## wpiea2023121-print-pdf - Section 3 derives the equilibria wherein the data monopolist sells the data to lenders and Section ## Source details **Canonical URL:** [wpiea2023121-print-pdf - Section 3 derives the equilibria wherein the data monopolist sells the data to lenders and Section](https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023121-print-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2023/english/wpiea2023121-print-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2023/english/wpiea2023121-print-pdf.pdf.json) --- ### Paper structure and focus - Section 3 derives the equilibria wherein the data monopolist sells the data to lenders. - Section 4 shows why the data monopolist chooses to sell data rather than engage in credit provision itself. - Section 5 analyzes welfare and socially optimal policy design. - Section 6 extends the baseline model to lender ownership of the data monopolist. - Proofs are in Appendix A; additional model extensions are in Appendix B (extensions B.1–B.8 referenced). ### Model overview (Section 2) - Agents: consumer-entrepreneurs ("households"), a monopolist digital currency issuer (DC issuer), and lenders; all agents are risk neutral. - Geography: N ≥ 3 connected islands; each island hosts one lender and a continuum of households with mass 1. - Lender definitions: a lender on the same island as a borrower is the borrower’s “home lender”; borrowers on the same island as a lender are the lender’s “on island” borrowers. - Payments: all islands share the same physical currency (cash) and the digital currency issuer. ### Households (Section 2.1) - Each household characteristics: creditworthiness q (private), preference for privacy ' (private), island location (public). - Project: yields payoff y if successful and 0 otherwise; success probability is q. - Privacy disutility: ' with > 0 and  the probability of revelation. - Distribution: on each island mass 1 of households uniformly on q ∈ [1/2,1] and ' ∈ [0,2]. - Payment choices: - Cash:  = 0 (fully protects privacy). - DC:  =  (revelation probability ). - Borrowing: limited liability; borrowing from off-island lender reduces successful project payoff by  > 0. - Household expected utility (type (q,')): - u(q;') = q max{y  I R; 0} ' - where I = 0 when borrowing from home lender and 1 otherwise;  =  if DC chosen and 0 if cash. - Baseline: privacy costs linear in probability of revelation; Appendix B.3 considers quadratic privacy costs. Baseline omits an ease-of-transactions benefit of DC; Appendix B.8 recalculates when including such a term. ### Digital currency issuer (Section 2.2) - Choice variables: intrusiveness/design parameter  ∈ [0,1]; data access fees ij charged to lender i for data on island j. - Lender demand for data on island j: Dij(; ij) ∈ {0,1}. - DC issuer objective: - max ∈[0,1], {ij} Σi Σj ij Dij(; ij) - Baseline: DC issuer sells data and does not itself engage in lending (Section 4 analyzes integration). ### Lenders (Section 2.3) - Funding: infinitely elastic at cost c ≥ 1; Bertrand competition for loans. - Differentiation: home lender advantage  > 0 and access to purchased data. - Lender i expected profit: - m i (E[q ki] R ki(q ki) c) +  i (E[q ju] R ui c) Σj ij Dij(; ij) - where m i mass of revealed borrowers choosing lender i, E[q ki] expected q among revealed borrowers, R ki individualized rates,  i mass of unrevealed borrowers choosing lender i, and E[q ju] expected q among unrevealed. - Value of purchasing data: enables price discrimination and attracting higher-q borrowers. ### Parameter conditions (Section 2.4) - Three conditions linking parameters to funding cost c: - y > c -  < 1 3 c - < 1 6 c - These ensure q = 1 borrower positive NPV, off-home payoff reduction  not too large, and households with q = 1 prefer to disclose. ### Timing (Section 2.5) - Stage 1: DC issuer chooses  and fees ij. - Stage 2: Households choose DC or cash and consume. - Stage 3: For DC users, creditworthiness revealed with probability . - Stage 4: Lenders decide on data purchases. - Stage 5: Lenders announce loan rates (individualized for revealed; pooled for unrevealed). - Stage 6: Households choose lender. - Stage 7: Project returns realize and repayment occurs. - Equilibrium solved by backward induction. ### Equilibrium partition by y (Table 1 summary) - Definition: "good" borrowers qy > c; "bad" borrowers qy < c. - Parameterization of share of bad borrowers and y ranges: - High y — Small share of bad borrowers — y > 2c +  - Medium y — Intermediate share of bad borrowers — y ∈ [4/3 c; 2c + ) - Low y — Large share of bad borrowers — y ∈ [c/ (??) ; 4/3 c] (source preserves notation: Low y — y 2 c; 4/3 c  — interpreted in text as y ∈ ( ? ; 4/3 c]; see propositions for exact bounds) - Section 3 provides propositions for these regimes; the Intermediate y case is derived in Appendix B.7. --- ### Equilibrium with few bad borrowers (Section 3.1) - Proposition 1 (when y > 2c + ): - DC issuer sets  = 1. - Charges each lender fee ii > 0 for on-island households (closed form in (28)). - Offers each lender free information about off-island households (ij = 0 when j ≠ i). - Households sort into DC use if: - ' ≤ c  q E[qju] 1 (equation (7)), with E[qju] closed-form in (26). - Home lenders buy on-island data and set: - R_k(q) = c/q +  (equation (8)) to revealed borrowers. - R_u = c E[qju] +  (equation (9)) to unrevealed borrowers. - All households borrow from home lenders. - Proof: Appendix A (p.33). - Households’ behavior (3.1.1): - If  = 0 (cash only): E[qju] = E[q] = 3/4; R_u = 4/3 c + . - Introducing DC: high-q households choose DC, lowering E[qju] and raising R_u; cascades of disclosure occur as lower E[qju] makes DC condition (7) hold for more households. - Welfare: households between dashed and unbroken lines in Figure 2 can be worse off than cash-only economy—privacy is a public good. - Bad borrowers self-select into cash because revelation precludes loans; when y > 2c +  enough good borrowers remain unrevealed to keep unrevealed loan market open. - Lenders’ strategies (3.1.2): - Bertrand competition with home-lender advantage  enables limit pricing and the rates in (8) and (9). - Purchasing data permits price discrimination and attracting higher-q borrowers. - DC issuer’s strategy (3.1.3): - Charges home lender a fee marginally below the lender’s total earnings on revealed households; closed-form fee in (28). - Offers off-island data free to maximize adverse selection on uninformed home lenders and extract value from on-island information. - Sets  = 1 to enlarge revealed set and induce disclosure cascades; @b̄/@ > 0. - Robustness: Appendix B.3 (quadratic privacy costs) — mass of DC users may decline with  but  = 1 remains optimal for DC issuer. --- ### Equilibrium with many bad borrowers (Section 3.2) - Proposition 2 (when y ∈ [c/?, 4/3 c] per source notation y2 c; 4/3 c ): - If  > y c, no credit provision and all households use cash. - If  < y c, households with q ≥ ' + c / (y ) choose DC and, if revealed, borrow from home lender with R_k(q) as in (8). - DC issuer sets  = 1 and charges positive on-island fees (expression in (36)); off-island data free. - Proof: Appendix A (p.39). - Households’ behavior (3.2.1): - When y ∈ [..], unrevealed households cannot obtain loans because bad borrowers dominate unrevealed pool and E[qju] y c < 0. - Loan market for unrevealed closed; interaction across households via E[qju] ceases to affect cash users. - Lenders’ strategies (3.2.2): - If  < y c: competition yields limit pricing (8); high-q households obtain loans and choose DC. - If  > y c: home lender monopoly extracts full surplus (R_k(q) = y), anticipating which households avoid DC. - For remainder, authors focus on case  < y c (equation (10)). - DC issuer’s strategy (3.2.3): - Provides off-island data free and charges on-island home lenders; sets  = 1 to maximize revealed borrowers and fees. - Home lender must buy revealed-household data to avoid losses from attracting all