metadata: authors: 0: author: Loeser,John Ashton
count: World
docna: 0: docna: Consumer Surplus with Incomplete Markets :
            Applications to Savings and Microfinance
docty: Policy Research Working Paper
abstracts: cdata!: The household welfare gains from
            financial inclusion are empirically elusive. This paper
            establishes that household welfare gains from a financial
            technology are equal to the area under dynamically
            compensated demand in a household model with incomplete
            financial markets, and general technology, preferences, and
            choice sets. This paper then estimates compensated demand
            for financial technologies leveraging three randomized
            control trials that introduce experimental variation in
            interest rates. Welfare gains per dollar lent or saved are
            small as compensated demand elasticities are large, but
            still correspond to large aggregate welfare gains from
            financial inclusion.
display_title: Consumer Surplus with Incomplete
            Markets : Applications to Savings and Microfinance
listing_relative_url: /research/2023/06/34078470/consumer-surplus-incomplete-markets-applications-savings-microfinance
topicv3: Financial Inclusion m1327456 363,Access to Finance m1327464 1249,Access to Finance for Households and
            Individuals m1327690 1250
publishtoextweb_dt: 2023-06-12T00:00:00Z
dois: 10.1596/1813-9450-10481
origu: Off of Sr VP Dev Econ/Chief Econ (DECVP)
author_keys: 0: author_key: 1594783
model_extractions: Policy Research Working Paper 10481 Consumer Surplus with Incomplete Markets Applications to Savings and Microfinance _John Ashton Loeser_ Development Economics Development Impact Evaluation Group June 2023
Policy Research Working Paper 10481

# **Abstract**

The household welfare gains from financial inclusion are empirically elusive. This paper establishes that household welfare gains from a financial technology are equal to the area under dynamically compensated demand in a household model with incomplete financial markets, and general technology, preferences, and choice sets. This paper then estimates compensated demand for financial technologies leveraging three randomized control trials that introduce experimental variation in interest rates. Welfare gains per dollar lent or saved are small as compensated demand elasticities are large, but still correspond to large aggregate welfare gains from financial inclusion.

This paper is a product of the Development Impact Evaluation Group, Development Economics. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The author may be contacted at jloeser@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
Consumer Surplus with Incomplete Markets: Applications to Savings and Microfinance<sup>∗</sup> John Loeser<sup>†</sup> JEL Classification Codes: D11, D14, G51, O1, O12, O16 Keywords: Consumer surplus, Incomplete financial markets, Financial inclusion

> ∗ `jloeser@worldbank.org` . This research benefited from suggestions and comments from Daniel Agness, Katy Bergstrom, Alfredo Burlando, Kieran Byrne, Alain de Janvry, Emma Frankham, Xavier Gine, Sean Higgins, Dahyeon Jeong, Erin Kelley, Cynthia Kinnan, Florence Kondylis, Greg Lane, Arianna Legovini, Jeremy Magruder, Aprajit Mahajan, David McKenzie, Elisabeth Sadoulet, Hee Kwon Seo, Ursula Wiriadinata, Brian Wright, and seminar audiences at NEUDC, UC Berkeley, WEFIDEV, and the World Bank. Financial support from the CEGA-Visa Financial Inclusion Lab is gratefully acknowledged. The views expressed do not reflect the views of the World Bank. All errors are my own.

> †Development Impact Evaluation, World Bank
# **Contents**

|**1**<br>**Intr**|**oduction**|**2**|
|---|---|---|
|**2**<br>**Con**|**sumer surplus with incomplete fnancial markets**|**6**|
|2.1|Model<br>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|.<br>7|
||2.1.1<br>Environment<br>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|.<br>7|
||2.1.2<br>Existence of model solutions . . . . . . . . . . . . . . . . . . . . . .|.<br>9|
||2.1.3<br>Graphical representation of the model. . . . . . . . . . . . . . . . .|.<br>9|
|2.2|Roy’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|.<br>11|
|2.3|Consumer surplus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|.<br>12|
||2.3.1<br>Equivalent variation<br>. . . . . . . . . . . . . . . . . . . . . . . . . .|.<br>12|
||2.3.2<br>Marshallian surplus . . . . . . . . . . . . . . . . . . . . . . . . . . .|.<br>14|
||2.3.3<br>Uncompensated and compensated demand elasticities . . . . . . . .|.<br>14|
|2.4|Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|.<br>16|
|**3**<br>**Pra**|**ctical guidance on consumer surplus from fnancial technologies**|**16**|
|3.1|Defning, estimating, and graphing demand for fnancial technologies<br>. . .|.<br>17|
|3.2|Simplifying assumptions on dynamically compensated demand . . . . . . .|.<br>18|
|3.3|Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|.<br>21|
|**4**<br>**Con**|**sumer surplus from savings and microfnance**|**23**|
|4.1|Retirement savings (Dufo et al., 2006) . . . . . . . . . . . . . . . . . . . .|.<br>23|
||4.1.1<br>Context and experimental design<br>. . . . . . . . . . . . . . . . . . .|.<br>23|
||4.1.2<br>Demand elasticities and consumer surplus<br>. . . . . . . . . . . . . .|.<br>25|
|4.2|Commitment savings (Karlan & Zinman, 2018)<br>. . . . . . . . . . . . . . .|.<br>27|
||4.2.1<br>Context and experimental design<br>. . . . . . . . . . . . . . . . . . .|.<br>27|
||4.2.2<br>Demand elasticities and consumer surplus<br>. . . . . . . . . . . . . .|.<br>28|
|4.3|Microfnance (Karlan & Zinman, 2019) . . . . . . . . . . . . . . . . . . . .|.<br>30|
||4.3.1<br>Context and experimental design<br>. . . . . . . . . . . . . . . . . . .|.<br>30|
||4.3.2<br>Demand elasticities and consumer surplus<br>. . . . . . . . . . . . . .|.<br>32|
|4.4|Discussion of results<br>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .|.<br>36|
|**5**<br>**Con**|**clusion**|**38**|

1
# **1 Introduction**

What are the household welfare gains from microfinance? More than 200 million households borrow from microfinance institutions annually. Despite this scale, a growing empirical consensus holds that microfinance has failed to meet its original promise, instead offering positive but modest impacts on borrowers (Banerjee et al., 2015b; Meager, 2019). However, estimating welfare gains is difficult. Households may benefit from microfinance through a – broad range of mechanisms microfinance enables productive investments and lumpy purchases (Banerjee et al., 2021), consumption smoothing (Lane, 2020), and acts as a gateway to traditional credit (Rigol & Roth, 2021). As a result, the sign and magnitude of impacts of microfinance on common welfare proxies, including household consumption, may be inconsistent with impacts on welfare (Banerjee et al., 2015b), and theory accounting for these diverse mechanisms is necessary to quantify impacts on welfare (Karlan & Zinman, 2019).

In this paper, I propose a method to estimate the household welfare gains from financial technologies, including microfinance, that builds on the following result: consumer surplus is equal to the area under dynamically compensated demand. To capture the complexity of household consumption and investment decisions in developing countries, I develop a dynamic household model with incomplete financial markets and general technology, preferences, and choice sets. I show that the area under dynamically compensated demand is equal to consumer surplus. I then estimate compensated demand and consumer surplus for financial technologies leveraging experimental variation in prices from three randomized – control trials consumer surplus per dollar lent or saved is small, as compensated demand elasticities are large, but still corresponds to large aggregate consumer surplus from financial technologies.

The intuition underlying the area under compensated demand as a measure of consumer surplus from financial technologies is captured by the following stylized example, visualized in Figure 1: consider a borrower that takes a loan with $1000 principal, for which the borrower will need to make 12 monthly payments of $100. The borrower has a minimum – “willingness-to-accept” (WTA) the minimum acceptable principal, holding fixed monthly payments, at which the borrower would take the loan. The borrower’s surplus from the loan is simply the difference between the $1000 principal (the “price”) and their WTA – although the borrower discretely changes their behavior when the offered principal falls below their WTA, as the borrower declines the loan, an envelope theorem holds and these responses have no first order effects on welfare. In canonical consumer theory, demand for a good as a function of price, holding fixed product attributes, traces out willingness-to-pay; for loans, demand for monthly payments as a function of the offered principal traces out WTA. This 2
<!-- Start of picture text --> Would you borrow -$p to be repaid in 12 monthly payments of $100?<br>-500<br>a<br>a -750<br>oc1o} Borrower<br>a<br>-1000<br>Loan<br>demand<br><!-- End of picture text -->
Estimation of dynamically compensated demand requires estimates of price elasticities and income effects; in contrast to the static household model, rather than a single income effect, the full matrix of intertemporal income effects on demand is necessary. I take a sufficient statistics approach (Chetty, 2009) and show this matrix of intertemporal income effects is an inner product of two terms – the intertemporal matrix of marginal propensities to spend out of temporary income, and growth in demand. Restrictions on this matrix yield two common frameworks: with complete financial markets, the intertemporal matrix of marginal propensities to spend is rank 1, while absent financial markets and with separable utility (i.e., the static household model), it is diagonal. In applications, I use empirical estimates of growth in demand, and I suggest theory-motivated bounds on intertemporal marginal propensities to spend.

A number of strong assumptions are embedded in my modeling framework. Common to many frameworks used to analyze the impacts of financial technologies, I assume households have full information about the technology and are not behavioral. To focus on household welfare, I do not consider general equilibrium effects or producer surplus. However, by reducing the estimation of consumer surplus from financial technologies to the estimation of demand, I provide theoretical grounding for the application of the broad array of models and methods that build on demand in static models.

I next estimate consumer surplus from financial technologies in three randomized control trials (Duflo et al., 2006; Karlan & Zinman, 2018, 2019). Each experiment introduced variation in prices: for deposits into a retirement savings account, for deposits into a commitment savings account, and for microfinance loans, respectively. As in the stylized example, the experimental treatment either reduced savers’ deposits conditional on their end-of-period balance, or increased borrowers’ loan amount conditional on their repayment schedule; in turn, this increased deposits and loan amount. I then estimate price elasticities of deposits and loan amount ranging from 4-to-14 across the three experiments; these are simply appropriate rescalings of estimated impacts from each paper. I then construct bounds on compensated price elasticities of deposits and loan amount: in general, these bounds are narrow compared to statistical uncertainty. Across the three experiments, the compensated price elasticities of deposits and loan amount correspond to annual consumer surplus of $7-to-$27 per $100 of deposits or loan amount.

I conclude by discussing the interpretation of my estimates of consumer surplus. I compare my estimates to existing experimental and quasi-experimental studies of the impacts of access to savings and credit technologies on household consumption, a common proxy for welfare; standard errors are large relative to my estimates, suggesting these studies are ex-post underpowered to detect impacts equal to my estimates of consumer surplus. De4
spite this, my estimates imply large welfare gains from financial technologies when scaled by takeup of household credit and savings in low income countries.

**Related literature** This paper estimates the household welfare gains from financial technologies using a sufficient statistics approach, complementing estimates applying reduced form and structural approaches. A growing literature has applied experimental and quasiexperimental methods to estimate the household impacts of access to microfinance (e.g., Pitt & Khandker, 1998; Karlan & Zinman, 2011; Kaboski & Townsend, 2012; Banerjee et al., 2015b; Meager, 2019; Lane, 2020); this work has identified consumption smoothing, durable consumption, and productive investment motives for borrowing, and has found mostly positive but imprecise impacts on proxies for household welfare. However, theory is necessary to quantify household welfare: the sign of impacts of financial technologies on common welfare proxies, including household consumption, need not be consistent with impacts on welfare, such as when borrowing complements lumpy investments (Banerjee et al., 2015b). Kaboski & Townsend (2011) estimate household welfare in a structural model of microfinance borrowing that allows consumption smoothing, productive investment, and strategic default in response to microfinance; I show my point estimates of consumer surplus from microfinance are comparable to theirs. While reduced form and structural approaches shed light on and quantify the mechanisms underlying borrower impacts of microfinance, the sufficient statistics approach I employ complements these approaches by enabling estimation of consumer surplus under a general neoclassical household model.

