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Consumer Demand for Credit Card Services


We apply a demand-based approach to study consumer behavior in the credit card market. Using a national database of U.S. card accounts, we find consumers internalize both rates and fees when making purchasing, borrowing, and late payment decisions on their card. Moreover, price effects broadly align with a rational model of card use. An exception is less borrowing in response to declining late-fees among subprime consumers. Extension of the rational model based on “focusing theory” explains this behavior. It also implies substantial indirect benefits of the CARD Act’s late-fee cap from subprime users’ re-focusing toward reducing their debt.

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  1. The Supreme court decision upheld a US Office of the Comptroller of the Currency regulation and was in effect an extension of the Marquette v. First of Omaha decision that did the same for interest rates. In the a decade following the decision, fees grew rapidly from around $5-$10 to $39 for almost all large issuers (United States Government Accountability Office 2006).

  2. An example was lenders changing late fees ex-post without warning or reason, even when the initial contract specified that the issuer has the unilateral right to do so.

  3. Fees in 2009 seemed to be effectively unchanging for years, and not customized like interest rates had been for decades, despite massive changes in economic conditions.

  4. This reduced the proportion of accounts charged an over-limit fee at least once in a year from 12 percent to about 1 percent (Consumer Financial Protection Bureau 2013)

  5. The rule also allowed issuers to charge consumers a higher late fee if the consumers previously paid late within the last six months.

  6. The general purpose portfolio comprises the bulk of lenders’ business. There are also small-business, student, and private label cards (e.g. store cards) ). These are important markets in the own right. However, they are much smaller and distinct from the general purpose market. Their analysis falls outside the scope of this study.

  7. It is important to note that not all balances are the same. Cards frequently offer promotional rates for a limited period, which are much lower than the stated contract rate. Moreover, certain transactions, such as cash withdrawals, are assessed a higher interest. A unique feature of the CCDB is that these different types of balances are categorized separately. The analysis focuses on how usage responds to the regular rate, often referred to as the go-to-rate. As a result, we consider balances to which this rate applies. These are by far the most common balance type.

  8. Though not shown here, the distribution of interest rates was mostly stable during the period of analysis.

  9. Note these are not directly available to lenders either when pricing.

  10. Users may have more than one credit card at the same or different lenders. During our sample period, the typical consumer had between 2 and 2.5 cards in their wallet. The CCDB does not tie these together and therefore is not well suited to understanding how users choose among cards. An analysis of the user’s broader credit card portfolio problem is beyond the scope of this paper.

  11. The interest rate and late fee are fixed in the user’s problem. This precludes any possibility of users eliciting a change to their interest rate and or late fee through their usage choices. We address the endogeneity of prices due to lender behavior empirically in Section 5.2.

  12. The timing here is such that the user is not charged interest on her purchases until they become debt, at the end of the month once her repayment choice is made. This timing approximates the grace period given to users when making purchases on their card.

  13. Note that \(\overline {K}_{ij}\) is not differentiable at around g = gmin due to the discontinuous reduction in costs of borrowing when approaching gmin from below. Kij is differentiable elsewhere. For the sake of simplicity in expressions we will use derivatives of this function whenever we consider a point ggmin. Note also that \(\overline {K}_{ij}\) also incorporates possible and costly future default.

  14. Note that we interpret broadly the value of making purchases and the costs of repayment/debt. This includes, for example, the user’s alternative payment environment. These factors may be fixed, affecting the shapes of \(u_{ij}, k_{ij},\overline {K}_{ij}\), or idiosyncratic, in which case they are captured by the unobserved state \(\eta ^{d}_{ijm}\). Moreover, we include in Xj card characteristics that do not change over time like rewards and annual fees.

  15. Given the late payment cutoff, gmin, which generates a discontinuity in the objective, a correct statement of the first-order condition requires a detailed set of cases. These are “summarized” here using the indicator in order to simplify the exposition and lend focus to the application at hand.

  16. Another way to write this is that \(V(g^{min},LF_{0}) < V(g^{INT},LF_{0}) \iff -{\varDelta } \overline {K}_{g^{INT}\rightarrow g^{min}, LF_{0}} < {\varDelta } k_{g^{INT} \rightarrow g^{min},LF_{0}}\). This is because \(V(g^{min},LF_{0}) < V(g^{INT},LF_{0}) \iff u - k(g^{min}) - \overline {K}(g^{min},LF_{0}) < u - k(g^{INT}) - \overline {K}(g^{INT},LF_{0}) \iff -(\overline {K}(g^{min},LF_{0})-\overline {K}(g^{INT},LF_{0})) < k(g^{min})-k(g^{INT})\). While a rise in LF increases the benefits of paying at least the minimum due, it does not change the cost of moving from gINT to gmin.

  17. In the figure, marginal values are drawn as linear. This is done only for clarity of exposition and does not reflect any additional restriction in the model.

  18. Note here we assume that gmin and LF do not depend on r. For individuals with sufficiently high month end balance, the minimum payment is calculated as 1% of the balance due, which can be a function of r. Also, their late payment fee can be higher for exceptionally high balances. In such cases the effect can in fact be ambiguous as the cost of repayment can be higher in this case. However, this applies to a very small number of cases, whereby this latter cost of repayment effect does not seem to dominate in the data.

  19. Other characteristics of the card, such as rewards and other bundled products alter the value a cardholder receives from using their card and thus form part of their state XjΩijm. As detailed below, we control for these aspects in large part via account fixed effects.