unrevealed households at unprofitable rates. --- ### DC issuer as lender and exclusive access (Section 4) - DC issuer integrated with lending (4.1): - Timing: DC design and household sorting (stages 1–2); monopoly sets loan rates (stage 3); households borrow (stage 4). - Outcome: all households choose cash; monopoly sets R_k(q) = y and makes no profit. Proposition 3: a single monopoly of payments and credit fails to make profit because households’ outside option of cash drives zero DC take-up. - Exclusive data access (4.2): - If only one lender has exclusive access, that lender charges revealed households R_u (given by (9)), making DC use strictly dominated by cash due to privacy cost; as a result, no household uses DC and DC issuer profit is zero. --- ### Welfare analysis: regulation (Section 5.1) - Policy maker sets long-term cap ̄ with  ∈ [0, ̄]; an uncertain state y realizes after policy: - With probability ∈ [0,1], y = y_h with y_h > 2c +  (good state). - With probability (1 ), y = y_l with y_l ∈ (2 4/3 c; 4/3 c] (bad state). - Welfare definition: sum of household payoffs and total profits of lenders and DC issuer; depends on value added qy c and privacy costs. - Aggregate welfare W and planner problem: - W = ∫_{borrowers} (qy c) f(q | "borrowing") dq ∫_{DC users} ( ') g(' | "DC use") d' - max_{} E[W] = max_{} { W_h + (1 ) W_l } (equation (12)) - Closed-form welfare pieces: - W_h = 3/4 y_h c 2  [ 1  1 + 4 /(3c) ] E[q | u] (equation (13)) - W_l =   (equation (14)) with  > 0 shown in (40) - E[q | u] closed-form in (26). - Socially optimal regulation (main insights): - If = 1 (good state certain): privacy costs dominate ⇒ ̄ = 0. - If = 0 (bad state certain): inclusion dominates ⇒ ̄ = 1. - For 0 < < 1, interior ̄ ∈ (0,1) may be optimal for some parameterizations; solved in (41). - Distributional effects: - Bad state: DC users’ disclosure is Pareto improving; no disclosure externalities. - Good state: disclosure cascades create winners (high q, low ') and losers (unrevealed facing higher R_u); corporate profits invariant to  in good state. - Comparative statics (three mechanisms): - Privacy valuation : - For → 0, ̄ = 1. - For very high , cash dominates; ̄ may be 0 in some parameterizations. - Intermediate can lead to interior ̄ or 0 due to cascades. - Market power : - ∂̄/∂ < 0 (higher  reduces optimal ̄). -  does not enter W_h in (13); effect on ̄ arises via bad state. - Economy type (developing vs advanced): - Lower (bad state more likely) ⇒ ∂̄/∂(1 ) > 0 ⇒ looser optimal regulation. - Second instrument: tax/subsidy on DC use (T) - T > 0 tax; T < 0 subsidy. - T shifts extensive margin only. - Proposition 6: regulation combined with T weakly improves expected welfare; specific gains depend on parameters (numerical examples: y_h = 2:2, y_l = 1:3, c = 1, = 0:05,  = 0:1). - Examples: - = 0:95: regulation alone ̄ = 0; a tax can allow limited DC to sustain credit in bad state. - = 0:5: a subsidy can increase DC uptake in bad state; instruments can be substitutes or complements. - Lender consortium (ownership of DC issuer) - Good state: each lender profit = 3/4  independent of ; consortium indifferent and private  indeterminate. - Bad state: profits rise with ; consortium sets  = 1. - Proposition 7: consortium does not exploit disclosure externalities in good state; optimal  indeterminate there. - Principal policy takeaway: - DC monopolist sets  = 1 to amplify disclosure cascades for profit. - Socially optimal regulation trades off credit inclusion in downturns vs privacy and cascade harms in upturns. - Optimal regulation tighter when lending market power  is strong and when both payment systems have solid take-up. - Combining regulation with tax/subsidy on DC use can improve expected welfare. --- ### Appendix A and B: proofs, closed forms, and extensions (key formulas and numeric thresholds) - Limit-pricing and breakeven arguments use R_k(q) = c/q +  and R_u = c E[q|u] +  with "!0 for captive pricing. - Sufficient condition for non-deviation:  < c/3. - Closed-form expressions (preserved from source): - E[q|u] (equation (26)): - E[q|u] = c/ [3c(2 - 1) + sqrt{3 sqrt{3c^2 (1 - ) + 4 ^2  (4 - 3)} (6 c^2 + 8 ^2)}] / (6 c^2 + 8 ^2) - (E[q|u] ∈ [1/2, 3/4]; negative root discarded) - m() (equation (27)): - m() = [3c(c+ 3 (1 - 2)) - 8 ^2  - ( +c) sqrt{3 sqrt{3c^2 (1 - ) + 4 ^2  (4 - 3)}}] / (3 c^2 - 4 ^2) - DC issuer fee b̄ = m() E[q|dc]  with closed-form in (28); @b̄/@ > 0. - R_u() (equation (29)): - R_u() = [6 c^2 + 8 ^2] / [3 c(2 - 1) + sqrt{3 sqrt{3c^2 (1 - ) + 4 ^2  (4 - 3)}] +  - Comparative statics (equation (30)): - @R_u()/@ > 0 - @^2 R_u()/@^2 > 0 - @R_u()/@ < 0 - @R_u()/@c > 1 - @R_u()/@ = 1 - Proposition 2 and alternative y ranges: - When y < 4/3 c no breakeven R_u exists; DC issuer fee expression differs (equation (36)); b̄ increases linearly in ;  = 1 privately optimal. - Welfare closed forms (selected): - W_h (equation (37)): - W_h = 3/4 y_h c 2  [1  (1 + 4 /(3 c))] E[q|u] - W_l depends on y_l relative to 2 + c +  (equations (38)–(40)). - Proposition 5: W_h maximized at  = 0; W_l maximized at  = 1; interior * solves @E[W]/@ = 0 (equation (41)). - Extensions (B): - B.1: pre-existing public information or alternative payment tech: DC issuer still privately chooses  = 1. - B.2: perfect positive correlation ' = 4 q 2 collapses dimensions;  = 1 remains privately optimal; W_h linear in  so planner picks 0 or 1. - B.3: quadratic privacy costs u(q,') = q max{y  I R, 0} '^{2}/2 (equation (42));  = 1 remains optimal for DC issuer in key cases. - B.4: fees/subsidies F at Stage 1; with  = 1 still privately optimal for DC issuer; properties: @m(;F)/@F < 0. - B.5: monetary policy setting c after state realization can prevent bad-state limited-credit equilibrium by making y > 2c + ; monetary policy cannot eliminate disclosure cascades in good state. - B.6: portable data (household-owned k) under quadratic privacy costs yields interior household choices (equations (46)–(49)); data portability reduces overdisclosure relative to monopoly but does not achieve socially optimal 0 disclosure in good state. - B.7: intermediate y regime can exhibit threshold e  with regime switch; DC issuer optimally chooses  = 1 where Proposition 2 prevails. - B.8: social planner with two instruments ( and T) — closed-form E[qju], W_h, W_l recorded in source. - Key numeric thresholds and parameter bounds (preserved exactly): - "!0 (limit pricing) - f(q) support q ∈ [1/2, 1] - E[q] = 3/4 in baseline when all unrevealed - Sufficient condition for non-deviation:  < c/3 - Parameter restrictions: < 1/6 c,  ∈ (0, 1/3 c] in various derivations - R_u() derivatives: @R_u()/@ > 0, @^2 R_u()/@^2 > 0, @R_u()/@ < 0, @R_u()/@c > 1, @R_u()/@ = 1 *Italic: wpiea2023121-print-pdf - Section 3 derives the equilibria wherein the data monopolist sells the data to lenders and Section (canonical PDF).* ### Section 3 derives the equilibria wherein the data monopolist sells the data to lenders and Section ### wpiea2023121-print-pdf - Section 3 derives the equilibria wherein the data monopolist sells the data to lenders and Section ### Paper structure and focus - Section 3 derives the equilibria wherein the data monopolist sells the data to lenders. - Section 4 shows why the data monopolist chooses to sell data rather than engage in credit provision itself. - Section 5 analyzes welfare and socially optimal policy design. - Section 6 extends the baseline model to lender ownership of the data monopolist. - Proofs are in Appendix A; additional model extensions are in Appendix B (extensions B.1–B.8 referenced). ### Model overview (Section 2) - Economy populated by three agent sets: consumer-entrepreneurs ("households"), a monopolist digital currency issuer (DC issuer), and lenders. - All agents are risk neutral. - The economy is distributed across N3connected islands. Each island hosts one lender and a continuum of households with mass1. - A lender on the same island as a borrower is the borrower’s “home lender”; borrowers on the