To produce estimates of household welfare gains from microfinance, this paper extends textbook calculation of consumer surplus to models with incomplete financial markets. In this regard, it is closest to Farhi et al. (2022), who provide formulas for consumer surplus from small price changes with incomplete financial markets, and in addition produce a more comprehensive price theory; in contrast, I focus only on consumer surplus, and as a result I am able to derive results for large price changes and with more general preferences, non-linear technology, and endogenous choice sets. I build most directly on results on exact consumer surplus in a static model (Willig, 1976; Hausman, 1981). In particular, I represent exact consumer surplus as the solution to a differential equation. In contrast to the static model, multiple distinct solutions exist; for a given path of price changes, I select the solution that does not require estimation of marginal utilities. I complement existing approaches that estimate consumer surplus under fully parametrized structural models with incomplete financial markets; Kaboski & Townsend (2011) estimates equivalent variation to improved access to credit as an initial period transfer, while Lagakos et al. (2018) estimates dynamic compensating consumption transfers for seasonal migration. I do so by proposing sufficient 5
statistics for equivalent transfers in models with incomplete financial markets; these sufficient statistics can be estimated from experimental variation in prices and income.

These results contribute to the analysis of welfare gains in dynamic household models with incomplete financial markets. I show that dynamically compensated demand elasticities are directly informative of welfare gains from general intertemporal substitution technologies, including financial technologies. For studies that isolate experimental or quasi-experimental variation in the price of financial technologies (Duflo et al., 2006; Karlan & Zinman, 2008, 2009, 2018, 2019; Alan & Loranth, 2013; Karlan et al., 2014; Ponce et al., 2017; Banerjee et al., 2018; Cai et al., 2020; Benetton, 2021; Benetton et al., 2021; Robles-Garcia, 2022), this enables transparent welfare analysis, as in Berkouwer & Dean (2022) and DeFusco et al. (2022). For studies that estimate welfare gains from financial technologies leveraging estimates of structural dynamic household models (Kaboski & Townsend, 2011), it suggests model implied demand elasticities should be assessed to evaluate the plausibility of welfare counterfactuals, as in Bryan et al. (2014), Lagakos et al. (2018), and Allcott et al. (2022).

**Outline** This paper is organized as follows. In Section 2, I present a household model with incomplete financial markets, in which I establish that consumer surplus is equal to the area under dynamically compensated demand. In Section 3, I provide practical guidance to researchers on estimating consumer surplus from financial technologies. In Section 4, I estimate uncompensated demand and bound compensated demand for financial technologies in three randomized control trials, which I apply to calculate consumer surplus. Section 5 concludes.

# **2 Consumer surplus with incomplete financial markets**

In this section, I define and characterize consumer surplus with incomplete financial markets. In Section 2.1, I present a model of households facing incomplete financial markets across period-states, with general technologies, preferences, and choice sets. In Section 2.2, I derive Roy’s identity on the welfare impacts of small price changes. In Section 2.3, I apply Roy’s identity to calculate period-state-contingent equivalent variation from price changes, and approximations thereof. In Section 2.4, I discuss extensions featuring unobserved heterogeneity across households and unobserved shocks, and ranking welfare changes under incomplete financial markets.

6
# **2.1 Model**

## **2.1.1 Environment**

A utility maximizing household faces a menu of technologies for substitution across periodstates _s ∈S ≡{_ 1 _, . . . , S}_ . The household makes an investment choice _bs ∈_ R<sup>_N_</sup> +<sup>ineach</sup> period-state _s_ , for which it faces prices _p ∈_ R<sup>_N_</sup> ; I let **b** _≡_ ( _b_ 1 _, . . . , bS_ ) denote the full vector of investment choices, and I assume investment choices satisfy **b** _∈B ⊂_ R<sup>_SN_</sup> +<sup>.</sup> In each period-state _s_ , the household holds wealth _ys_ , plus its period-state production as a function of investment choices _fs_ ( **b** ). The household selects consumption _cs ∈_ R, subject to the period-state budget constraint: consumption _cs_ plus investment expenditures _p · bs_ is weakly less than wealth _ys_ plus production _fs_ ( **b** ).

The household solves This setup treats time and uncertainty in a general manner. The set of period-states _S_ enumerates both time and all potential histories of discrete shocks to the household. Transition probabilities are flexibly embodied in a utility function _U_ , period-state production technology _fs_ , and investment choice set _B_ that depend flexibly on the full vector of consumption choices _c_ and investment choices **b** . While this level of generality is unlikely to be necessary in applications, the resulting notation is more parsimonious than treating time and uncertainty separately, and it underscores which assumptions are necessary for the theoretical results in Sections 2.2 and 2.3. Note that this flexibility also implies that any choice variables with the subscript _s_ may therefore depend on the full history of shocks; I discuss implications of this for the empirical analysis of consumer surplus in Section 2.4, as conditioning on full histories of shocks may not be empirically plausible.

**Price** _p_ For all results below on consumer surplus from price changes, it is crucial that prices _p_ enter the household problem only through the period-state budget constraint. Specifically, prices and wealth are perfect substitutes holding fixed investment choices. Importantly, changes in prices do not directly affect the investment choice set _B_ .

**Additional assumptions** I make the following additional assumptions, each of which I discuss below:

7
- Technology _fs_ is continuous _∀ s ∈S_ .

- The investment choice set _B_ is compact.

- Utility _U_ is continuous, and its derivatives with respect to consumption _∇cU_ are continuous and strictly positive.

**Technology** _fs_ ( **b** ) Production may depend flexibly on multiple previous period-states’ investment choices. Continuous technology over a compact choice set allows for lumpy investment technologies, and complementarities across mixed discrete and continuous investments.

For instance, a choice to default in period-state _s_ would contemporaneously manifest as increased output _fs_ ( **b** ) (i.e., less repayment). However, it may affect future constraints on the household’s subsequent investment choices through the investment choice set _B_ , and output _fs′_ ( **b** ) in future period-states _s_<sup>_′_</sup> both positively (e.g., through reduced subsequent repayment) and negatively (e.g., through seized assets). Simultaneously, past investments in soil fertility on the households’ agricultural land may shape agricultural productivity, and may complement the application of modern agricultural inputs, as multiple past investment choices may interact to determine state-dependent output _fs_ ( **b** ).

**Investment choice set** _B_ That the full vector of investment choices **b** lies in a compact set _B_ provides substantial flexibility. It allows for discrete and continuous choice, and allows for broad dynamic incentives; it nests the case where each period-state’s investment choice set is continuous in previous choices and compact-valued, that is _bs ∈Bs_ ( _b_ 1 _, . . . , bs−_ 1). For instance, a household’s access to credit, and its credit limit, may depend on its credit history including any missed payments or default.

That investment choices are positive is without loss of generality, as an additional investment choice can be added with negative prices to capture negative investment choices, as in the case of saving and borrowing at a constant interest rate.

**Utility** _U_ ( _c,_ **b** ) Utility need not be separable in consumption across period-states and in investment choices. This allows for deviations from both exponential discounting and expected utility maximization, as in Epstein & Zin (1989), and non-pecuniary investment motives (e.g., social stigma against default). That consumption choices _c_ are unrestricted within R<sup>_S_</sup> , and that utility is strictly increasing in consumption, are not without loss of – generality in particular, they imply that the household does not consume at the edge of its consumption choice set (i.e., the household does not consume at a subsistence constraint) and that each period-state’s budget constraint binds with equality, which I make use of and discuss in Section 2.2.

8
# **2.1.2 Existence of model solutions**

The household problem in Equation 1 has consumption and investment solutions. Intuitively, as household utility is strictly increasing in consumption, the budget constraint binds in each period, and the household problem reduces to an investment choice problem. As technology is continuous in investments, the investment choice problem involves maximizing a continuous function on a compact set _B_ , which is guaranteed to have a solution.

# **Proposition 1.** _A solution exists to the household problem in Equation 1._

I denote the consumption and investment solutions to the household problem in Equation 1 as ( _C_ ( _p, y_ ) _,_ **B** ( _p, y_ )) : R<sup>_N_</sup> _×_ R<sup>_S_</sup> _→_ 2<sup>R</sup><sup>_S_</sup> _×_ 2<sup>_B_</sup> . I also let _Cs_ ( _p, y_ ) and _Bs_ ( _p, y_ ) denote periodstate _s_ consumption and investment solutions. By construction, utility _U_ ( _c,_ **b** ) evaluated at these solutions is equal to indirect utility _V_ ( _p, y_ ).

# **2.1.3 Graphical representation of the model**

In a version of the model with just two period-states and in which investment choices do not enter directly into utility, the model admits a simple graphical representation, which I present in Figure 2a. The household faces a consumption possibilities frontier, that is the set of feasible consumption choices as a function of prices, wealth, technology, and the investment choice set. With incomplete financial markets, the consumption possibilities frontier is nonlinear; this is a result of constraints in the investment choice set (e.g., the household can save but not borrow) and nonlinear technology (e.g., decreasing returns to scale). The household chooses its utility maximizing consumption bundle within its consumption possibilities frontier, which occurs at the tangency of the household’s indifference curves to its consumption possibilities frontier; this optimal consumption bundle is feasible at the household’s optimal investment bundle.

In Section 2.2, we will consider the impact of price changes on household welfare; changes in prices affect household utility by shifting the consumption possibilities frontier. This intu– ition suggests an application of the envelope theorem the first order impact of a change in prices on household welfare is equal to the impact of the shift in the consumption possibilities frontier at the household’s optimal consumption bundle.

9
<!-- Start of picture text --> <s+y.+f,(b)pibs \ U()=V( ply)<br>° Ca(phy) poop seer ooo<br>> U()=V( p’,y)<br>< y.+f,(b)<br>Co (ply) prea r ance centr neces<br>i<br>| rN<br>i<br>Ci (p°.y)= Ci (p'y)<br>Cy<br><!-- End of picture text -->

<!-- Start of picture text --> Ci(p,<br>y+ EV") | - - +s 22---------- a=<br>)<br>Ci(p,y+EV") Ci(p.y+EV')<br>cy<br><!-- End of picture text -->
# **2.2 Roy’s identity**

Although the model in Section 2.1 flexibly captures a broad range of technologies, preferences, and choice sets, it restricts that prices enter the household problem only through investment expenditures. As in Auclert (2019) and Farhi et al. (2022), this permits a generalization – of Roy’s identity to the setting with incomplete financial markets the impact of a small change in the price of an investment on utility is proportional to the sum of the marginal utility of period-state wealth times period-state investment.

**Theorem 1.** _Indirect utility V_ ( _p, y_ ) _is absolutely continuous. In addition, suppose that the investment choice_ **B** ( _p, y_ ) _is a singleton; then the marginal utility of wealth is equal to the marginal utility of consumption_

_and the impact of a small change in prices on utility is equal to its impact holding fixed investment choices._ _Proof._ I consider the following investment choice problem equivalent to the household problem in Equation 1, derived from substituting for consumption with the binding budget constraints in Equation 1.<sup>1</sup> I then apply an envelope theorem (Corollary 4 of Milgrom & Segal, 2002) to Equation 5; the following immediately hold.<sup>2</sup> First, _V_ ( _p, y_ ) is absolutely continuous. Second, if **B** ( _p, y_ ) is a singleton, then _∇pV_ ( _p, y_ ) = _−_<sup>�</sup><sup>_S_</sup> _s_ =1<sup>_Bs_(</sup><sup>_p, y_)</sup><sup>_dU_</sup><sup><u>(</u></sup><sup>_C_</sup><sup><u>(</u></sup><sup>_<u>p,y</u>_</sup> _dc_<sup><u>)</u></sup> _s_<sup>_<u>,</u>_</sup><sup>**B**</sup><sup><u>(</u></sup><sup>_<u>p,y</u>_</sup><sup><u>))</u></sup> and _∇yV_ ( _p, y_ ) = _∇cU_ ( _C_ ( _p, y_ ) _,_ **B** ( _p, y_ )) (Equation 3). Substituting yields Equation 4.