  20. During our period of analysis, approximately 13 percent of active accounts had an over-the-limit instance in any particular quarter (Consumer Financial Protection Bureau 2013). Given that about 60 percent of accounts are active in any given month, this implies a 13% × 60% × (1/3) ≈ 2.6 percent over the limit incidence.

  21. This bias is exacerbated in short panels, small T, with a large number of fixed effects, large N. It arises from sampling error in the estimation of the fixed effects carrying over to the estimates of the parameters of interest, whereby the resulting asymptotic distribution of the estimator is no longer centered around 0, even if T grows at the same rate as N.

  22. Simpler “automatic” forms of bias reduction, such as jackknife methods, are not suitable to our application because late fees change only once during the sample period. As a result, we appeal to the shape of the objective function and use an analytic reduction technique. Nevertheless, this solution is more generally applicable to other situations meriting fixed effects, which often may be hampered by limited variation in at least one explanatory variable.

  23. An account’s interest can change if the account is in arrears. This is referred to as penalty repricing. Note that the estimation sample focuses on accounts that are not delinquent, eliminating this possibility (Section 3).

  24. Note that credit scores are not included as a state in the model, and as such are not controlled for in the empirical analysis. This is because users have more information about their future default than is included in the credit score. From an empirical perspective, including this score as an explanatory variable is inconsistent with the model and generates additional endogeneity concerns.

  25. Specifically, we define a group as containing accounts for all lenders within a 10 credit score point band as of the beginning of the sample, and that are originated within the same 2 year period. Our results are robust to variations of this definition.

  26. This logic follows a similar argument used for demand estimation in product markets. The idea is that variation in the price of other products, which are driven by cost shifts and excluded from a consumers’ utility, are correlated via firms’ optimal pricing. Functions of “other” products are frequently used as instruments to identify price effects as distinct from unobserved product characteristics entering utility (Berry 1994; Berry et al. 1995; Gandhi and Houde 2019). Existing work identifying price response in credit card exploit the timing of repricing (Gross and Souleles 2002) or a one time portfolio level change in the interest rate (Nelson 2017). All of these studies use data from a single lender to identify these effects. Moreover, the latter relies on a one time rise for all accounts at once, rather than a richer set of price movements over time and over lenders. Complementing these approaches, the current setting leverages both upward and downward within account price changes over all lenders and at different points in time based on movements in the cost of funding those accounts.

  27. We use Michael Stepner’s binscatter function. See

  28. We have considered that other factors may be driving our observed price responses. These can include liquidity and/or other budget constraints, as-well-as unobserved components of income, spending, and borrowing. First, we note that the liquidity constraints story is unlikely simply because all of the users in our sample have excess liquidity on their credit card - by definition. At times this is substantial liquidity - even in the subprime sample. Moreover, though not explicitly modeled, broader balance sheet constraints, relating to income, spending, and/or other borrowing, are in fact incorporated into model through the shapes of the value from purchasing (uij) and cost of repayment/debt (\(k_{ij} / \overline {K}_{ij}\)) curves. Since we do not observe the users entire balance sheet, we of course cannot rule those factors out definitively. However, we were unable to come up with a clear mechanism whereby these factors systematically drive our observed price response over such a large population of users.

  29. As noted in Hacker et al. (2014), “ESI measures the proportion of individuals who lose at least 25 percent of their available household income, due to either changes in income or changes in out-of-pocket medical spending, and who lack sufficient liquid financial wealth to fully cushion the loss.” We exclude cardholders in the six states singled out to be most affected: Florida, Georgia, Alabama, Mississippi, Arkansas, and California.

  30. Keys and Wang (2016) documents a mass of repayment at the minimum amount due. This is especially so among subprime users. Their analysis similarly attributes this fact to consumer inattention, though in a different form.

  31. Further, note that the extended model also lends scope to potential rise in purchasing as a response to lower late fees. Though much less pronounced than for debt outstanding, we do find some evidence of this prediction in the data. See Fig. 4.

  32. Note also that

    $$ \lim\limits_{w \rightarrow 0}\frac{\partial^{2}b^{\ast}}{\partial (1+r)\partial w} = \beta \overline{K}_{i}^{\prime} \left( \frac{1}{u_{ij}^{\prime\prime}} - \frac{1}{k_{ij}^{\prime\prime}} \right) < 0. $$

    For small w, a re-focusing away from the late fee raises borrowing response.

  33. As additional evidence note that, absent greater focusing, the model suggests that subprime users in low income counties likely pay late more often as they more frequently struggle to make ends meet - e.g. they are more often close to the kink. However, in our subprime sample, cardholders in low income counties in fact pay late less often than those residing in high income (top quintile) counties - 7.4% vs. 7.7%.

  34. Note that non-borrowers may have held debt before and/or after the price change, but not during.

  35. Very few users reach their limit. Even among subprime users, average utilization is bellow 50 percent. This leaves potentially thousands of dollars in excess liquidity - which accounts for several weeks of wages as shown in the data.

  36. Since \(\overline {S}\) is annualized, and assuming \(\hat {P}\) is independent and fixed across months, we calculate the expected late fees paid by an account in a given year as: \(\kappa (\cdot ) = \displaystyle {\sum \limits _{m = 1}^{12}} {12 \choose m} \hat {P}^{m}(1-\hat {P})^{12-m} \cdot (m \cdot LF). \)

  37. Currently with this method we can estimate the adjusted Tobit with \(\sim 200,000\) fixed effects and 13 parameters in approximately 10 minutes. Using a commercially available optimization routine this would take several days of computation time.