same island as a lender are the lender’s “on island” borrowers. - All islands share the same physical currency (cash) and the digital currency issuer. ### Households (Section 2.1) - Each household has three characteristics: - creditworthiness q (private), - preference for privacy ' (private), - island location (public). - Each household is born with an investment project that yields payoff y if successful and 0 otherwise; success probability is q. - Privacy disutility is ' and households attach disutility ' to loss of privacy where >0 and  is the probability of revelation. - On each island, the mass1of households is uniformly distributed on a two-dimensional plane with q2  1 2 ;1  and '2[0;2]. - Households choose between cash and the digital currency (DC) for consumption: - Cash fully protects privacy ( =0). - DC use implies revelation with probability  (i.e.,  =) and generates privacy disutility. - Households are protected by limited liability. - Borrowing from an off-island lender reduces the payoff of a successful project by  >0. - Household expected utility (type (q,')): - u(q;') = q max{yIR; 0} ' - where I is an indicator equal to 0 when borrowing from home lender and 1 otherwise; R is the gross loan interest rate;  = if the household chooses DC and 0 if it chooses cash. - Baseline omits an ease-of-transactions benefit of DC; Appendix B.8 recalculates expressions when including such a term. - Privacy costs are linear in the probability of revelation in the baseline; Appendix B.3 considers quadratic privacy costs. ### Digital currency issuer (Section 2.2) - The DC issuer chooses the intrusiveness/design parameter  ( in [0;1]) that governs the probability of learning households’ creditworthiness for DC users. - The DC issuer is a monopolist in digital payments and sets data access fees ij charged to lender i for data on households on island j. - Lender i’s demand for data on island j is Dij(; ij) and takes value 0 or 1 depending on pricing and value of the data. - DC issuer maximizes data fee revenue: - max 2[0;1], {ij} ΣiΣj ij Dij(; ij) - The DC issuer in the focus of Sections 3 sells data to lenders and does not itself engage in lending in the baseline (Section 4 analyzes the alternative). ### Lenders (Section 2.3) - Lenders have infinitely elastic funding at cost c1 and engage in Bertrand competition for loans. - Two sources of differentiation: - home lender advantage parameterized by  >0, - access to information about households’ credit quality via purchased data. - Lender i’s expected profit: - m i (E[q ki] R ki(q ki) c) + i (E[q ju] R ui c) Σj ij Dij(; ij) - where m i is mass of revealed borrowers choosing lender i, E[q ki] is expected q among revealed borrowers, R ki(q ki) are individualized rates,  i mass of unrevealed borrowers choosing lender i, and E[q ju] expected q among unrevealed. - Purchasing data allows a lender to price-discriminate and attract higher-q borrowers, leaving others pooled. ### Parameter conditions (Section 2.4) - Three conditions relating parameters to lenders’ funding cost c: - y > c - ensures q = 1 borrower always has positive NPV project. -  < 1 3 c - ensures off-home payoff reduction  is not too large so competitive mechanisms operate. - < 1 6 c - ensures households with q = 1 will always want to disclose their type; aids tractability. ### Timing (Section 2.5) - Stage 1: DC issuer chooses  and data access fees ij. - Stage 2: Households decide to use DC or cash and consume. - Stage 3: For DC users, creditworthiness is revealed with probability . - Stage 4: Lenders decide on data purchases. - Stage 5: Each lender announces loan rates (individualized for revealed borrowers; pooled rate for unrevealed). - Stage 6: Households choose lender. - Stage 7: Project returns realize and repayment occurs. - The equilibrium is solved by backward induction. ### Equilibria (Section 3) — partition by y and Table 1 summary - Equilibria are derived in three propositions corresponding to different credit market