean, imo, important and helpful to think through for class on policy!)
  • Ex: Conditional on antecedents [(Coase) minus (perfect competition) plus (thing must be allocated via markets)], \(\uparrow\) Competition \(\leadsto\) More efficient allocations
• Like how Gauss-Markov Assumptions \(\Rightarrow\) OLS is BLUE, yet our whole field (at least, a whole class, DSAN 5300) built on what to do when GM Assumptions don’t hold
• For policy development, helpful to think through
  • which cases “break” FFT (more honored in the breach)
  • How each violation might be “fixed” through policy
• Our violation: No externalities assumption
  • Possible policy “fixes”: property rights, market-socialist nationalization

Part 2 Suddenly Collides with Part 1: Property Rights

• Rawlsian Rights: Vetos on societal decisions; Constitution can make some inalienable (can’t sell self into slavery), some alienable
• Property rights: alienable. You can gift or sell the rights if you want (veto is over society just, like, taking your property if someone else would be happier with it)

Case : Society decides Right to Clean Air \(\prec\) Right to Smoke \(\Rightarrow\) Start at \(E\)

• \(A\) can pay \(B\) to alienate right (Pay $50/month, can smoke 5 ciggies) \(\leadsto\) \(X\)
• Movement along light blue curve: giving up \(x\) money for \(y\) smoke, equally happy. \(u_A(p)\) identical for \(p\) on curve
• Movement to higher light blue curve () \(\Rightarrow\) greater utility \(u_A' > u_A\)

Case Society decides Smoke \(\prec\) Clean Air \(\Rightarrow\) Repeat for \(E' \leadsto X'\)

“Edgeworth Box” for Right to Smoke vs. Right to Clean Air: \(A\), \(B\) are roommates; \(A\) loves smokin, \(B\) loves clean air

Externalities \(\Leftrightarrow\) Costs of Actions Paid by Someone Else!

• Firm \(S\) produces amount of steel \(s\), pollution \(x\)
• Firm \(F\) “produces” amount of fish \(f\)
• \(S\) optimizes

\[ s^*_{\text{Priv}}, x^*_{\text{Priv}} = \argmax_{s,x}\left[ p_s s - c_s(s, x) \right] \]

• While \(F\) optimizes

\[ f^*_{\text{Priv}} = \argmax_{f}\left[ p_f f - c_f(f, x) \right] \]

• If [Yugoslavia-style] nationalized, new optimization of joint steel-fish venture is

\[ s^*_{\text{Yugo}}, f^*_{\text{Yugo}}, x^*_{\text{Yugo}} = \argmax_{s, f, x}\left[ p_s s + p_f f - c_s(s, x) - c_f(f, x) \right] \]

• Can prove/“prove” that \(o(s^*_{\text{Yugo}}, f^*_{\text{Yugo}}, x^*_{\text{Yugo}})\) Pareto-dominates \(o(s^*_{\text{Priv}}, x^*_{\text{Priv}}, f^*_{\text{Priv}})\)

Social Welfare Functionals

• (Next week… let’s do this in-depth next week!)

Functionals?

• You probably know what a function \(f(x)\) is; a functional is a function of functions: \(\mathscr{G}(f)\)
• It’s from math, which is scary, but it’s just notation to remind us that we’re analyzing functions of functions
• In our case, they “work the same way” as regular functions, e.g., \(\mathscr{G}(f,g) = f^2 + g^2\), so \(f(x) = x, g(x) = 2x \Rightarrow \mathscr{G}(f,g)(x) = x^2 + 4x^2 = 5x^2\)

We Live In A Society, Part 2

• Utilitarianism, Kantian Ethics, and Rawls can all be modeled as Social Welfare Functionals

\[ W(\mathbf{u}) = W(u_1, \ldots, u_n) \Rightarrow W(\mathbf{u})(x) = W(u_1(x), \ldots, u_n(x)) \]