> 1As mentioned in Section 2.1.1, the assumption that utility is well defined over any real values of consumption is consequential – transforming the household problem in Equation 1 to the investment choice problem in Equation 5 relies on this assumption. Absent this assumption, the investment choice problem is defined only for investment choices that result in feasible (e.g., positive) consumption, and that set of investment choices might be affected by the price of investments. In that case, the choice set for the investment choice problem would be affected by prices, invalidating the application of the envelope theorem. However, this is only a concern if households select consumption at the boundary of their consumption choice set; in this case, a small change in prices could cause a discrete investment that has large impacts on utility to result in infeasible consumption, causing households that can no longer feasibly make that investment to experience a large change in utility. Note this is not a concern under common restrictions (e.g., Inada conditions) on utility that guarantee households do not consume at the boundary of their consumption choice set; these conditions may be met even if households face a subsistence constraint (e.g., Stone-Geary preferences).

> 2Details of the application of the envelope theorem are in Appendix A.

11
# **2.3 Consumer surplus**

## **2.3.1 Equivalent variation**

In the static household model, or models with complete financial markets, it is natural to define consumer surplus in units of permanent income; in contrast, with incomplete financial markets, permanent income need not exist. If a household cannot transfer income across period-states, it may not be possible to compensate a household for a change in prices in one period-state exclusively with transfers to another period-state. As an alternative, Farhi et al. (2022) consider Slutzky compensation, that is providing transfers that compensate for small price changes in each period-state.

I extend consumer surplus to models with incomplete financial markets by defining it as a vector of period-state-contingent transfers that holds utility fixed following a price change. I consider a path of price changes _p_<sup>_k_</sup> : [0 _,_ 1] _→_ R<sup>_N_</sup> , and I implicitly define the equivalent variation for a change in prices from _p_<sup>_k_</sup> to _p_<sup>1</sup> as where EV( _p_<sup>_k_</sup> _, p_<sup>1</sup> ; _y_ ) : R<sup>_N_</sup> _×_ R<sup>_N_</sup> _×_ R<sup>_S_</sup> _→_ 2<sup>R</sup><sup>_S_</sup> . By construction, the household is indifferent between prices changing from _p_<sup>_k_</sup> to _p_<sup>1</sup> , and receiving the associated equivalent variation as period-state-contingent transfers. In contrast to the static household model, equivalent variation has multiple solutions, as there are many potential vectors of period-state-contingent transfers across which the household is I present a graphical representation of equivalent variation with incomplete financial markets in Figure 2b, in a version of the model with just two period-states and in which investment choices do not enter directly into utility. Changes in prices change the shape of the household’s consumption possibilities frontier, which affects the household’s optimal consumption bundle and its indirect utility. Similarly, income transfers shift the household’s consumption possibilities frontier; equivalent variation is the set of shifts of the consumption possibilities frontier which enable the household to achieve the same level of utility as from the change in prices. Note that equivalent variation shifts the consumption possibilities frontier to a tangency with the household’s indifference curve after the price change. As both indifference curves and the consumption possibilities frontier are nonlinear, different shifts to the consumption possibilities frontier will result in different points of tangency; in contrast to a model with complete financial markets, the choice of period-states in which to compensate the household is consequential.

To characterize equivalent variation, I consider solutions to the following differential equa12
tion derived from the definition of equivalent variation in Equation 6. I posit the existence of a differentiable solution to Equation 6, EV<sup>_∗_</sup> ( _p_<sup>_k_</sup> _, p_<sup>1</sup> ; _y_ ) _∈_ EV( _p_<sup>_k_</sup> _, p_<sup>1</sup> ; _y_ ), which I construct below in Equation 9.<sup>3</sup> Differentiating Equation 6 with respect to _k_ , that is along the path of price changes, and substituting Roy’s identity (Equation 4) yields In general, solutions to Equation 7 may require knowledge of the full vector of marginal utilities of income. When markets are complete, asset prices determine relative marginal utilities of income; in other cases, estimating marginal utilities of income may require estimation of preferences, technology, and beliefs.

Note that a solution to Equation 7 is specific to the path of price changes. However, as solutions to this system solve Equation 6, they are exact measures of consumer surplus (Hausman, 1981).

I focus on a particularly tractable solution to Equation 7 that does not require knowledge of marginal utilities, by solving the system of differential equations This is simply the term inside the sum in Equation 7, and states that the change in periodstate equivalent variation from a small price change is equal to the negative of the direct effect of the price change on compensated expenditures.

There are at least two reasons to normatively prefer solutions to Equation 8 over other solutions to equivalent variation. First, when demand is separable across period-states, that is _Bs_ ( _p, y_ ) is only affected by income in period-state _s_ , then static equivalent variation in each period-state solves Equation 8.<sup>4</sup> Second, solutions to Equation 8 are suggested naturally by Roy’s identity (Equation 4): equivalent variation for small price changes maintains the feasibility of consumption choices holding fixed investment choices.

I solve Equation 8 for equivalent variation by integrating over the path of price changes

> 3I impose the initial condition that equivalent variation is 0 when prices do not change, that is that EV _s_ ( _p_<sup>1</sup> _, p_<sup>1</sup> ; _y_ ) = 0 for all _s ∈S_ .

> 4Relatedly, Blackorby et al. (1984) and Keen (1990) demonstrate that static equivalent variation in a model with complete financial markets is equivalent variation only if there are no across period-state income

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from the initial price _p_<sup>1</sup> .

This solution is the integral over compensated demand with respect to price changes. It differs from static equivalent variation in that demand has across-period-state income effects.

**Theorem 2.** _EV_<sup>_∗_</sup> ( _p_<sup>_k_</sup> _, p_<sup>1</sup> ; _y_ ) _∈ EV_ ( _p_<sup>_k_</sup> _, p_<sup>1</sup> ; _y_ )

# **2.3.2 Marshallian surplus**

In the static household model, it is common to use Marshallian surplus, that is the integral over demand with respect to a change in prices, as a measure of consumer surplus when income effects are small (Willig, 1976; Vives, 1987). With incomplete financial markets, just as in the static household model, Marshallian surplus is exact when there are no income effects, as uncompensated and compensated demand are identical.

**Corollary 1.** _Suppose there are no income effects, so B_ ( _p_<sup>_k_</sup> _, y_ ) = _B_ ( _p_<sup>_k_</sup> _, y_<sup>_′_</sup> ) _for all k ∈_ [0 _,_ 1] _, y, y_<sup>_′_</sup> _∈_ R<sup>_S_</sup> _. Define Marshallian surplus in period-state s as_

_Then MS_ ( _p_<sup>_k_</sup> _, p_<sup>1</sup> ; _y_ ) _∈ EV_ ( _p_<sup>_k_</sup> _, p_<sup>1</sup> ; _y_ ) _._

# **2.3.3 Uncompensated and compensated demand elasticities**

In practice, log linear approximations of uncompensated and compensated demand are commonly used, in part because they yield tractable approximations of Marshallian surplus and equivalent variation.<sup>5</sup> For this purpose, and consistent with the empirical application in Section 4, I focus on changes in a single price in this section; for convenience, I use _Bs_ to refer to demand for the single investment choice for which prices are changing.<sup>6</sup>

> 5A log linear approximation of a household’s investment demand is unlikely to be appropriate, as investment choices are often discrete or discontinuously shaped by other discrete choices. However, a log linear approximation of average investment demand across households may be reasonable even when each household makes a discrete investment choice. I discuss the extension of the results in this section to allow for heterogeneous households in Section 2.4.

> 6I extend these results to the case with multiple price changes in Online Appendix B; demand for each investment choice is affected by the full vector of price changes, and income effects are generated by all price changes.

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I define the uncompensated price elasticity of demand The term _εs_ can be interpreted as the price elasticity of expenditures on the investment choice; this corresponds to empirical estimates of impacts on deposits and loan amount in Section 4.

The corresponding compensated price elasticity of demand is The difference between uncompensated and compensated price elasticities of demand is _−φs_ , the income effect of the price change. Expanding the derivative of compensated demand in Equation 12 yields The negative of the income effect is simply the inner product of the household’s marginal propensity to make investment expenditures from period-state income MPBY _s ∈_ R<sup>_S_</sup> , and the growth rate of investment expenditures _Gs ∈_ R<sup>_S_</sup> . In contrast to the static household model, income effects are both within and across period-states. In Section 4, I estimate _Gs_ directly, and consider approaches to bound MPBY _s_ using the marginal propensity to consume.

These elasticities can be used to approximate Marshallian surplus and equivalent variation. If uncompensated and compensated demand are in fact log linear, then These approximations depend only on the prices before and after the price change, and estimates of the price elasticity of investment choice expenditures _εs_ and income effects _φs_ . I estimate the price elasticity of investment choice expenditures _εs_ and bound income effects _φs_ across three empirical applications in Section 4.

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## **2.4 Extensions**

The model and results above considered a single household, and did not normatively interpret consumer surplus.

In Online Appendix C, I follow Hausman & Newey (2016) and consider bounds on average consumer surplus under unobserved heterogeneity across households and shocks; I construct these bounds by substituting bounds on the matrix of average intertemporal marginal propensities to spend into Equation 12.<sup>7</sup> In Online Appendix D, I consider the application of consumer surplus to ranking policies, focusing on the case with unobserved heterogeneity across households and states of the world. I discuss ranking policies using the net present value (NPV) of average consumer surplus, aggregated using the social planner’s cost of transferring income across periods and states. I note it retains a similar interpretation, and limitations, to the NPV of average consumer – surplus in the static model it implicitly assumes the contrasted policies benefit the same groups of households conditional on fixed characteristics (Finkelstein & Hendren, 2020) or income (Hendren, 2020), a strong assumption in the presence of unobserved heterogeneity in preferences across households (Sallee, 2022).

# **3 Practical guidance on consumer surplus from financial technologies**

In Section 2, I established that consumer surplus from a financial technology is equal to the area under dynamically compensated demand; I this section, I lay out key considerations for applied researchers estimating consumer surplus from financial technologies. In Section 3.1, I discuss the definition of prices, and therefore quantities, used to construct demand. In Section 3.2, I describe assumptions, and associated tests, on dynamically compensated demand, that permit the use of statically compensated or uncompensated demand. In Section 3.3, I consider common extensions and potential sources of bias in the estimation of consumer surplus from financial technologies, presented with comparisons to the static case.

> 7I allow unobserved heterogeneity in two steps. First, in Online Appendix C.1, I allow for unobserved heterogeneity across households. Second, in Online Appendix C.2, I allow for unobserved heterogeneity across states of the world, such as unobserved aggregate or household shocks; I make the additional assumption that household intertemporal marginal propensities to spend do not systematically differ out of income received across histories of shocks with respect to compensated demand.

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# **3.1 Defining, estimating, and graphing demand for financial technologies**

I break down the estimation of consumer surplus from a financial technology, or consumer surplus more generally, into three steps: defining demand, estimating demand, and graphically representing surplus as the area under demand. I describe deviations from the static model at each step for the model with incomplete financial markets.

**Defining demand** In Section 2, demand for a financial technology was defined as periodstate specific quantities purchased as a function of prices and income. In the static model for traditional consumer goods, it is typically clear how to define quantities, prices, and income. For financial technologies, this is less obvious. However, just as in the static model, the essential ingredients are that prices and income are perfect substitutes holding fixed quantity, and that price times quantity equals expenditures. I discuss three examples below, and I present three additional examples in the applications in Section 4.

