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- 2026
- Working Paper
A 9/16 Exponent for the Ball–Nayar–Tkocz Reverse Entropy Power Inequality

By: Scott Duke Kominers

For a centrally symmetric log-concave random vector X, let N (v) = exp h( v, X ) for v 0, with N (0) = 0, where h denotes differential entropy. Ball, Nayar, and Tkocz proved that N ^(1/5) is subadditive and conjectured that N is a norm. We prove that the map v N (v)^ is subadditive for every 0 < 9/16; in particular, N ^(1/2) is subadditive.
- 2026
- Working Paper
A 9/16 Exponent for the Ball–Nayar–Tkocz Reverse Entropy Power Inequality

By: Scott Duke Kominers

For a centrally symmetric log-concave random vector X, let N (v) = exp h( v, X ) for v 0, with N (0) = 0, where h denotes differential entropy. Ball, Nayar, and Tkocz proved that N ^(1/5) is subadditive and conjectured that N is a norm. We prove that the map v N (v)^ is subadditive for every 0 < 9/16; in particular, N ^(1/2) is subadditive.
- 2026
- Working Paper
When Microenterprises Grow, Are Consumers Better Off? Evidence from Large Loans to Microenterprises in Chile

By: Dean Karlan, Natalia Rigol and Benjamin N. Roth

Evaluations of microenterprise credit typically measure effects only on borrowing firms. But what about their customers? If microenterprises sell relatively undifferentiated goods and services, as is often hypothesized, credit may simply reallocate sales across firms and create little consumer benefit. In a randomized controlled trial in Chile, large loans increased treated firms’ profits by USD 292 per month, a 13.4% increase. Customer survey data indicate even larger benefits for customers: a gain of USD 494 per month in consumer surplus. Furthermore, using a sample of more than 125,000 non-treated firms operating in the same markets, we find little evidence of business stealing. The welfare gains from credit expansion thus extend well beyond the borrowers themselves.
- 2026
- Working Paper
When Microenterprises Grow, Are Consumers Better Off? Evidence from Large Loans to Microenterprises in Chile

By: Dean Karlan, Natalia Rigol and Benjamin N. Roth

Evaluations of microenterprise credit typically measure effects only on borrowing firms. But what about their customers? If microenterprises sell relatively undifferentiated goods and services, as is often hypothesized, credit may simply reallocate sales across firms and create little consumer benefit. In a randomized controlled trial in Chile,...
- August 2026
- Case
Campaign Finance: 2024 Pennsylvania Senate Race

By: Robert F. White

The case details how campaign financing works through a close look at the 2024 Pennsylvania Senate race between incumbent Bob Casey and challenger David McCormick. The case traces the various kinds of campaign funding and their sources, and helps lay out the many layers in U.S. politics funding.
- August 2026
- Case
Campaign Finance: 2024 Pennsylvania Senate Race

By: Robert F. White

The case details how campaign financing works through a close look at the 2024 Pennsylvania Senate race between incumbent Bob Casey and challenger David McCormick. The case traces the various kinds of campaign funding and their sources, and helps lay out the many layers in U.S. politics funding.

About the Unit

The Entrepreneurial Management Unit strives to raise the level of academic work in the field of entrepreneurship, in methodological rigor, conceptual depth, and managerial applicability. We also strive to improve the odds of entrepreneurial success for our students and for practitioners worldwide.

Because it is such a complex phenomenon, entrepreneurship must be studied through multiple lenses. We use three.

The process of entrepreneurship - We seek to understand the processes of entrepreneurial activity in start-ups and established firms by examining the antecedents and consequences of various forms of entrepreneurial opportunity identification and opportunity pursuit for individuals, organizations, and industries. We see experimentation and innovation in products, services, processes, and business models as central to entrepreneurial activity.
The finance of entrepreneurship - We seek to understand the financing of entrepreneurial ventures by studying the antecedents and consequences of entrepreneurial funding decisions both domestically and internationally.
The context of entrepreneurship - We seek to understand the ways in which entrepreneurs both respond to and shape the context in which they operate, by examining the history of entrepreneurship across time and national borders and by analyzing the legal and cultural contexts for managerial action.

Please also visit the Arthur Rock Center for Entrepreneurship.