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The views expressed in this paper are those of the authors and do not necessarily reflect those of the Consumer Financial Protection Bureau, Office of the Comptroller, U.S. Treasury Department, or the United States. We would like to thank Loretta Mester (Co-Editor), Haluk Unal (Chief Editor), and two anonymous referees for their help improving this work. We would also like to thank Ron Borzekowski, Paul Heidhues, Johannes Johnen, and David Silberman for thoughtful comments and suggestions. This paper has benefited immensely from participants and our discussants at the International Industrial Organization Conference, European Economic Association, European Association for Research in Industrial Economics, American Economic Association conference, and Boulder Summer Conference on Consumer Financial Decision Making and at the Consumer Financial Protection Bureau’s, European School of Management and Technology’s, the Pennsylvania State University’s, and Boston Federal Reserve Bank’s internal seminars. The authors are solely responsible for any remaining errors. Declarations of interest: The authors have no relevant financial or non-financial interests to disclose.

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Appendix A: Full Regression Results

Tables A.1A.2A.3, and A.4,show the full regression results. In each table, columns (1)-(5) report results separately for each quintile or starting credit score. For computational stability, balances, purchase volume, and wages are re-scaled to $1000s and LF and r are measured in 10 percentage points (r = 14.99% → 1.499) or dollar (LF = $39 → 3.9) units, respectively. Account age is measured in years and the trend is the number of years from September 2010. In other words, it takes value \(\frac {1}{12}\) in October 2010 and value \(-\frac {1}{12}\) in August 2010. Bootstrapped standard errors, in parentheses, are at the account level and reported based on 500 draws.

Table A.1 First Stage Regressions
Table A.2 Second Stage Regressions (Payments)
Table A.3 Second Stage Regressions (Borrowing)
Table A.4 Second Stage Regressions (Late)

Table A.1 reports first stage results.

Table A.2 reports the full (second stage) results for the payments equation.

Table A.3 reports the full (second stage) results for the borrowing equation.

Table A.4 reports the full (second stage) results for the late payment equation. Note that, since it is a binary equation, the second moment of the normal distribution, σ, is normalize to 1.

Appendix B: Pass through of treasury yields to interest rates

In this section, I provide evidence on the pass through of the cost of funds to the interest rate on accounts. As has been shown in previous work, the cost of funding accounts is closely tied to the 1-Year T-Bill rate (Ausubel 1991). As I show below, lender decisions to change existing accounts’ interest rates are associated with movements in the 1-Year T-Bill rate. Specifically, we estimate:

$$ {\varDelta} rate_{jm} = \alpha_{0} + \alpha_{1} {\varDelta}_{-1} T-Bill_{m} + \epsilon_{jm}, $$

where alpha gives the percentage point pass through of the T-Bill rate to the individual account’s interest rate.

Table B.1 shows pass through regressions relating the change in an account’s interest rate with a past change in the Treasury yield. Column (1) shows the average within group effect, whereby the group is defined as in Section 5.2. Columns (2)-(6) shows these effects separately by credit score quintile. Standard errors are clustered at the account level.

Table B.1 Pass Through of the Cost of the 1-Year T-Bill Rate

As can be seen in the table, lenders pass through between 1 and 8 percent of the change in the 1-Year T-Bill rate to the price of existing accounts, about 6 percent on average. This is a monthly change, and does not include any effects on the interest rate from new accounts - as was shown in Ausubel (1991). While substantial, note that lenders are heterogeneous in their funding strategies, whereby not all would respond to a change in this rate. Moreover, funding heterogeneity is likely larger in this period as compared to that described in Ausubel (1991). This may then under-count the pass-through of lenders costs to the price on existing accounts. Nevertheless, it provides evidence in favor of this view.

Appendix C: Model example with quadratic utility

C.1 Baseline model

In this section, we provide an example of the model with a quadratic specification. This exposition gives a clearer illustration of the model’s mechanics by giving with closed form solutions for purchases (\(c^{\ast }_{ijm}\)), the repayment amount (\(g^{\ast }_{ijm}\)), and late payment, \(\ell ^{\ast }_{ijm}\). Suppose that

$$ \begin{array}{@{}rcl@{}} && u_{i}(c_{ijm}) = \alpha_{i} \cdot c_{ijm}-\frac{\gamma_{i}}{2} \cdot c_{ijm}^{2} \\ && k_{i}(g_{ijm}) = \theta_{i} \cdot g_{ijm} +\frac{\epsilon_{i}}{2} \cdot g_{ijm}^{2} \\ && \overline{K}_{i}[b_{ijm}] = \mathbb{E} \left[\mu_{i} (b_{ijm} + \mathbbm{1}[\cdot] LF) + \frac{\rho_{i}}{2}(b_{ijm} + \mathbbm{1}[ \cdot ] LF)^{2} |{\varOmega}_{ijm}\right], \end{array} $$
$$s.t. \alpha_{i},\gamma_{i}, \theta_{i}, \epsilon_{i}, \mu_{i}, \rho_{i} >0. $$

In addition, αi > γi, purchase utility shows diminishing returns, and αi > 𝜃i, the user prefers to make positive purchases on the card. The parameters also depend on the state, \({\varOmega }_{ijm}= \{b_{0}, X_{j}, y_{im},\eta ^{c}_{ijm},\eta ^{g}_{ijm}\}\), where b0 is the initial debt, Xj captures card js attributes, yim captures user is characteristics, \(\eta ^{c}_{ijm}\) is an idiosyncratic purchase shock, and \(\eta ^{g}_{ijm}\) is an idiosyncratic repayment shock.