- In Section 1, I presented an example of a static model of demand for a loan. Price is negative loan amount per $1200 of total monthly payments, quantity is total monthly payments divided by $1200, expenditures are negative loan amount, and income is temporary income in the period in which the loan is taken.

- Benetton (2021) present a static model of supply and demand for mortgages; on the demand-side, the borrower chooses one mortgage product from an offered menu. These mortgages have a period during which interest rates are fixed, after which they float with market rates; for tractability, suppose that households pay interest, but not principal, during the fixed rate period. In this case, prices are annual interest rates, quantities are initial loan balances, expenditures are annual loan payments during the fixed rate period, and income is expected annual income during the fixed rate period.

- Benetton et al. (2021) and Robles-Garcia (2022) expand on the static model of supply and demand in Benetton (2021); mortgage products also carry an origination fee. In this case, an alternative definition of prices are origination fees (applied in RoblesGarcia (2022)), quantities are binary take-up (with loan balance and interest rates treated as attributes that differentiate mortgage products), expenditures are the origination fee if the household takes up the loan, and income is temporary income the period in which the mortgage product is selected.

**Estimating demand** As in the static model, estimating demand for a financial technology principally involves isolating exogenous variation in its supply; each of the applications in

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Section 4 leverages a randomized control trial that introduces random variation in consumer prices. The source of random variation in prices determines the interpretation of consumer surplus for the financial technology.

When price changes are temporary and of known duration, consumer surplus is equivalent variation for access to the technology for the duration of the price change; when the duration of the price change is relatively short, a static model is appropriate. In Sections 4.1 and 4.2, I present applications where price changes were temporary and I apply a static model to the estimation of consumer surplus.

When price changes are permanent, consumer surplus is equivalent variation for access to the technology from the time of the price change. The dynamics of demand with respect to permanent price changes are essential to the interpretation of consumer surplus from permanent price changes. In Section 4.3, I present an application in which the price elasticity of demand is increasing over time, and I account for these dynamics when estimating and interpreting consumer surplus.

Compensated demand for financial technologies depends on intertemporal income effects. In Section 3.2, I discuss assumptions under which intertemporal income effects can be ignored, and income effects can be handled as in the static model. In three applications in Section 4, I consider approaches to bound intertemporal income effects in order to bound compensated demand relative to uncompensated demand.

**Graphing demand and consumer surplus** Consumer surplus from financial technologies can be represented as the area under compensated demand. In Section 4, I argue that uncompensated demand is a reasonable approximation of compensated demand for many financial technologies, and I discuss approaches to justifying the assumption that uncompensated demand is a reasonable approximation of compensated demand in Section 3.2. When variation in prices comes from a randomized control trial, as in Section 4, estimating uncompensated demand is simply fitting a curve through average quantities at each randomized price.

In Figure 3, I present estimates of uncompensated demand from each application in Section 4. In each application, the estimates of Marshallian surplus from treatment in Tables 1 and 2 correspond to the integral of demand over a change in prices; the difference between Marshallian surplus and equivalent variation is attributable to income effects.

# **3.2 Simplifying assumptions on dynamically compensated demand**

In this paper, I develop and demonstrate an approach to estimating consumer surplus from financial technologies, building on tools for the estimation of consumer surplus in the static 18
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model. The approach builds on three key sufficient statistics for consumer surplus from financial technologies, defined in Section 2.3.3. The first is the vector of price elasticities _ε_ : I discuss defining, estimating, and visualizing _ε_ in Section 3.1. The second is the growth in demand for the financial technology _G_ . The third is the matrix of marginal propensities to spend on the financial technology in period-state _s_ from temporary income in period-state _s_<sup>_′_</sup> , MPBY. Estimating MPBY is particularly challenging, especially for future temporary income shocks; instead, I propose approaches to constructing plausible bounds on MPBY in the applications in Section 4. These latter two sufficient statistics _G_ and MPBY are used to construct the income effects from dynamic price changes, as one period’s demand may be affected by income in other period-state. It is only through these generalized income effects that estimation of consumer surplus in a dynamic model differs from the estimation of consumer surplus in the static model.

As an alternative to bounding intertemporal marginal propensities to spend MPBY, two approaches are common in literature estimating household welfare gains from a financial technology. The first approach implicitly assumes that there are no intertemporal income effects; that is, that past and anticipated income shocks do not affect use of the financial technology, or that the financial technology is only used once. In this case, consumer surplus from a financial technology is the area under statically compensated demand. The second approach implicitly assumes that there are no income effects, that is, income shocks do not affect use of the financial technology. In this case, consumer surplus from a financial technology is the area under uncompensated demand.<sup>8</sup>

**No intertemporal income effects** As seen in Equation 13, two conditions are required for intertemporal income effects to arise: households must make the investment choice in multiple periods, and income in other periods must affect demand. When households make the investment choice in a single period, or when income in other periods does not affect demand, then a static model of demand is appropriate.

Households do not make an investment choice in multiple periods for many financial technologies. Benetton et al. (2021) consider the case of mortgages by first-time home buyers, which by construction are only made once. As a consequence, the static model they employ is appropriate for calculating consumer surplus from mortgages for first-time home buyers.<sup>9</sup> In addition, their consumer surplus calculations are likely robust to parametric assumptions,

> 8As discussed in Section 3.1, the duration of price changes, and dynamic responses of demand during the price change, remain important to the interpretation of consumer surplus under these simplifying assumptions.

> 9If households can change the timing of their first-time home purchase in response to changes in prices, this approach implicitly sums undiscounted consumer surplus within household across periods. However, this is a general limitation of applying a static framework to the estimation of consumer surplus.

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conditional on their estimated demand matching the appropriate empirical price elasticity of demand.

Household income in other periods may not affect demand for many financial technologies. For example, when investments are made infrequently in contexts with severe credit constraints, income effects are unlikely to affect demand in other periods. Berkouwer & Dean (2022) evaluate consumer surplus from improved cookstoves, which are replaced approximately every two years, and find that credit constraints starkly limit willingness-to-pay. In this environment, a transfer equal to consumer surplus from the cookstove is unlikely to affect demand for the cookstove two years before or after.

**No income effects** In other cases, income effects may not be first order; as demonstrated in Corollary 1, in this case Marshallian surplus (the area under uncompensated demand) is equal to consumer surplus.

As discussed in Section 2.3.2, and seen in Equation 12 and 13, income effects are small when the investment choice represents a small share of income, and a plausible marginal propensity to spend is therefore small, the uncompensated price elasticity of demand is – large. Karlan et al. (2014) study demand for index insurance in Ghana they estimate a price elasticity of demand of -2, while the value of index insurance purchased does not exceed 0.05 as a share of annual agricultural income. These values differ by an order of magnitude, suggesting Marshallian surplus closely approximates consumer surplus from index insurance.

Alternatively, when intertemporal income effects can be ignored and the investment choice is discrete, as in Berkouwer & Dean (2022) discussed above, Marshallian surplus is equal to compensating variation for a price decrease (Bhattacharya, 2015).

# **3.3 Extensions**

The application of consumer surplus as a measure of welfare gains from financial inclusion rests on the strong assumptions laid out in Section 2. Many of these assumptions are also often applied in the static model; a large literature has relaxed many of these assumptions, which can be similarly applied to the incomplete financial markets setting.

- **Information** Households are often not aware of the price schedules they face for financial technologies; Duflo et al. (2007) present evidence that complexity of incentives for retirement savings in the United States may limit takeup, while Karlan & Zinman (2019) argue that their finding that price elasticities of demand for microfinance in Mexico are increasing in time since an interest rate decrease may reflect slow diffusion of information about the decrease. In static settings, estimates of demand under full

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information can be used to recover consumer surplus under incomplete information (Chetty et al., 2009; Allcott & Taubinsky, 2015); my theoretical results suggest these approaches could be extended to the estimation of consumer surplus under incomplete information with incomplete financial markets.

- **Behavioral biases** Households may make investment choices subject to behavioral biases; in the case of microfinance, Banerjee (2013) and Zinman (2014) present a number of models which generate such biases, including present-focused preferences.<sup>10</sup> Mullainathan et al. (2012), Farhi & Gabaix (2020), and Lerva (2022) provide examples of how the estimation of consumer surplus can be augmented with additional sufficient statistics to account for internalities and externalities. These methods build on estimates of compensated demand, such as those I produce in this paper, and permit many comparative statics on welfare to be represented graphically using both observed and “de-biased” demand. In recent work, Allcott et al. (2022) estimate consumer surplus from payday lending in the US, building on their finding that borrowers are sophisticated but present-focused; the intuition above suggests that their welfare analysis is likely robust to many of their modeling assumptions, conditional on implied demand and their estimates of internalities.

- **Producer surplus and supply** Both consumer surplus and producer surplus are necessary for a full accounting of the welfare gains from financial inclusion. However, estimating producer surplus in this setting is complicated by the presence of selection and moral hazard (Zinman, 2014). Recent work has directly estimated the impacts on microfinance lender profits of price changes or the introduction of new products (Karlan & Zinman, 2019; Lane, 2020; DeFusco et al., 2022). Impacts of price changes on profits can be used to construct average and marginal cost curves, accounting for adverse selection and moral hazard, and to characterize the market equilibrium when combined with estimates of demand, leveraging tools applied to the analysis of markets for health insurance Einav et al. (2010), as in DeFusco et al. (2022). With additional instruments, more flexible estimation of supply is possible (Benetton, 2021; Benetton et al., 2021; Robles-Garcia, 2022). My theoretical results in Section 2 clarify that the current practice of abstracting from households’ complex multidimensional decision making environment, and instead focusing on matching empirical demand for the financial technology, does not bias estimates of consumer surplus from price changes, as in the static case.

- **General equilibrium** An immediate consequence of incomplete financial markets is

> 10An analogous concern is that the preferences of a paternalistic social planner may differ from the preferences of the household.

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that households vary in their marginal returns to investment, and reallocation across households in general equilibrium may therefore have first order impacts on efficiency, in addition to the usual distributional consequences. Buera et al. (2021) estimate a model of microfinance in general equilibrium, in which entry and exit of and reallocation across heterogeneous entrepreneurs are triggered by microfinance and its impacts on equilibrium wages, while Breza & Kinnan (2021) provide evidence that equilibrium impacts of microfinance on wages are large. As a complementary approach, recent work has decomposed the general equilibrium impacts of microeconomic shocks on aggregate welfare into a partial equilibrium effect (which I estimate), and an allocative efficiency effect (Baqaee & Farhi, 2020; Bau & Matray, 2020; Sraer & Thesmar, 2020); under these approaches, the impacts of financial inclusion on allocative efficiency depend crucially on its impacts on reallocation towards high marginal return entrepreneurs.

# **4 Consumer surplus from savings and microfinance**

In this section, I apply the framework in Section 2, and approach discussed in Section 3.1, to estimate consumer surplus from savings and microfinance. I build on three randomized control trials (Duflo et al., 2006; Karlan & Zinman, 2018, 2019) which introduce experimental variation in interest rates for retirement savings accounts in the United States (Section 4.1), commitment savings accounts in Philippines (Section 4.2), and microfinance loans in Mexico (Section 4.3), respectively.

## **4.1 Retirement savings (Duflo et al., 2006)**

### **4.1.1 Context and experimental design**

I leverage experimental variation in retirement savings match rates studied by Duflo et al. (2006) to estimate the saver surplus from retirement savings. The experiment was implemented with H&R Block, a large tax preparer in the United States, for a single retirement savings product, the X-IRA.