• \(u_i(x)\): Given bundle of resources \(x\), how much utility does \(i\) experience? \(u_i: \mathcal{X} \rightarrow \mathbb{R}\)
• \(W(\mathbf{u})\): Aggregates \(u_i(x)\) over all \(i\), to produce measure of overall welfare of society. \(W: (\mathcal{X} \rightarrow \mathbb{R})^N \rightarrow \mathbb{R}\).
• \(W(\mathbf{u}) = \sum_{i=1}^n \omega_iu_i(x)\). \(\omega_i\) is \(i\)’s welfare weight
• (Preview) Decomposition: evaluate policies by estimating marginal utility \(u'_i(x)\) compared to \(\omega_i\))

Alternative SWF Specifications

• Social values

\[ W(\underbrace{v_1, \ldots, v_n}_{\text{Values}})(x) \overset{\text{e.g.}}{=} \omega_1\underbrace{v_1(x)}_{\text{Privacy}} + \omega_2\underbrace{v_2(x)}_{\mathclap{\text{Public Health}}} \]

• Stakeholder Analysis

\[ W(\underbrace{s_1, \ldots, s_n}_{\text{Stakeholders}})(x) = \omega_1\underbrace{u_{s_1}(x)}_{\text{Teachers}} + \omega_2\underbrace{u_{s_2}(x)}_{\text{Parents}} + \omega_3\underbrace{u_{s_3}(x)}_{\text{Students}} + \omega_4\underbrace{u_{s_4}(x)}_{\mathclap{\text{Community}}} \]

• (Adapted from this great intro video!)

Utilitarian SWF

• Easy mode (possibly/probably your intuition?): Everyone’s welfare weight should be equal, \(\omega_i = \frac{1}{n}\)

\[ W(u_1, \ldots, u_n)(x) = \frac{1}{n}u_1(x) + \cdots + \frac{1}{n}u_n(x) \]

• \(\implies\) Utilitarian Social Welfare Functional!
• The Silly Problem of Utilitarian SWF: What if everyone is made happy by \(u_{\text{Jeef}} = -\infty\)?

The Hard Problem of Utilitarian SWF

While the rhetoric of “all men [sic] are born equal” is typically taken to be part and parcel of egalitarianism, the effect of ignoring the interpersonal variations can, in fact, be deeply inegalitarian, in hiding the fact that equal consideration for all may demand very unequal treatment in favour of the disadvantaged (Sen 1992)

• \(\implies\) “Equality of What?”
• What is the “thing” that egalitarianism obligates us to equalize (the equilisandum/equilisanda): Utility? Opportunity? Resources? Money? Freedom?

Policy Evaluation via Inverse Fairness

We Can Finally Understand This Image from Week 1!

References

Berlin, Isaiah. 1959. Two Concepts of Liberty: An Inaugural Lecture Delivered Before the University of Oxford on 31 October 1958. Clarendon Press.

Douglass, Frederick. 1855. My Bondage and My Freedom. Yale University Press.

Ibsen, Henrik. 1879. The Doll’s House: A Play. D. Appleton & Company.

Jacobs, Jeff, and Suresh Naidu. 2025. “Operationalizing Freedom as Non-Domination in the Labor Market.” In Perspectives on Labor Republicanism, edited by Philip Pettit. Cambridge: Cambridge University Press.

Lerner, Abba P. 1972. “The Economics and Politics of Consumer Sovereignty.” The American Economic Review 62 (1/2): 258–66. https://www.jstor.org/stable/1821551.

Lovett, Frank. 2022. The Well-Ordered Republic. Oxford University Press.

Pettit, Philip. 1997. Republicanism: A Theory of Freedom and Government. Oxford University Press.

Sen, Amartya. 1992. Inequality Reexamined. Clarendon Press.

Skinner, Quentin. 1998. Liberty Before Liberalism. Cambridge University Press.

Smith, Adam. 1776. The Wealth of Nations. Random House Publishing Group.

Thucydides. 2013. The War of the Peloponnesians and the Athenians. Cambridge University Press.