In optimality, we have that

$$ \begin{array}{@{}rcl@{}} {}\alpha_{i} - \gamma_{i}c^{\ast}_{ijm} \!&=&\! \theta_{i}+\epsilon_{i}g^{\ast}_{ijm} \\ {}\theta_{i} +\epsilon_{i}g^{\ast}_{ijm} \!&=&\! \beta_{i}(1 + r)E\left[\mu_{i} + \rho_{i} \left( (b_{0}+c^{\ast}_{ijm} - g^{\ast}_{ijm})(1+r)+\mathbbm{1}(\cdot)LF\right) |{\varOmega}_{ijm}\right] \end{array} $$

Note from Eq. C.2, if the utility gains from credit card purchases decline slower in the purchase amount, e.g., when the user generates exceptional value from making purchases on their card, then trade-off in purchase, repayment trends towards purchases. The amount of unpaid balances may rise or fall dependent on whether (1) repayment today is costly, and/or (2) the future cost of debt is high. Alternatively, when a user’s marginal cost of future debt increases, ρi , or when βi , the user purchases less in equilibrium and repays more of her debt. Put together, this illustrates that purchases decline when the rate increases, and so does the the amount of debt held on the card.

Whether the user pays late or not depends on her equilibrium repayment amount without late payment (\(\mathbbm {1}(\cdot )=0\)) and how this compares to the minimum payment amount gmin. The probability of paying late for the account can then be calculated as:

$$ Pr(Late)_{ijm} = \Pr (\beta_{i}(1+r){\varDelta}\overline{K}_{i}|_{g^{min}}<\theta_{i}+\epsilon_{i}.g^{min}) $$


$$ {\varDelta}\overline{K}_{i}|_{g^{min}}=LF \times \mathbb{E}\left[\mu_{i}+\rho_{i}\left[\left( b_{0} + c^{\ast}_{ijm} - g^{min}\right)(1+r)+\frac{LF}{2}\right] | {\varOmega}_{ijm}\right]. $$

When LF decreases, the probability of paying late increases for a given account. Conditional on paying late, the amount of credit card purchases and the amount of unpaid balances both decrease in the level of late payment fee. Thus, the model predicts a higher late payment propensity, a higher volume of purchasing, and greater debt levels when the late fee declines.

C.2 Extended model: Focusing

We also extend the quadratic example to illustrate how the focusing theory prediction could be obtained if we change the utility function of an account. Suppose that there is a positive weight, 0 ≤ wi ≤ 1, in front of the interest rate costs of future debt The utility function of the account becomes:

$$ \begin{array}{@{}rcl@{}} && {}\alpha_{i} - \gamma_{i}c^{\ast}_{ijm} = \theta_{i}+\epsilon_{i}g^{\ast}_{ijm} \\ &&{} \theta_{i} +\epsilon_{i}g^{\ast}_{ijm} = \beta_{i}(1+r)E\left[\mu_{i}+\rho_{i} \left( w_{i}(b_{0}+c^{\ast}_{ijm} - g^{\ast}_{ijm})(1+r) + \mathbbm{1}(\cdot)LF\right) |{\varOmega}_{ijm}\right] \end{array} $$

Consider a decline in LF. Instead of constituting a shift in the right hand side of the optimality condition, like for the baseline case, it represents a shift in the c/g relationship. The reduction in LF increases wi and so leads the user to internalize a larger fraction of the cost of her debt. For users borrowing more, the latter effect should dominate the former. Further, different levels of credit card borrowing are affected differently by a decline in LF. A user responds more to changes in rate and this increase in response becomes larger when she carries more credit card debt. Aggregating across users, we may find an lower unpaid balances (debt) in response to reduced LF. Moreover, this reaction is more pronounced among those with a high level of unpaid balances.

Appendix D: Robustness

In this section we provide more details of the robustness checks for our analysis. One important worry is that much of our results are colored by the effects of the Great Recession, which is at its height during our period of analysis. To assuage these concerns we identify states highly exposed to the crisis using the Economic Security Index of (Hacker et al. 2012; Hacker et al. 2014). Specifically, we compare our results in the full sample to those in which we exclude areas highly impacted by the crisis. These results are shown in Table D.1.

Table D.1 Comparison to Low Exposure States

Columns (1) and (2) of Table D.1 report results on rate and late fee responses, respectively, for the full sample. Columns (3) and (4) report analogous results for the low exposure sample. As is evident in the table, our results are remarkably resilient to this pruning. This holds across the entire credit score distribution and for all three of payments, borrowing, and late payment. While there may be effects of the Great Recession no accounted for in our analysis, the results of this check indicates that they are not obviously first order to our results.

Another concern is the impact of our sample selection. Namely, our exclusion of accounts moving into non-repayment (> 60 days overdue) and new accounts created during our period of analysis. While we cannot directly add these into our framework, we instead re-create our summary statistics using the full sample to get a better understanding of how these two sample differ based on observables used in the estimation.