**Retirement savings product** I briefly summarize the description of the retirement savings product from Duflo et al. (2006), and incorporate it into the model in Section 2. At the time households enrolled in the experiment prepare their taxes with H&R Block, they are offered the opportunity to make a one-time deposit into an X-IRA with their forthcoming tax refund, with a minimum deposit of 300 USD. The deposit into the X-IRA may or may not be tax deductible, and households may or may not face tax penalties for withdrawing

23
from their X-IRA before age 59.5. Lastly, households enrolled in the experiment are offered a match rate _r_ on deposits up to 1000 USD; in practice, 85% of deposits in treatment arms with a match occurred below or at the 1000 USD threshold. The experiment, including the match offer, was implemented during a single tax preparation season in 2005.

I therefore represent the retirement savings product in the household problem in Equation 1 as where <u>1</u> 1+ _r_<sup>_b_1 and</sup><sup>_b_1 are the household’s deposit into the retirement savings account, excluding</sup> and including the match, respectively.

**Experimental variation in match rates** Households enrolled in the experiment were randomized into a 0%, 20%, and 50% match rate, revealed at the time of the offer to deposit into the savings account. I let _D ∈{_ 0 _,_ 1 _,_ 2 _}_ correspond to a household’s treatment assignment in order of increasing match rate. I normalize _p_<sup>0</sup> = 1 to be the price of deposits under 0% <u>1</u> match rate, yielding prices of deposits under a 20% and 50% match rate of _p_<sup>1</sup> = 1+0 _._ 2<sup>= 0</sup><sup>_._833</sup> and _p_<sup>2</sup> = 0 _._ 667, respectively.

**Prices and investment choice set** One limitation of this modeling approach for this experiment is that prices enter the household problem both through the budget constraint and the investment choice set. Note that we can rewrite the household’s investment choice set as

In this case, changes in prices now have an additional welfare impact through shifting the households’ choice set of deposits including the match (that is, _b_ 1). However, note that households can always choose to deposit more than 1000 USD in the X-IRA, albeit with no match; this suggests a revealed preference approach to place an upper bound on how much households value the impact of price decreases on their investment choice set, which could be implemented with microdata from the experiment.

**Additional modeling considerations** Key aspects of the tax incentives and withdrawal penalties around the savings product above complicate modeling, yet can be flexibly accommodated in the model in Section 2. Not only is the timing of withdrawals from the savings product endogenous to deposits, but tax deductions and penalties will vary both

24
with deposits and also other behavioral responses endogenous to deposits. I cam model tax deductions, withdrawals, and tax penalties flexibly through _fs_ ( **b** ), which allows these outcomes to depend not only on the amount the household deposited and its previous withdrawals, but also on any other investment choices made by the household that affect its tax status.

# **4.1.2 Demand elasticities and consumer surplus**

**Estimation strategy** I leverage the experimental variation in match rates to estimate the price elasticity of demand for commitment savings. Let _mi_ denote observed deposits by household _i_ into the offered X-IRA retirement savings account, and let _Di ∈{_ 0 _,_ 1 _,_ 2 _}_ indicate the treatment assignment for household _i_ . I estimate

This fits a log linear model of average deposits as a function of prices through estimates of average deposits conditional on treatment assignment from Duflo et al. (2006); absent microdata from the experiment, I estimate this model by generalized method of moments.

**Price elasticity of deposits** _ε_ I present my estimate of the price elasticity of deposits for retirement savings in Table 1, along with associated estimates of compensated demand and consumer surplus from commitment savings. As deposits correspond to expenditures in the model in Section 2, my estimate corresponds to _ε_ (rather than the price elasticity of demand _ε −_ 1) in the log linearization of demand in Section 2.3.3. I estimate _ε_ = _−_ 4 _._ 2, with a standard error of 0 _._ 3; the large variation in match rates introduced by the experiment results in a highly precise estimate of the price elasticity of deposits.

**Bounding income effects** _φ_ **and the compensated deposits elasticity** As the experiment consists of a one-shot offer, I model the match rate as lasting for a single period, and the income effect term in the compensated elasticity of deposits _φ_ simplifies to _φ_ = MPBY, the marginal propensity to deposit into the retirement savings account from temporary income. Using a one year MPC of 0.5, and allowing for the possibility that households have access to multiple savings accounts, I upper bound MPBY _≤_ 1 _−_ MPC = 0 _._ 5. I further assume that households would not decrease their retirement savings deposits in response to a temporary positive income shock, implying MPBY _≥_ 0. This yields _φ ∈_ [0 _,_ 0 _._ 5], and in turn yields bounds on the compensated price elasticity of deposits _φ_ + _ε ∈_ [ _−_ 4 _._ 2 _, −_ 3 _._ 7].

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Table 1: Price elasticities of savings demand and saver surplus

|(Dufo et al., 2006)|(Karlan & Zinman, 2018)|
|---|---|
|(1)<br>|(2)|
|**Panel A: Regression estimates of demand elasticities**||
|log de<br>|posit amount<br>|
|(USD)|(PHP)|
|Intercept<br>3.37|5.64|
|(0.09)<br>[0.000]|(0.08)<br>[0.000]|
|log price (_ε_)<br>-4.22|-9.74|
|(0.26)<br>[0.000]|(8.16)<br>[0.233]|
|# of household observations<br>13,962|6,696|
|**Panel B: Estimates of compensated demand elasticities **|**and consumer surplus**|
|Compensated deposit elasticity (_φ_+_ε_)<br>-4.22<br>(Lower bound)|-9.74|
|Compensated deposit elasticity (_φ_+_ε_)<br>-3.72<br>(Upper bound)|-9.24|
|_Surplus from high interest rate treatment_<br>_per $100 control group deposit amount_||
|Marshallian surplus<br>$107.43<br>($7.29)|$1.58<br>($0.10)|
|Equivalent variation (Lower bound)<br>$94.58|$1.57|
|Equivalent variation (Upper bound)<br>$107.43|$1.58|
|_Surplus from savings_<br>_per $100 deposit amount_||
|Marshallian surplus<br>$23.70<br>($1.48)|$10.27<br>($8.61)|
|Equivalent variation (Lower bound)<br>$23.70|$10.27|
|Equivalent variation (Upper bound)<br>$26.89|$10.83|

_Notes:_ Regression coefficients and consumer surplus are presented in this table, with standard errors in parentheses and p-values in brackets. In Panel A, regression coefficients are presented; Column 1 is estimated by GMM matching average outcomes conditional on randomly assigned prices reported in Duflo et al. (2006), while Column 2 is estimated by Poisson PML with heteroskedasticity-robust standard errors. In Panel B, bounds on compensated deposit elasticities and consumer surplus are presented.

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**Marshallian surplus from and equivalent variation bounds for high interest rate treatment and commitment savings** I apply Equations 14 and 15 for Marshallian surplus and equivalent variation, respectively, to calculate the welfare gains from the high match rate treatment and from retirement savings.

The high match rate treatment, expressed per $100 of control group deposits, generated Marshallian saver surplus of $107.4, with a standard error of $7.3. As the high match rate treatment generated a large increase in deposits through a large decrease in prices, its associated surplus is closely tied to the estimated demand elasticity.

Marshallian saver surplus from retirement savings is simply _−_ 1 _/ε_ , implying that retirement savings generates saver surplus of $23.7 per $100 of deposits, with a standard error of $1.5. Bounds on equivalent variation are of similar width to the 95% confidence interval for the estimate of Marshallian saver surplus. Although estimates of the demand elasticity for retirement savings are the most precise that I analyze in this paper, due to the large sample and large variation in match rates, statistical uncertainty in the point estimate remains similar in magnitude to bias in Marshallian saver surplus as a measure of equivalent variation due to income

# **4.2 Commitment savings (Karlan & Zinman, 2018)**

## **4.2.1 Context and experimental design**

I leverage experimental variation in interest rates on commitment savings accounts studied by Karlan & Zinman (2018) to estimate the saver surplus from commitment savings. The experiment was implemented with First Valley Bank in the Philippines, for a commitment savings product, Gihandom Savings.

**Commitment savings product** I briefly summarize the description of the commitment savings product from Karlan & Zinman (2018), and incorporate it into the model in Section 2. Households were recruited for the experiment in a door-to-door marketing campaign, and randomized into three commitment savings account offers, two of which we focus on which vary the annual interest rate _r_ – one with _r_ = 1 _._ 5%, and the other with _r_ = 3 _._ 0%. Following the offer, households could open the account from the bank with a minimum deposit of 100 PHP; households would set a goal amount of at least 2,000 PHP, and a goal term of between 3 months and 2 years, and could withdraw only after both the goal term and amount were met.

I therefore represent the commitment savings product in the household problem in Equa27
tion 1 as where 1+1 _r_<sup>_bt_and</sup><sup>_bt_arethehousehold’sdepositsandend-of-periodbalance,respectively,in</sup> the commitment savings account, and _r_ is the annual interest rate. I model the deposit decision as annual, to align the model with the outcome variable I use in the analysis in Section 4.2.2 (deposits during the 12 months following the introduction of the commitment savings account).

**Experimental variation in interest rates** Households were randomized at the time of the marketing visit into a 1.5% and 3% offered interest rate. I let _D ∈{_ 0 _,_ 1 _}_ correspond to the household’s treatment assignment in order of increasing interest rate. I normalize _p_<sup>0</sup> = 1 to be the price of deposits under the 1.5% interest rate, yielding the price under a 3% interest rate of _p_<sup>1</sup> =<sup><u>1+0</u></sup> 1+0<sup>_<u>.</u>_</sup> _._<sup><u>015</u></sup> 030<sup>= 0</sup><sup>_._985.</sup>

**Prices and investment choice set** One limitation of this modeling approach for this experiment relates to the interaction of interest rates and the commitment aspect of the savings accounts leading to changes in the investment choice set. When a household receives additional interest, its balance increases, and as a result the household is committed to maintaining a higher balance until they reach their goal, corresponding to a higher lower end of the interval _Bt,_ 1( _bt−_ 1). This tightening of the investment choice set would reduce my estimates of the welfare gains from commitment savings in Section 4.2.2.

**Additional modeling considerations** Key aspects of the commitment savings product above and the household’s deposit and withdrawal decisions complicate modeling; despite this complexity, these can be flexibly accommodated in the model in Section 2. First, households may withdraw early in the event of hardship. Borrowers’ investment choice set _Bt_ ( _bt−_ 1) may be period-state-specific, permitting withdrawals in response to shocks. In addition, if early withdrawal comes with some ordeal costs or punishment, the early withdrawal decision may directly enter into utility. Second, households face a minimum initial deposit; this may be modeled similarly to the minimum initial deposit for retirement savings in Section 4.1.

# **4.2.2 Demand elasticities and consumer surplus**

**Estimation strategy** I leverage the experimental variation in interest rates to estimate the price elasticity of demand for commitment savings. Let _mi_ denote observed deposits

28
by household _i_ during the 12 months following the introduction of the commitment savings account, and let _Di ∈{_ 0 _,_ 1 _}_ indicate the treatment assignment for household _i_ . I estimate This is similar to the estimation of impacts in Karlan & Zinman (2018), with three differences. First, I transform treatment assignment into log prices, with control group log price normalized to 0 (log _p_<sup>0</sup> = 0); this rescales the treatment effect of high interest rates on deposits by the inverse of the treatment group log price, that is 1 _/_ log _p_<sup>1</sup> _≈−_ 67. Second, I do not include any control variables, as including controls does not meaningfully affect the results; the intercept _α_ therefore equals the log of average deposits at control group interest rates.

**Price elasticity of deposits** _ε_ I present my estimate of the price elasticity of deposits for commitment savings in Table 1, along with associated estimates of compensated demand and consumer surplus from commitment savings. As deposits correspond to expenditures in the model in Section 2, my estimate corresponds to _ε_ (rather than the price elasticity of demand _ε −_ 1) in the log linearization of demand in Section 2.3.3. I estimate _ε_ = _−_ 9 _._ 7, with a standard error of 8 _._ 2; while the point estimate is not significantly different from 0, it is sufficiently large to rule out much larger price elasticities of deposits, which would correspond to much smaller consumer surplus from savings.