Recent Publications

A 9/16 Exponent for the Ball–Nayar–Tkocz Reverse Entropy Power Inequality

By: Scott Duke Kominers

2026 |
Working Paper |
Faculty Research

For a centrally symmetric log-concave random vector X, let N (v) = exp h( v, X ) for v 0, with N (0) = 0, where h denotes differential entropy. Ball, Nayar, and Tkocz proved that N ^(1/5) is subadditive and conjectured that N is a norm. We prove that the map v N (v)^ is subadditive for every 0 < 9/16; in particular, N ^(1/2) is subadditive.

Keywords: Mathematical Methods

Citation

Read Now

Finite-Family Avoidance and Isolation Gaps for Quadratic Forms

By: Scott Duke Kominers

2026 |
Working Paper |
Faculty Research

An isolation of a positive definite integral lattice f represents every proper full-rank sublattice of f but not f itself. We prove that every isolation of f has rank at least rank(f) + 2. In fact, every finite family of rank-(rank(f) + 1) lattices that miss f must miss a common proper full-rank sublattice; this finite-family avoidance property passes to orthogonal sums. A spinor-genus condition gives the corresponding result for rank-(rank(f) + 2) lattices when rank(f) 3, and hence an isolation gap of at least 3 for every lattice with such a summand.

Citation

Read Now

Unpacking Feature Requests: Should Linear Reconsider How Labels Work?

By: Sara McKinley Torti

September 2026 |
Supplement |
Faculty Research

Citation

Purchase

Unpacking Feature Requests: Should Linear Reconsider How Labels Work?

By: Sara McKinley Torti

September 2026 |
Exercise |
Faculty Research

Citation

Fazeshift: AI for AR

By: Rembrand Koning and Sarah Sasso

September 2026 |
Teaching Note |
Faculty Research

Teaching Note for HBS Case No. 826-181.

Citation

Educators

When Microenterprises Grow, Are Consumers Better Off? Evidence from Large Loans to Microenterprises in Chile

By: Dean Karlan, Natalia Rigol and Benjamin N. Roth

2026 |
Working Paper |
Faculty Research

Evaluations of microenterprise credit typically measure effects only on borrowing firms. But what about their customers? If microenterprises sell relatively undifferentiated goods and services, as is often hypothesized, credit may simply reallocate sales across firms and create little consumer benefit. In a randomized controlled trial in Chile, large loans increased treated firms’ profits by USD 292 per month, a 13.4% increase. Customer survey data indicate even larger benefits for customers: a gain of USD 494 per month in consumer surplus. Furthermore, using a sample of more than 125,000 non-treated firms operating in the same markets, we find little evidence of business stealing. The welfare gains from credit expansion thus extend well beyond the borrowers themselves.

Keywords: Credit; Profit; Demand and Consumers

Citation

Read Now

Bounded Capacity Makes Substitutability Locally Testable

By: Scott Duke Kominers

2026 |
Working Paper |
Faculty Research

We study exact choice-oracle testing of substitutability in the setting of matching with contracts. Our first result sharpens the known exponential lower bound: At the identity choice function C (A) = A on n contracts, every substitutability certificate relative to the class of choice functions rationalized by strict preference orders over bundles and satisfying the law of aggregate demand must include the queries at all 2 2 nonempty proper menus. Hence, every randomized tester for the stated class with two-sided error at most makes at least (1 2 )(2 2) expected queries at C . By contrast, under irrelevance of rejected contracts and bounded capacity at most q, every failure of substitutability is witnessed on a menu of size at most q + 2, which means that substitutability testing is polynomial in n for fixed q.

Keywords: Matching; Matching Markets; Matching Platform; Matching With Contracts; Markets; Substitutability; Algorithms; Mathematical Methods; Market Design

Citation

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Redistricting, Gerrymandering, and U.S. Election Structural Issues (2026)

By: Robert F. White

August 2026 |
Technical Note |
Faculty Research

In 2026, in the United States, the process of redistricting, the drawing of district boundaries for elections, had become one of the most contentious aspects of government because in a country closely divided among Democratic and Republican voters, how the district lines were drawn helped control who was elected and therefore who made the laws. The case explores the history of redistricting, the Supreme Court cases that apply to redistricting, and the wide-spread politically motivated mid-decade redistricting occurring in 2025 and 2026. It also discusses other structural issues related to elections including voter registration, voter identification, mail-in and absentee ballots, vote counting, and state efforts to change voting laws.

Citation

Educators

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