Table D.2 shows our summary statistics for the full population of general purpose cards in the CCDB. As compared to our estimation sample, the table shows that the broader population consists of accounts that are a bit older, with slightly higher credit scores, and that face lower interest rates. As might be expected, these differences are most apparent at the lower end of the credit score distribution. One particularly apparent difference is the higher propensity of late payment among those at the bottom quintile of the credit score distribution, 5.55 vs. 9.72 percent. Nevertheless, the broad patterns of usage established in Table 1 remain largely unchanged when including all accounts in the CCDB’s general purpose portfolio.

Table D.2 Summary Statistics for Full Population

Appendix E: Estimation details

In this section we detail the procedure described in Section 5 for estimating Eqs. 910, and 11. Equations 910 correspond to the purchase and balance equations for B ∈{U,C}, respectively, and are a type I Tobit. Equation 11, corresponding to the late payment equation L, is a binomial Probit. For each equation, we first posit its likelihood function and respective Score (S) and Hessian (H). We then derive the analytical bias adjustment term (Hahn and Newey 2004; Arellano and Hahn 2007). Finally, we describe the iterative Efficient Newton-Raphson (ENR) procedure used for estimation (Hospido 2012) and relevant standard error calculations.

E.1 Likelihood equations

The Tobit likelihood, corresponding to the purchase volume and revolving balance Eqs. 910, respectively, takes the following form:

$$ \mathscr{L}^{B}_{it} = c_{it} \cdot \left[\!-ln(\sqrt{2\pi}) - ln(\sigma) - \frac{(y - x\theta - \alpha_{i})^{2}}{2\sigma^{2}}\! \right] + (1 - c_{it}) \cdot ln\left[{\varPhi} \left( \frac{lb - x\theta - \alpha_{i}}{\sigma}\right)\right] $$

where cit = 𝟙 (yit > lb) and lb is the censored lower bound, in this case lb = 0. Moreover, 𝜃 is a (J − 1) × 1 vector of parameters and αi (i = 1,...,N) and σ are scalars. As shown in Olsen (1978), the above likelihood is not globally concave in the parameter space. To ensure global concavity, we transform \({\mathscr{L}}^{B}_{it}\) as follows

$$ \begin{array}{@{}rcl@{}} \mathscr{L}^{B,O}_{it} &=& c_{it} \cdot \left[-ln(\sqrt{2\pi})+ln({\varDelta}) - \frac{1}{2}({\varDelta} y-x\beta -\eta_{i})^{2}) \right] \\&&+ (1-c_{it}) \cdot ln\left[{\varPhi}(0-x\beta-\eta_{i})\right] \end{array} $$

where \(\beta = \frac {\theta }{\sigma }\), \(\eta _{i} = \frac {\alpha _{i}}{\sigma }\), and \({\varDelta } = \frac {1}{\sigma }\). Then we can write the Score (SB,O) and Hessian (HB,O) of the likelihood as (J + N) × 1 and (J + N) × (J + N) matrices, respectively with

$$ S^{B, O} = \left[\begin{array}{c} \frac{\partial \mathscr{L}^{B, O}}{\partial {\upbeta}_{j}} \\\\ \frac{\partial \mathscr{L}^{B, O}}{\partial {\varDelta}} \\\\ \frac{\partial \mathscr{L}^{B, O}}{\partial \eta_{i}} \end{array}\right] = \left[\begin{array}{c} {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} x^{j}_{it} \cdot \left[c_{it} \cdot (y-x\upbeta-\eta_{i}) - (1-c_{it}) \cdot {\varLambda}(\varphi_{it})\right] \\\\ {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} c_{it} \cdot \left[\frac{1}{{\varDelta}} - y_{it}({\varDelta} y_{it} - x_{it}\upbeta - \eta_{i})\right] \\\\ {\sum\limits_{t=1}^{T}} \cdot \left[c_{it} \cdot (y-x\upbeta-\eta_{i}) - (1-c_{it}) \cdot {\varLambda}(\varphi_{it})\right] \end{array}\right] $$
$$ H^{B,O} =\left[ \begin{array}{ccc} \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\upbeta}_{j}\partial\beta_{k}} & \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial \beta_{j}\partial{\varDelta}} & \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial \beta_{j}\partial\eta_{i}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\varDelta}\partial\beta_{j}} & \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\varDelta}^{2}} & \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\varDelta} \partial \eta_{i}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial \eta_{i}\partial\beta_{j}} & \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial \eta_{i} \partial {\varDelta} } & \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial \eta_{i} \partial\eta_{j}} \end{array}\right], $$