**Bounding income effects** _φ_ **and the compensated deposits elasticity** Absent data on future years of commitment savings deposits, I model the change in interest rates as lasting for just one year. In this case, the income effect term in the compensated elasticity of deposits _φ_ simplifies to _φ_ = MPBY, the marginal propensity to deposit into the commitment savings account from temporary income. Using a one year MPC of 0.5, and allowing for the possibility that households have access to multiple savings accounts, I upper bound MPBY _≤_ 1 _−_ MPC = 0 _._ 5. I further assume that households would not decrease their commitment savings deposits in response to a temporary positive income shock, implying MPBY _≥_ 0. This yields _φ ∈_ [0 _,_ 0 _._ 5], and in turn yields bounds on the compensated price elasticity of deposits _φ_ + _ε ∈_ [ _−_ 9 _._ 7 _, −_ 9 _._ 2].

**Marshallian surplus from and equivalent variation bounds for high interest rate treatment and commitment savings** I apply Equations 14 and 15 for Marshallian surplus and equivalent variation, respectively, to calculate the welfare gains from the high interest rate treatment and from commitment savings.

29
The high interest rate treatment, expressed per $100 of control group deposits, generated Marshallian saver surplus of $1.6, with a standard error of $0.1. This estimate is much more – precise than the estimated demand elasticity the interest rate variation induced by the experiment is sufficiently small that the impacts of the high interest rate treatment on saver welfare are well approximated by the reduced deposits savers can make holding fixed their end-of-year balance. Closely related to this point, bounds on equivalent variation to the high interest rate treatment are not meaningfully different from Marshallian surplus.

For larger price changes, the estimated elasticity plays a much larger role; Marshallian saver surplus is simply _−_ 1 _/ε_ , implying that commitment saving generates saver surplus of $10.3 per $100 of deposits, with a standard error of $8.6. Relatedly, bounds on equivalent variation are also wider, at [$10.3, $10.8]; however, the width of these bounds is small relative to statistical uncertainty in the point estimate for Marshallian saver surplus.

# **4.3 Microfinance (Karlan & Zinman, 2019)**

## **4.3.1 Context and experimental design**

I leverage experimental variation in microfinance interest rates studied by Karlan & Zinman (2019) to estimate the borrower surplus from microfinance. The experiment was implemented with Mexico’s largest microfinance lender, Compartamos Banco, for a single microfinance lending product, Crédito Mujer.

**Microfinance product** I briefly summarize the description of the microfinance credit product from Karlan & Zinman (2019), and incorporate it into the model in Section 2. Each period, borrowing groups of women are evaluated based on observed characteristics, including general creditworthiness and past borrowing and repayment behavior with the microfinance lender. Borrowing groups are offered the opportunity to take a non-collateralized loan up to a threshold based on these characteristics; offered thresholds range from 4,000 MXN to 24,000 MXN. Borrowers are obligated to repay the loan in identical weekly installments over 16 weeks, which are calculated based on three factors: the initial principal, value-added tax on interest payments, and “add-on” interest calculated based on the initial principal rather than the balance of the loan. Monthly add-on interest before the experiment ranged from 4.0% to 5.0%, with the interest determined geographically and on the basis of past borrowing and repayment behavior. I let _bt−_ 1 _∈_ R<sup>_N_</sup> +<sup>denotepastborrowingandrepayment</sup> behavior, _rt_ ( _bt−_ 1) : R<sup>_N_</sup> +<sup>_→{_0</sup><sup>_._040</sup><sup>_,_0</sup><sup>_._045</sup><sup>_,_0</sup><sup>_._050</sup><sup>_}_denotethepre-experimentalinterestrate,</sup> _bt_ ( _bt−_ 1) : R<sup>_N_</sup> +<sup>_→_[4000</sup><sup>_,_24000]denotetheofferedthreshold,and</sup><sup>_bt,_1denotetheprincipalon</sup> new loans taken in period _t_ .

30
A borrower who takes a loan of _bt,_ 1 _∈_ R+ is obligated to repay<sup>1+1</sup><sup>_._15</sup><sup>_∗_4</sup> 16<sup>_∗rt_</sup><sup><u>(</u></sup><sup>_bt−_1)</sup> _bt,_ 1 in each of the 16 weeks following period _t_ , which includes 15% value-added tax on the 4 months of repayments of monthly add-on interest. I represent the lending product in the household problem in Equation 1 as

**Additional modeling considerations** Key aspects of the lending product above and the household’s borrowing decisions complicate modeling; crucially, these can be flexibly accommodated in the model in Section 2. First, default and delayed repayment do occur (1% and 10% of loans, respectively), so households repayments are also a function of their past and current decisions to delay repayment or default. I can capture these responses by incorporating these delay and default decisions into the vector of investment choices _bt_ , and by allowing flexible technology _ft_ ( _bt_ ). Second, there may be additional extensive margin borrowing costs, such as group formation, and default costs, such as social pressure from other group members. I can represent these costs by allowing borrowing and default decisions in _bt_ to enter directly into utility _U_ ( _c,_ **b** ). Third, borrowers are ostensibly required to make deposits into a personal savings account proportional to their borrowing, but these requirements were partially enforced. These partially enforced requirements may generate period-state-specific interactions across the borrower’s investment choices in the borrower’s choice set _Bt_ ( _bt−_ 1), or alternatively in the borrower’s utility function _U_ ( _c,_ **b** ).

**Experimental variation in interest rates** The experiment was implemented in the context of the microfinance lender decreasing its monthly add-on interest rates. At the start of the experiment, the microfinance lender’s geographic regions had their monthly add-on interest rates decreased across all borrowers; this decrease was 0.5pp for randomly assigned control regions, and 1pp for randomly assigned treatment regions. Let _D_ be an indicator for treatment assignment; following the start of the experiment, weekly repayment obligations 1+1 _._ 15 _∗_ 4 _∗_ <u>(</u> _rt_ <u>(</u> _bt−_ 1) _−_ <u>(1+</u> _D_ <u>)</u> _∗_ 0 _._ 005) (as a fraction of the initial principal) were decreased to 16 . This decrease is well approximated by a constant proportion: repayment obligations fell by an additional 1.91% to 1.98% (1.95% average) in treatment regions relative to control regions. The experiment was implemented in 78 regions and lasted for 29 months from 2007 through 2009, after which the lender decreased control region interest rates to the level in treatment regions.

31
I therefore model the experiment as permanently shifting the price _p_ of the microfinance lending product; prices are negative as households borrow, with more negative prices (price decreases) corresponding to larger principal holding fixed repayment obligations. I normalize _p_<sup>0</sup> = _−_ 1 to be the price in the control group, and I calculate low interest rate treatment group prices _p_<sup>1</sup> = _−_ 1 _−_ 01 _._ 0195<sup>=</sup><sup>_−_1</sup><sup>_._020.Similarly,Ilet</sup><sup>_bt,_1(</sup><sup>_pD_)representmicrofinanceborrowing</sup> under assigned prices _p_<sup>_D_</sup> . The monthly loan amount I observe in the data under treatment assignment _D_ are therefore equal to _−p_<sup>_D_</sup> _bt,_ 1( _p_<sup>_D_</sup> ); microfinance borrowing choices _bt,_ 1( _p_<sup>_D_</sup> ) are proportional to repayment obligations, while prices _p_<sup>_D_</sup> shift principal holding fixed repayment obligations.

This modeling choice implicitly imposes additional assumptions. First, it treats the price change as the same across counterfactual add-on interest rates, while in fact price changes differ slightly; this error is an order of magnitude smaller than other sources of estimation error, and I therefore ignore it. Second, it imposes that the investment choice set _Bt_ ( _bt−_ 1) is not affected by prices; that is, it imposes that the maximum borrowing threshold is proportional to repayment obligations, and therefore _bt,_ 1, rather than loan amount _−p_<sup>_D_</sup> _bt,_ 1. When maximum borrowing thresholds are on loan amount _−p_<sup>_D_</sup> _bt,_ 1, price decreases also reduce welfare by tightening the constraint on microfinance borrowing choices _bt,_ 1; at the extreme, a constrained household that does not value the future does not benefit from such a price decrease, as they do not value their reduced repayment obligations. In practice, Karlan & Zinman (2019) find that only 19% of loans are at the maximum borrowing threshold, suggesting that bias due to overestimation of the welfare gains from price decreases to constrained borrowers is likely to be small.

# **4.3.2 Demand elasticities and consumer surplus**

**Estimation strategy** I leverage the experimental variation in interest rates to estimate the price elasticity of demand for microfinance. Let _lg,n_ denote total observed loan amount in region _g_ in month _n_ , and let _t_ ( _n_ ) _∈{_ 1 _,_ 2 _,_ 3 _}_ correspond to the year of the experiment. I let _lg,_ 0 be the pre-experimental loan amount in region _g_ . Further, I let _Dg ∈{_ 0 _,_ 1 _}_ indicate the treatment assignment for region _g_ . I estimate

This is similar to the estimation of impacts in Karlan & Zinman (2019), with two differences. First, I transform treatment assignment into log prices, with control group log price normalized to 0 (log( _−p_<sup>0</sup> ) = 0); this rescales the treatment effect of low interest rates on log loan amount by the inverse of the treatment group log price, that is 1 _/_ log( _−p_<sup>1</sup> ) _≈_ 50. Second, 32
I control for log total loan amount from the two pre-experimental months for which data is available, log _lg,_ 0, to improve precision (McKenzie, 2012). In addition, I demean baseline values of the dependent variable _lg,_ 0; the intercept _αt_ ( _n_ ) therefore equals average log loan amount at control group interest rates in Year _t_ .

**Price elasticity of loan amount** _ε_ I present my estimates of the price elasticity of loan amount for the microfinance lending product in Table 2, along with associated estimates of compensated demand and consumer surplus from the microfinance lending product. As loan amount corresponds to negative expenditures in the model in Section 2, my estimate corresponds to _ε_ (rather than the price elasticity of demand _ε −_ 1) in the log linearization of demand in Section 2.3.3. I estimate the price elasticity of demand grows over time, with _ε_ 1 = 6 _._ 4 with a standard error of 3.2 in Year 1 and growing to _ε_ 3 = 14 _._ 1 with a standard error of 5.3 in Year 3. These estimates are consistent with those in Karlan & Zinman (2019), albeit with a scale.

**Bounding income effects** _φ_ **and the compensated loan amount elasticity** In contrast to Sections 4.1 and 4.2, I estimate dynamic _εt_ . To produce bounds on income effects, I therefore consider bounds on both _−_ MPBY _t,t′_ , the marginal propensity to take loan amount in period _t_ from temporary income in period _t_<sup>_′_</sup> (negative because loan amount corresponds to negative expenditures), and on _Gt,t′_ , the growth rate of loan amount.