outcomes, delineated by the parameter y. - Definition: "good" borrowers have positive NPV (qy > c); "bad" borrowers have negative NPV (qy < c). - Table 1 parameterization of share of bad borrowers and corresponding y ranges: - High y — Small share of bad borrowers — y > 2c +  - Medium y — Intermediate share of bad borrowers — y 2 4 3 c; 2c +   - Low y — Large share of bad borrowers — y 2 c; 4 3 c  - Section 3.1 discusses the "High y" case where households often have positive NPV projects (qy > c) except at very low q. - Section 3.2 analyzes the "Low y" case where many borrowers are bad (qy < c) unless q is high. - The "Intermediate y" case is derived in Appendix B.7 and is an extension of the other two cases. *Italicized source: wpiea2023121-print-pdf - Section 3 derives the equilibria wherein the data monopolist sells the data to lenders and Section (canonical PDF).* ### 3.1 Equilibrium with few bad borrowers ### 3.1 Equilibrium with few bad borrowers ### Main proposition and equilibrium characterization - Proposition 1: When y > 2c + , the DC issuer sets  = 1, charges each lender data access fees ii > 0 for information about its on-island households (closed form in (28)), and offers each lender free information about off-island households ( ij = 0 when j 6= i). - Households sort into DC use if - ' c  q E[qju] 1  (equation (7)), with the closed-form for E[qju] displayed in (26). - Home lenders buy data access and offer differentiated loan rates - R_k(q) = c/q +  (equation (8)) to revealed borrowers, - R_u = c E[qju] +  (equation (9)) to unrevealed borrowers. - All households borrow from their respective home lenders. - Proof referenced: Appendix A (p.33). ### Households’ optimal strategies (3.1.1) - Without DC (equivalently,  = 0): - All households use cash so that E[qju] = E[q] = 3/4. - R_u = 4/3 c + . - Introduction of DC: - Some households (those to the right of and below the dashed line in Figure 2) with highest credit quality choose DC when offered, lowering E[qju] and raising R_u as they become revealed. - A lower E[qju] makes condition (7) hold for more households, inducing further DC adoption — producing a cascade of disclosure. - Welfare implication: - Households between the dashed and unbroken lines in Figure 2 can be worse off than in the cash-only economy, highlighting a public good nature of privacy. - Bad borrowers: - Households at the left end of Figure 2 (bad borrowers) self-select into cash because revelation precludes them from receiving loans. - When y > 2c + , there remain enough good borrowers including high-privacy good borrowers who choose cash, so the loan market for unrevealed borrowers remains open. ### Lenders’ optimal strategies (3.1.2) - Lenders engage in Bertrand competition on loan rates, but home lender advantage permits households to pay a premium up to  on loans from home lenders. - Limit pricing yields equations (8) and (9). - Equilibrium outcomes result from both the pricing game and the information acquisition game among lenders. ### The DC issuer’s optimal strategy (3.1.3) - The DC issuer maximizes the difference in expected profits between an informed and an uninformed home lender to induce lenders to buy data. - Key observations: - Symmetric Bertrand competition implies lenders break even; data about off-island borrowers has zero value to each lender in that symmetric information game. - By providing data on off-island borrowers for free, the DC issuer maximizes adverse selection for an uninformed home lender and thus the price it can charge for on-island data. - Incentives and pricing: - A home lender that refuses to buy on-island data would need to set a single loan rate below (9) to attract revealed borrowers, reducing profits on unrevealed borrowers and suffering adverse selection on the revealed segment. - Consequently, the home lender purchases data as long as it makes any positive profit on revealed households; the DC issuer charges a fee marginally below the lender’s total earnings on