$$ \left[\begin{array}{c} \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\upbeta}_{j}\partial{\upbeta}_{k}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\upbeta}_{j}\partial{\varDelta}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\upbeta}_{j}\partial\eta_{i}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\varDelta}^{2}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial {\varDelta} \partial \eta_{i}} \\\\ \frac{\partial^{2}\mathscr{L}^{B,O}}{\partial \eta_{i} \partial\eta_{j}} \end{array}\right] = \left[\begin{array}{c} {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} x^{j}_{it} x^{k}_{it} \cdot \left[-c_{it} - (1-c_{it}) \cdot {\varLambda}(\varphi_{it}) \cdot (\varphi_{it} + {\varLambda}(\varphi_{it}))\right]\\\\ {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} c_{it} \cdot x^{j}_{it} \cdot y_{it} \\\\ {\sum\limits_{t=1}^{T}} x^{j}_{it} \left[ -c_{it} - (1-c_{it}) \cdot {\varLambda}(\varphi_{it}) \cdot (\varphi_{it}+ {\varLambda}(\varphi_{it})) \right]\\\\ {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} - c_{it} \cdot \left( \frac{1}{{\varDelta}^{2}} + y^{2}_{it} \right) \\\\ {\sum\limits_{t=1}^{T}} c_{it} \cdot y_{it} \\\\ \mathcal{1}(i=j) \cdot {\sum\limits_{t=1}^{T}} \left[-c_{it} - (1-c_{it}) \cdot {\varLambda}(\varphi_{it}) \cdot (\varphi_{it} + {\varLambda}(\varphi_{it}))\right] \end{array}\right], $$

where φit ≡−(xitβ + ηi) and \({\varLambda }(x) \equiv \frac {\phi (x)}{\Phi (x)}\).

The Probit likelihood, corresponding to the late payment equation, takes the following form

$$ \mathscr{L}_{it}^{L} = d_{it} \cdot ln\left( {\varPhi}(x\theta+\alpha_{i})\right) + (1-d_{it}) \cdot ln\left( 1-{\varPhi}(x\theta+\alpha_{i})\right) , $$

where, because it is not identified, the variance of the Φ is normalized to σ = 1 and Ψitxit𝜃 + αi. The Score (SL) is given by

$$ S^{L} =\left[ \begin{array}{c} \frac{\partial \mathscr{L}^{L}}{\partial \theta_{j}} \\\\ \frac{\partial \mathscr{L}^{L}}{\partial \alpha_{i}} \end{array}\right] =\left[ \begin{array}{c} {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} x^{j}_{it} (d_{it} \cdot {\varLambda}({\varPsi}_{it})- (1-d_{it}) \cdot {\varLambda}(-{\varPsi}_{it})) \\\\ {\sum\limits_{t=1}^{T}} \left[d_{it} \cdot {\varLambda}({\varPsi}_{it})- (1-d_{it}) \cdot {\varLambda}(-{\varPsi}_{it})\right] \end{array}\right], $$

where, as above, \({\varLambda }(x) \equiv \frac {\phi (x)}{\Phi (x)}\). We write the Hessian (HL) as a (K + N) × (K + N) symmetric matrix with the following form

$$ H^{L} =\left[ \begin{array}{cc} \frac{\partial^{2}\mathscr{L}^{L}}{\partial \theta_{j}\partial\theta_{k}} & \frac{\partial^{2}\mathscr{L}^{L}}{\partial \theta_{j}\partial\alpha_{i}} \\ \frac{\partial^{2}\mathscr{L}^{L}}{\partial \alpha_{i}\partial\theta_{k}} & \frac{\partial^{2}\mathscr{L}^{L}}{\partial \alpha_{i} \partial\alpha_{j}} \end{array}\right] =\left[ \begin{array}{cc} H^{L}_{\theta\theta} & H^{L}_{\theta\alpha} \\ H^{L}_{\alpha\theta} & H^{L}_{\alpha\alpha}. \end{array}\right]. $$

Then let

$$ {\varGamma}_{it} = \left[ d_{it}\cdot {\varLambda}({\varPsi}_{it}) \cdot (-{\varPsi}_{it} - {\varLambda}({\varPsi}_{it})) - (1 - d_{it})\cdot {\varLambda}(-{\varPsi}_{it}) \cdot (-{\varPsi}_{it}+{\varLambda}(-{\varPsi}_{it}))\right], $$

and write

$$ \left[ \begin{array}{c} \frac{\partial^{2}\mathscr{L}^{L}}{\partial \theta_{j}\partial\theta_{k}} \\\\ \frac{\partial^{2}\mathscr{L}^{L}}{\partial \theta_{j}\partial\alpha_{i}} \\\\ \frac{\partial^{2}\mathscr{L}^{L}}{\partial \alpha_{i} \partial\alpha_{j}} \end{array}\right] =\left[ \begin{array}{c} {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} x^{j}_{it} x^{k}_{it} \cdot {\varGamma}_{it} \\\\ {\sum\limits_{t=1}^{T}} x^{j}_{it} \cdot {\varGamma}_{it} \\\\ \mathcal{1}(i=j) \cdot {\sum\limits_{t=1}^{T}} {\varGamma}_{it} \end{array}\right] $$

These above equations thus characterize the un-adjusted likelihoods of interest and their respective Score and Hessian expressions.