First, I bound _−_ MPBY _t,t′_ . I do so in two steps – I make an assumption on intertemporal marginal propensity to consumes, and I then bound the marginal propensity to take loan amount relative to the marginal propensity to consume. In the first step, I assume that the marginal propensity to consume in period _t_ from shocks to temporary income in period _t_<sup>_′_</sup> is MPC _t,t′_ = MPC(1 _−_ MPC)<sup>_|t−t′|_</sup> . I argue this functional form is a reasonable starting point. For shocks to future income ( _t_<sup>_′_</sup> _> t_ ), it is the unique functional form that yields a proportionally declining marginal propensity to consume from future income, and also a marginal propensity to consume from permanent income of 1 (Gelman et al., 2016; Ralston et al., 2017). For shocks to past income ( _t_<sup>_′_</sup> _< t_ ), it imposes that households consume a constant fraction MPC of the temporary income shock in each period, and save the remainder. In the second step, I construct lower and upper bounds in the following manner. For an upper bound, I assume that households borrow no more than their marginal propensity to consume from temporary income, that is _−_ MPBY _t,t′ ≤_ MPC _t,t′_ . For a lower bound for shocks to future income, I assume that households will not decrease borrowing in response to a positive shock to future income, that is _−_ MPBY _t,t′ ≥_ 0 when _t_<sup>_′_</sup> _> t_ . For a lower bound for shocks to past or current income, I assume that borrowing will not decrease

33
Table 2: Price elasticities of microfinance demand and borrower surplus

|(Karla|n & Zin|man, 2019)|
|---|---|---|
|Year 1|Year 2|Year 3|
|(1)|(2)|(3)|
|**Panel A: Regression estimates of demand elastici**|**ties**||
|log loan|amount|(’000 MXN)|
|Intercept<br>8.61|8.95|9.16|
|(0.04)|(0.05)|(0.06)|
|[0.000]|[0.000]|[0.000]|
|log loan amount,<br>0.72|0.53|0.43|
|Year 0, demeaned<br>(0.04)|(0.04)|(0.05)|
|[0.000]|[0.000]|[0.000]|
|log price (_εt_)<br>6.40|10.50|14.08|
|(3.22)|(4.30)|(5.25)|
|[0.047]|[0.015]|[0.007]|
|# of regions<br>78|78|78|
|# of region-month observations<br>936|936|390|

**Panel B: Estimates of compensated demand elasticities and consumer surplus**

|Compensated loan amount elasticity (_φt_+_εt_)<br>(Lower bound)|4.40|8.25|11.70|
|---|---|---|---|
|Compensated loan amount elasticity (_φt_+_εt_)<br>(Upper bound)|6.90|11.25|14.95|
|_Surplus from low interest rate treatment_<br>_per $100 control group loan amount_||||
|Marshallian surplus|$2.11<br>($0.07)|$2.20<br>($0.10)|$2.28<br>($0.12)|
|Equivalent variation (Lower bound)|$2.07|$2.15|$2.23|
|Equivalent variation (Upper bound)|$2.12|$2.22|$2.30|
|_Surplus from microfnance_<br>_per $100 loan amount_||||
|Marshallian surplus|$15.62<br>($7.84)|$9.53<br>($3.90)|$7.10<br>($2.65)|
|Equivalent variation (Lower bound)|$14.48|$8.89|$6.69|
|Equivalent variation (Upper bound)|$22.71|$12.13|$8.55|

_Notes:_ Regression coefficients and consumer surplus are presented in this table, with standard errors in parentheses and p-values in brackets. In Panel A, regression coefficients are presented; Columns 1 through 3 are estimated by OLS with robust standard errors clustered at the region-level. In Panel B, bounds on compensated loan amount elasticities and consumer surplus are reported.

34
by more than the household’s marginal propensity to save from temporary income, that is _−_ MPBY _t,t′ ≥−_<sup><u>1</u></sup><sup>_−_</sup> MPC<sup><u>MPCMPC</u></sup><sup>_t,t′_when</sup><sup>_t′≤t_.Insummary,thisyields</sup> Second, I bound _Gt,t′ ∈_ [1 _,_ 1 _._ 5], with 1.5 corresponding to the ratio of Year 2-to-Year 1 loan amount in the low interest rate treatment group.

Using a one year MPC of 0.5, this yields These bounds on income effects are much wider than the bounds on static income effects in – Section 4.1 and 4.2 this is because the price reduction results in income effects from past, current, and future periods. Despite this, these bounds remain relatively tight compared to the demand elasticities estimated in Table 2. For the estimated Year 1 price elasticity of loan amount of _ε_ 1 = 6 _._ 4, the bounds on the compensated price elasticity of loan amount are _φ_ 1 + _ε_ 1 _∈_ [4 _._ 4 _,_ 6 _._ 9], while for the estimated Year 3 price elasticity of loan amount of _ε_ 3 = 14 _._ 1, the bounds on the compensated price elasticity of loan amount are _φ_ 3 + _ε_ 3 _∈_ [11 _._ 7 _,_ 15 _._ 0]. As in the previous two examples, statistical uncertainty is larger than potential bias from using uncompensated price elasticities as a measure of compensated price elasticities.

**Marshallian surplus from and equivalent variation bounds for low interest rate treatment and microfinance** I apply Equations 14 and 15 for Marshallian surplus and equivalent variation, respectively, to calculate the welfare gains from the low interest rate treatment and from the microfinance lending product.

The low interest rate treatment, expressed per $100 of control group loan amount, generated Marshallian borrower surplus of $2.1 in Year 1, with a standard error of $0.1, and $2.3 in Year 3, with a standard error of $0.1. This estimate is much more precise than – the estimated demand elasticity the interest rate variation induced by the experiment is sufficiently small that the impacts of the low interest rate treatment on borrower welfare are well approximated by the increased loan amount savers can receive holding fixed their repayments. Closely related to this point, bounds on equivalent variation to the low interest rate treatment are not meaningfully different from Marshallian surplus.

For larger price changes, the estimated elasticity plays a much larger role; Marshallian borrower surplus is simply 1 _/ε_ , implying that the microfinance lending product generates borrower surplus of $15.6 per $100 of deposits in Year 1, with a standard error of $7.8, and 35
$7.1 per $100 of deposits in Year 3, with a standard error of $2.7. Relatedly, bounds on equivalent variation are also wider, at [$14.5, $22.7] in Year 1 and [$6.7, $8.6] in Year 3; however, the width of these bounds is small relative to statistical uncertainty in the point estimate for Marshallian borrower surplus.

# **4.4 Discussion of results**

**Interpretation of magnitude of surplus** Leveraging experimental estimates of demand for savings and microfinance, across a range of contexts, I estimate Marshallian surplus from investment technologies ranging from $7 per $100 of loan amount from microfinance in Mexico to $24 per $100 of deposits for retirement savings in the United States. One natural question in response to these estimates is whether we should think of them as small or large.

To interpret my estimates of Marshallian surplus, I begin by comparing them to comparable existing estimates of welfare gains from savings and microfinance. In Table 3, I present experimental and quasi-experimental estimates of the annual welfare gains from the introduction of new savings and microfinance technologies, expressed per $100 of loan amount or deposits. As most of the estimates are for microfinance, I focus my discussion on the comparison to and the interpretation of my estimates of Marshallian surplus from microfinance. First, and most comparable to my estimates, Kaboski & Townsend (2011) estimate a structural model of savings, borrowing, and investment that closely matches quasi-experimental impacts of microfinance in Thailand; they use the estimated model to calculate equivalent variation from microfinance, albeit as a single initial period transfer. I use this estimate to construct a conservative lower bound on annual equivalent variation of $4 per $100 of loan amount, somewhat smaller than my estimates of Marshallian surplus from microfinance in Mexico.

Second, I compare my estimates to a more common approach: using household consumption as a proxy for household welfare. The prevalence of this approach is despite the well known limitation that impacts of savings or microfinance on household consumption may fail to recover the correct sign of welfare gains (Kaboski & Townsend, 2011; Banerjee et al., 2015b). In Table 3, I highlight another limitation: these estimates tend to be extremely imprecise; with two exceptions, the standard errors on the estimated impacts of microfinance or savings on household consumption are at least twice as large as my largest estimate of Marshallian surplus, suggesting even a metaänalysis across these estimates would be underpowered to reject that they are, on average, the same as my estimates of Marshallian surplus. The two exceptions inform future research using household consumption as a proxy for household welfare. The first exception, Attanasio et al. (2015), estimates impacts of mi36
Table 3: Comparing surplus from microfinance and savings to their impacts on consumption

|Source<br>Country|Outcome|Year|Estimate|(SE)|
|---|---|---|---|---|
|Consumption impacts of savings|||||
|Dupas et al. (2018)<br>MWI|Household expenditures|2011|$129|($260)|
|UGA|Household expenditures|2011|$37|($164)|
|Surplus from savings|||||
|Table 1<br>USA|Marshallian surplus|2005|$24|($1)|
|Table 1<br>PHL|Marshallian surplus|2007|$10|($8)|
|Consumption impacts of microfnance|||||
|Angelucci et al. (2015)<br>MEX|Food expenditures|2011|$66|($248)|
|Attanasio et al. (2015)<br>MNG|Household expenditures|2009|$25|($14)|
|Augsburg et al. (2015)<br>BIH|Household expenditures|2010|-$111|($56)|
|Banerjee et al. (2015a)<br>IND|Household expenditures|2008|$26|($95)|
||Household expenditures|2010|-$188|($198)|
|Crépon et al. (2015)<br>MAR|Household consumption|2009|-$68|($71)|
|Kaboski & Townsend (2011)<br>THA|Household consumption|2002|$139|($39)|
|Surplus from microfnance|||||
|Table 2<br>MEX|Marshallian surplus|2007|$16|($8)|
||Marshallian surplus|2009|$7|($3)|
|Kaboski & Townsend (2011)<br>THA|Equivalent variation (Lower bound)|2002|$4||
||Equivalent variation, present value|2002|$69||

_Notes:_ Experimental estimates of the impacts of savings and microfinance on consumption and estimates of saver and borrower surplus from microfinance, per $100 of deposits and loan amount, are reported in this table, with standard errors in parentheses. The construction of estimates from other papers is described in Online Appendix A.

crofinance in Mongolia on household expenditures of $25 (with a standard error of $14) per $100 loan amount, significantly different from $0 but not from my estimates of Marshallian surplus from microfinance in Mexico. Their estimate is relatively precise despite a small number of observations for two primary reasons: they have a strong first stage, with treated households 51pp more likely to take a loan from the MFI than control households, and the loans were large, with an average loan size of two months of household consumption. The second exception, Kaboski & Townsend (2011), estimate impacts of microfinance in Thailand on household consumption of $139 (with a standard error of $39) per $100 loan amount. As they also find these large positive impacts persist over time, this estimate on its own implies much larger welfare gains than their structural model; in Online Appendix E, I demonstrate that because the impact of microfinance on consumption does not discount uncertain returns to risky investments, it may be biased upwards as a measure of consumer surplus.

With this context, I then ask if we should think of the magnitudes of Marshallian surplus 37
from savings and microfinance as small or large. First, I note that my estimates of Marshallian surplus are not significantly different from estimates of the impacts of microcredit loans to women in Bangladesh on household consumption from Pitt & Khandker (1998) ($18 per $100 loan amount) and from Khandker (2005) ($15 and $21 per $100 loan amount); while the internal and external validity of these estimates has been widely debated, the magnitudes themselves were interpreted as implying that microfinance cost-effectively increases household consumption (Morduch, 1999). Second, I note that it is often the case that households’ borrowing and saving is large as a share of household consumption. Household credit as a share of GDP averages roughly 10% in low income countries where data is available, and rises to over 40% in high income countries (International Monetary Fund, 2006). In low income countries, my estimates of Marshallian surplus from microfinance in Table 2 would therefore correspond to consumer surplus from household credit of 1% - 2% of GDP. Lastly, I note that the price elasticities of demand I estimate for microfinance and savings are large relative to conventional estimates of demand for consumer goods. As consumer surplus is inversely proportional to the compensated price elasticity of demand, these large elasticities correspond to relatively small consumer surplus. However, as noted above, household borrowing and saving is large as a share of household consumption. The consumer surplus per dollar lent or saved may be small, and therefore often empirically elusive, but the intense use of financial technologies corresponds to large aggregate welfare gains from financial inclusion.

**Interpretation of accuracy and precision of elasticities** To visualize the variation in my estimates of price elasticities of demand, I plot the absolute values of my estimates of the price elasticities of deposits and loan amounts _ε_ , along with bounds on compensated elasticities _φ_ + _ε_ , in Figure 4. With the exception of Duflo et al. (2006), statistical imprecision in the estimates is substantial, as they introduced an order of magnitude more variation in prices than Karlan & Zinman (2018) and Karlan & Zinman (2019). In all cases, bounds – on compensated elasticities are narrower than 95% confidence intervals improvements in the efficiency of estimates of consumer surplus from financial technologies are likely to come primarily from more precise estimates of price elasticities of demand, which will likely require the introduction of large variation in prices.

# **5 Conclusion**

In this paper, I estimate the partial equilibrium consumer welfare gains from savings and credit. I show how experimental estimates of the impacts of changes in interest rates on savings and borrowing can be used to construct welfare-relevant price elasticities of demand.

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- Allcott, H. & Taubinsky, D. (2015). Evaluating behaviorally motivated policy: Experimental evidence from the lightbulb market. _American Economic Review_ , 105(8), 2501–38.

- Angelucci, M., Karlan, D., & Zinman, J. (2015). Microcredit impacts: Evidence from a randomized microcredit program placement experiment by compartamos banco. _American Economic Journal: Applied Economics_ , 7(1), 151–182.

- Attanasio, O., Augsburg, B., De Haas, R., Fitzsimons, E., & Harmgart, H. (2015). The impacts of microfinance: Evidence from joint-liability lending in mongolia. _American Economic Journal: Applied Economics_ , 7(1), 90–122.

- Auclert, A. (2019). Monetary policy and the redistribution channel. _American Economic Review_ , 109(6), 2333–67.

- Augsburg, B., De Haas, R., Harmgart, H., & Meghir, C. (2015). The impacts of microcredit: Evidence from bosnia and herzegovina. _American Economic Journal: Applied Economics_ , 7(1), 183–203.

- Banerjee, A., Breza, E., Duflo, E., & Kinnan, C. (2021). Can microfinance unlock a poverty trap for some entrepreneurs?

- Banerjee, A., Duflo, E., Glennerster, R., & Kinnan, C. (2015a). The miracle of microfinance? evidence from a randomized evaluation. _American economic journal: Applied economics_ , 7(1), 22–53.

- Banerjee, A., Duflo, E., & Hornbeck, R. (2018). How much do existing borrowers value microfinance? evidence from an experiment on bundling microcredit and insurance. _Economica_ , 85(340), 671–700.

- Banerjee, A., Karlan, D., & Zinman, J. (2015b). Six randomized evaluations of microcredit: Introduction and further steps. _American Economic Journal: Applied Economics_ , 7(1), 1–21.

- Banerjee, A. V. (2013). Microcredit under the microscope: what have we learned in the past two decades, and what do we need to know? _Annu. Rev. Econ._ , 5(1), 487–519.

- Baqaee, D. R. & Farhi, E. (2020). Productivity and misallocation in general equilibrium. _The Quarterly Journal of Economics_ , 135(1), 105–163.

- Bau, N. & Matray, A. (2020). _Misallocation and capital market integration: Evidence from India_ . Technical report, National Bureau of Economic Research.

40
- Benetton, M. (2021). Leverage regulation and market structure: A structural model of the uk mortgage market. _The Journal of Finance_ , 76(6), 2997–3053.

- Benetton, M., Gavazza, A., & Surico, P. (2021). Mortgage pricing and monetary policy.

- Berkouwer, S. B. & Dean, J. T. (2022). Credit, attention, and externalities in the adoption of energy efficient technologies by low-income households. _American Economic Review_ , 112(10), 3291–3330.

- Bhattacharya, D. (2015). Nonparametric welfare analysis for discrete choice. _Econometrica_ , 83(2), 617–649.

- Blackorby, C., Donaldson, D., & Moloney, D. (1984). Consumer’s surplus and welfare change in a simple dynamic model. _The Review of Economic Studies_ , 51(1), 171–176.

- Breza, E. & Kinnan, C. (2021). Measuring the equilibrium impacts of credit: Evidence from the indian microfinance crisis. _The Quarterly Journal of Economics_ , 136(3), 1447–1497.

- Bryan, G., Chowdhury, S., & Mobarak, A. M. (2014). Underinvestment in a profitable technology: The case of seasonal migration in bangladesh. _Econometrica_ , 82(5), 1671–1748.

- Buera, F. J., Kaboski, J. P., & Shin, Y. (2021). The macroeconomics of microfinance. _The Review of Economic Studies_ , 88(1), 126–161.

- Cai, J., De Janvry, A., & Sadoulet, E. (2020). Subsidy policies and insurance demand. _American Economic Review_ , 110(8), 2422–53.

- Chetty, R. (2009). Sufficient statistics for welfare analysis: A bridge between structural and reduced-form methods. _Annu. Rev. Econ._ , 1(1), 451–488.

- Chetty, R., Looney, A., & Kroft, K. (2009). Salience and taxation: Theory and evidence. _The American Economic Review_ , 99(4), 1145–1177.

- Crépon, B., Devoto, F., Duflo, E., & Parienté, W. (2015). Estimating the impact of microcredit on those who take it up: Evidence from a randomized experiment in morocco. _American Economic Journal: Applied Economics_ , 7(1), 123–50.

- DeFusco, A. A., Tang, H., & Yannelis, C. (2022). Measuring the welfare cost of asymmetric information in consumer credit markets. _Journal of Financial Economics_ , 146(3), 821–840.

- Duflo, E., Gale, W., Liebman, J., Orszag, P., & Saez, E. (2006). Saving incentives for low-and middle-income families: Evidence from a field experiment with h&r block. _The Quarterly Journal of Economics_ , 121(4), 1311–1346.

41
- Duflo, E., Gale, W., Liebman, J., Orszag, P., & Saez, E. (2007). Savings incentives for low-and moderate-income families in the united states: why is the saver’s credit not more effective? _Journal of the European Economic Association_ , 5(2-3), 647–661.

- Dupas, P., Karlan, D., Robinson, J., & Ubfal, D. (2018). Banking the unbanked? evidence from three countries. _American Economic Journal: Applied Economics_ , 10(2), 257–97.

- Einav, L., Finkelstein, A., & Cullen, M. R. (2010). Estimating welfare in insurance markets using variation in prices. _The quarterly journal of economics_ , 125(3), 877–921.

- Epstein, L. G. & Zin, S. E. (1989). Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. _Econometrica_ , 57(4), 937–969.

- Farhi, E. & Gabaix, X. (2020). Optimal taxation with behavioral agents. _American Economic Review_ , 110(1), 298–336.

- Farhi, E., Olivi, A., & Werning, I. (2022). _Price Theory for Incomplete Markets_ . Working Paper 30037, National Bureau of Economic Research.

- Feigenberg, B., Field, E., & Pande, R. (2013). The economic returns to social interaction: Experimental evidence from microfinance. _Review of Economic Studies_ , 80(4), 1459–1483.

- Finkelstein, A. & Hendren, N. (2020). Welfare analysis meets causal inference. _Journal of Economic Perspectives_ , 34(4), 146–167.

- Gelman, M., Gorodnichenko, Y., Kariv, S., Koustas, D., Shapiro, M. D., Silverman, D., & Tadelis, S. (2016). _The response of consumer spending to changes in gasoline prices_ . Technical report, National Bureau of Economic Research.

- Hausman, J. A. (1981). Exact consumer’s surplus and deadweight loss. _The American Economic Review_ , 71(4), 662–676.

- Hausman, J. A. & Newey, W. K. (2016). Individual heterogeneity and average welfare. _Econometrica_ , 84(3), 1225–1248.

- Hendren, N. (2020). Measuring economic efficiency using inverse-optimum weights. _Journal of public Economics_ , 187, 104198.

- International Monetary Fund (2006). Household credit growth in emerging market countries.

- Kaboski, J. P. & Townsend, R. M. (2011). A structural evaluation of a large-scale quasiexperimental microfinance initiative. _Econometrica_ , 79(5), 1357–1406.

42
- Kaboski, J. P. & Townsend, R. M. (2012). The impact of credit on village economies. _American Economic Journal: Applied Economics_ , 4(2), 98–133.

- Karlan, D., Osei, R., Osei-Akoto, I., & Udry, C. (2014). Agricultural decisions after relaxing credit and risk constraints. _The Quarterly Journal of Economics_ , 129(2), 597–652.

- Karlan, D. & Zinman, J. (2009). Observing unobservables: Identifying information asymmetries with a consumer credit field experiment. _Econometrica_ , 77(6), 1993–2008.

- Karlan, D. & Zinman, J. (2011). Microcredit in theory and practice: Using randomized credit scoring for impact evaluation. _Science_ , 332(6035), 1278–1284.

- Karlan, D. & Zinman, J. (2018). Price and control elasticities of demand for savings. _Journal of Development Economics_ , 130, 145–159.

- Karlan, D. & Zinman, J. (2019). Long-run price elasticities of demand for credit: evidence from a countrywide field experiment in mexico. _The Review of Economic Studies_ , 86(4), 1704–1746.

- Karlan, D. S. & Zinman, J. (2008). Credit elasticities in less-developed economies: Implications for microfinance. _American Economic Review_ , 98(3), 1040–68.

- Keen, M. (1990). Welfare analysis and intertemporal substitution. _Journal of Public Economics_ , 42(1), 47–66.

- Khandker, S. R. (2005). Microfinance and poverty: Evidence using panel data from bangladesh. _The world bank economic review_ , 19(2), 263–286.

- Lagakos, D., Mobarak, A. M., & Waugh, M. E. (2018). The welfare effects of encouraging rural-urban migration.

- Lane, G. (2020). Credit lines as insurance: Evidence from bangladesh.

- Lerva, B. (2022). The monetary value of externalities: Experimental evidence from ugandan farmers.

- McKenzie, D. (2012). Beyond baseline and follow-up: The case for more t in experiments. _Journal of development Economics_ , 99(2), 210–221.

- McKenzie, D. & Woodruff, C. (2008). Experimental evidence on returns to capital and access to finance in mexico. _The World Bank Economic Review_ , 22(3), 457–482.

43
- Meager, R. (2019). Understanding the average impact of microcredit expansions: A bayesian hierarchical analysis of seven randomized experiments. _American Economic Journal: Applied Economics_ , 11(1), 57–91.

- Milgrom, P. & Segal, I. (2002). Envelope theorems for arbitrary choice sets. _Econometrica_ , 70(2), 583–601.

- Morduch, J. (1999). The microfinance promise. _Journal of economic literature_ , 37(4), 1569–1614.

- Mullainathan, S., Schwartzstein, J., & Congdon, W. J. (2012). A reduced-form approach to behavioral public finance. _Annu. Rev. Econ._ , 4(1), 511–540.

- Pitt, M. M. & Khandker, S. R. (1998). The impact of group-based credit programs on poor households in bangladesh: Does the gender of participants matter? _Journal of political economy_ , 106(5), 958–996.

- Ponce, A., Seira, E., & Zamarripa, G. (2017). Borrowing on the wrong credit card? evidence from mexico. _American Economic Review_ , 107(4), 1335–61.

- Ralston, L., Andrews, C., & Hsiao, A. (2017). The impacts of safety nets in africa: what are we learning?

- Rigol, N. & Roth, B. N. (2021). _Loan Officers Impede Graduation from Microfinance: Strategic Disclosure in a Large Microfinance Institution_ . Working Paper 29427, National Bureau of Economic Research.

- Robles-Garcia, C. (2022). Competition and incentives in mortgage markets: The role of brokers.

- Rosenzweig, M. R. & Udry, C. (2020). External validity in a stochastic world: Evidence from low-income countries. _The Review of Economic Studies_ , 87(1), 343–381.

- Sallee, J. M. (2022). Pigou creates losers: On the implausibility of achieving pareto improvements from efficiency-enhancing policies.

- Sraer, D. & Thesmar, D. (2020). How to use natural experiments to estimate misallocation.

- Vives, X. (1987). Small income effects: A marshallian theory of consumer surplus and downward sloping demand. _The review of economic studies_ , 54(1), 87–103.

- Willig, R. D. (1976). Consumer’s surplus without apology. _The American Economic Review_ , 66(4), 589–597.

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