E.2 Analytically adjusted likelihood

To correct for the asymptotic bias resulting from small T and large N, the incidental parameters problem, we use an analytically derived bias correction of the concentrated likelihood as described in Hahn and Newey (2004) and Arellano and Hahn (2007). Although there are other simpler methods of bias reduction, such as automatic jackknife methods, we use the analytical method because we have only one change in the price of the late fees. As a result, we must rely more on the structure of the likelihood in our identification. Following Arellano and Hahn (2007), the asymptotic bias term of the concentrated likelihood is given by

$$ b_{i}(\theta) = \frac{1}{2} \cdot H^{-1}_{(\alpha) i}(\theta) \cdot {\Upsilon}_{(\alpha) i}(\theta) = \frac{1}{2} \cdot \mathbb{E}_{i}\left[ - \frac{\partial^{2}\mathscr{L}(\theta,\alpha(\theta))}{\partial \alpha^{2}} \right]^{-1} \cdot \mathbb{E}_{i}\left[\frac{\partial \mathscr{L}(\theta,\alpha(\theta))^{2}}{\partial \alpha}\right] $$

Then, following Hospido (2012), we reformulate the above expression in terms of the original, un-concentrated likelihood, as an input into the estimation. Given this approach, we write down the following estimator of the bias term

$$ \hat{b}_{i}(\theta,\alpha) = - \frac{1}{2} \cdot \left[{{\sum}_{t=1}^{T}} \frac{\partial^{2}\mathscr{L}_{it}(\theta,\alpha)}{\partial \alpha^{2}} \right]^{-1} \cdot \left[{{\sum}_{t=1}^{T}} \frac{\partial \mathscr{L}_{it}(\theta,\alpha)}{\partial \alpha}^{2} \right] $$

We then subtract the bias from the original likelihood to arrive at the adjusted likelihood

$$ \mathscr{L}^{A}(\theta,\alpha) = {\sum\limits_{i=1}^{N}\sum\limits_{t=1}^{T}} \mathscr{L}_{it}(\theta,\alpha) - {\sum\limits_{t=1}^{T}} b_{i}(\theta,\alpha) $$

From above, it follows that SA = SSb and HA = HHb where Sb and Hb are the Score and Hessian of the bias term, respectively. It follows that we can write

$$ S^{b} = {\sum\limits_{i=1}^{N}}\left[ \begin{array}{c} \frac{\partial b_{i}}{\partial \theta_{j}}\\\\ \frac{\partial b_{i}}{\partial \alpha_{i}} \end{array}\right] = - \frac{1}{2} \cdot {\sum\limits_{i=1}^{N}} \left[ \begin{array}{c} H^{-1}_{(\alpha) i} \cdot \left( \frac{\partial {\Upsilon}_{(\alpha) i}}{\partial \theta} - b_{i} \cdot \frac{\partial H_{(\alpha) i}}{\partial \theta}\right)\\\\ H^{-1}_{(\alpha) i} \cdot \left( \frac{\partial {\Upsilon}_{(\alpha) i}}{\partial \alpha} - b_{i} \cdot \frac{\partial H_{(\alpha) i}}{\partial \alpha}\right) \end{array}\right] $$


$$ H^{b}= -\frac{1}{2} {\sum\limits_{i=1}^{N}} \left[ \begin{array}{cc} \frac{\partial^{2}b_{i}}{\partial \theta_{j}\partial\theta_{k}} & \frac{\partial^{2}b_{i}}{\partial \theta_{j}\partial\alpha_{i}} \\\\ \frac{\partial^{2}b_{i}}{\partial \alpha_{i}\partial\theta_{j}} & \frac{\partial^{2}b_{i}}{\partial \alpha_{i}\partial\alpha_{j}} \end{array}\right] $$


$$ \left[ \begin{array}{c} \frac{\partial^{2}b_{i}}{\partial \theta_{j}\partial\theta_{k}} \\\\ \frac{\partial^{2}b_{i}}{\partial \theta_{j}\partial\alpha_{i}} \\\\ \frac{\partial^{2}b_{i}}{\partial \alpha_{i}\partial\alpha_{j}} \end{array}\right] = \left[ \begin{array}{c} H^{-1}_{(\alpha) i} \left[\left( \frac{\partial^{2}{\Upsilon}_{(\alpha) i}}{\partial \theta_{j}\partial\theta_{k}} - b_{i} \frac{\partial^{2}H_{(\alpha) i}}{\partial \theta_{j}\partial\theta_{k}}\right) - \left( \frac{\partial b_{i}}{\partial \theta_{j}} \cdot \frac{\partial H_{(\alpha) i}}{\partial \theta_{k}} + \frac{\partial b_{i}}{\partial \theta_{k}} \cdot \frac{\partial H_{(\alpha) i}}{\partial \theta_{j}}\right)\right] \\\\ H^{-1}_{(\alpha) i} \left[\left( \frac{\partial^{2}{\Upsilon}_{(\alpha) i}}{\partial \theta_{j}\partial\alpha_{i}} - b_{i} \frac{\partial^{2}H_{(\alpha) i}}{\partial \theta_{j}\partial\alpha_{i}}\right) - \left( \frac{\partial b_{i}}{\partial \theta_{j}} \cdot \frac{\partial H_{(\alpha) i}}{\partial \alpha_{i}} + \frac{\partial b_{i}}{\partial \alpha_{i}} \cdot \frac{\partial H_{(\alpha) i}}{\partial \theta_{j}}\right)\right] \\\\ \mathcal{1}(i = j) \cdot H^{-1}_{(\alpha) i} \left[\left( \frac{\partial^{2}{\Upsilon}_{(\alpha) i}}{\partial \alpha_{i}\partial\alpha_{j}} - b_{i} \frac{\partial^{2}H_{(\alpha) i}}{\partial \alpha_{j}\partial\alpha_{i}}\right) - \left( \frac{\partial b_{i}}{\partial \alpha_{j}} \cdot \frac{\partial H_{(\alpha) i}}{\partial \alpha_{i}} + \frac{\partial b_{i}}{\partial \alpha_{i}} \cdot \frac{\partial H_{(\alpha) i}}{\partial \alpha_{j}}\right)\right] \end{array}\right] $$

This completes the characterization the adjusted likelihood \({\mathscr{L}}^{A}(\theta ,\alpha )\) for the Probit and Tobit likelihoods.

E. 3 Efficient Newton-Raphson (ENR)

Given the smoothness and convexity of our objective functions, we estimate the model parameters using an efficient Newton-Raphson (ENR) algorithm laid out in Hospido (2012). This method exploits the block structure of the Hessian matrix of the log likelihood function and provides significantly increased estimation speed.Footnote 37 In what follows, denote Θ = (𝜃,α). The Kth step of the ENR algorithm is

$$ {\Theta}_{[K]}= {\Theta}_{[K-1]} - \left[ \frac{\partial^{2}\mathscr{L}({\Theta}_{[K-1]})}{\partial {\Theta}\partial{\Theta}^{\prime}}\right]^{-1} \cdot \left[\frac{\partial \mathscr{L}({\Theta}_{[K-1]})}{\partial {\Theta}} \right] $$

Where the Score and Hessian in their block form can be expressed as follows

$$ S^{A} = \frac{\partial \mathscr{L}^{A}}{\partial {\varTheta}} =\left[ \begin{array}{c} \underbrace{S^{A}_{\theta}}_{J \times 1}\\\\ \underbrace{S^{A}_{\alpha}}_{N \times 1} \end{array}\right] \text{and} H^{A} = \frac{\partial^{2}\mathscr{L}^{A}}{\partial {\Theta}\partial{\Theta}^{\prime}} = \left[\begin{array}{cc} \underbrace{H_{\theta\theta }}_{J \times J} & \underbrace{H_{\theta\alpha }}_{J \times N} \\\\ \underbrace{H^{\prime}_{\theta \alpha}}_{N \times J} & \underbrace{H_{\alpha\alpha }}_{N \times N} \end{array}\right] $$

Given the block nature of HA, re-write the Kth step of the ENR algorithm in two parts as

$$ \begin{array}{@{}rcl@{}} & \theta_{[K]} = \theta_{[K-1]} - \left[H^{A}_{\theta\theta} - H^{A}_{\theta\alpha} \left( H^{A}_{\alpha\alpha}\right)^{-1} H^{A}_{\alpha\theta}\right]^{-1} \cdot \left[S^{A}_{\theta} - H^{A}_{\theta\alpha}\left( H^{A}_{\alpha\alpha}\right)^{-1}S^{A}_{\alpha}\right] \\ \\ & \alpha_{[K]} = \alpha_{[K-1]}-H^{-1}_{\alpha\alpha} \cdot \left[S^{A}_{\alpha}+H_{\alpha\theta}(\theta_{[K]}-\theta_{[K-1]})\right] \end{array} $$

Note that in the Tobit case, the parameter 𝜃 includes the shape parameter of the normal distribution, whereas in the Probit case it does not as the variance is normalized to 1.

E.4 Standard error calculations

The above estimator is consistent and asymptotically normal under our assumption of i.i.d errors. It follows that

$$ \sqrt{T}\left( \begin{array}{c} \hat{\theta} - \theta_{0} \\ \hat{\alpha}_{i} - \alpha_{i0} \end{array}\right) \xrightarrow{d} N\left( 0,\left( \begin{array}{cc} I_{\theta,\theta} & I_{\theta,\alpha_{i}} \\ I_{\alpha_{i},\theta} & I_{\alpha_{i},\alpha_{i}} \end{array}\right)^{-1}\right), $$
$$ \sqrt{T}(\hat{\theta} - \theta_{0})\xrightarrow{d} N\left( 0,\left( I_{\theta,\theta}-I_{\theta,\alpha_{i}}I^{-1}_{\alpha_{i},\alpha_{i}}I_{\alpha_{i},\theta}\right)^{-1}\right), $$


$$ \left( \begin{array}{cc} I_{\theta,\theta} & I_{\theta,\alpha_{i}} \\ I_{\alpha_{i},\theta} & I_{\alpha_{i},\alpha_{i}} \end{array}\right) = \left( \begin{array}{cc} -\mathbb{E}\left( \frac{\partial^{2}\mathscr{L}}{\partial (\theta)^{2}}\right) & -\mathbb{E}\left( \frac{\partial^{2}\mathscr{L}}{\partial \theta\partial \alpha_{i}}\right) \\ -\mathbb{E}\left( \frac{\partial^{2}\mathscr{L}}{\partial \alpha_{i}\partial\theta}\right) & -\mathbb{E}\left( \frac{\partial^{2}\mathscr{L}}{\partial (\alpha_{i})^{2}}\right) \end{array}\right). $$

We use sample means as consistent estimators. We can then recover confidence intervals of the original parameters using the delta method. Nevertheless, we prefer not to assume i.i.d errors. Moreover, we use the control function approach of Smith and Blundell (1986), which gives an estimated first stage parameter. To account for this additional variation, we instead calculate standard errors using a block bootstrap, where we draw (with replacement) a sample of N accounts. This also makes it possible to generate standard errors for more complex functions of the parameters, such as the average price response.

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Grodzicki, D., Alexandrov, A., Bedre-Defolie, Ö. et al. Consumer Demand for Credit Card Services. J Financ Serv Res (2022).

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  • Financial decision making
  • Credit cards

JEL Classification

  • D12
  • D90
  • G50