revealed households ( times the mass of revealed households times their expected quality; closed form in (28)). - Optimal DC design: - The DC issuer sets  = 1 to make the revealed set as large as possible, increasing odds of revelation per DC user and expanding DC user mass via cascade effects. - As  increases, the mass of unrevealed DC users shrinks, E[qju] declines, inducing more households to opt for DC use. - Robustness note: - Appendix B.3: With quadratic privacy costs, mass of DC users may decline as  increases (depending on parameters) but  = 1 remains optimal for the DC issuer. --- ### 3.2 Equilibrium with many bad borrowers ### Main proposition and equilibrium characterization - Proposition 2: When y2 c; 4/3 c  , then - if  > y c, no credit provision takes place and all households use cash. - if  < y c, households with q  ' + c y  choose to use the DC and, if revealed, borrow from their home lender with loan rates given by (8). - The DC issuer optimally sets  = 1 and charges lenders a positive data access fee for on-island borrowers (shown in (36)), while offering each lender free data on its off-island borrowers. - Proof referenced: Appendix A (p.39). ### Households’ optimal strategies (3.2.1) - When y2 c; 4/3 c  , unrevealed households can never obtain loans: - Bad borrowers outweigh good borrowers among the unrevealed (E[qju] is low). - No viable breakeven loan rate exists for the unrevealed; households can at most pay y on a successful project, but E[qju] y c < 0 when y2 c; 4/3 c  . - Therefore no lender could break even on lending to such unrevealed households. - With the loan market for unrevealed closed: - Interaction between one household’s payment choice and others ceases. - Choice of DC users still affects E[qju], but E[qju] does not affect the return on choosing cash (cash users do not obtain loans). - No negative disclosure externalities remain. ### Lenders’ optimal strategies (3.2.2) - If  < y c: - Competition leads to limit pricing consistent with (8), enabling high-quality households to obtain attractive loan offers and choose DC over cash. - If  > y c: - Lenders effectively become monopolists: households may have positive NPV projects when borrowing from the home lender but negative NPV from other lenders. - Time inconsistency: at Stage 5 the home lender will charge revealed households R_k(q) = y, extracting the full profit; anticipating this at Stage 2, no household chooses DC. - For remainder of paper, the authors focus on the case where home-lender advantage is small enough to sustain competition and assume: -  < y c (equation (10)). - Note: (10) is tighter than (5) iff y < 4/3 c. ### The DC issuer’s optimal strategy (3.2.3) - Given (10), credit provision takes place and the DC issuer offers household data for free to other lenders to maximize pressure on home lenders to pay access fees that appropriate revealed-household profits. - The DC issuer sets  = 1 to maximize number of revealed borrowers and thereby maximize access fees. - If the home lender refuses to buy revealed-household data and attempts a single loan rate to attract revealed households, all unrevealed households (otherwise credit excluded) would rush to the home lender, causing losses; hence the home lender purchases data access. --- ### 4 Data usage strategies ### DC issuer as lender (4.1) - Timing (Figure 3): DC design and household sorting (stages 1–2), monopoly sets loan rates (stage 3), households borrow/invest/repay (stage 4). - Outcome: - All households choose cash and the monopoly makes no profit. - At stage 3 the monopoly sets R_k(q) = y, seizing expected value added of a revealed borrower’s project; anticipating this, households avoid DC due to privacy cost and no loan-rate benefit. - Proposition 3: A single monopoly of both payment data and credit provision fails to make profit, as households’ outside option of cash leads to zero credit provision when lending market power is too large. - Proof referenced: Stage 2 in proof of Proposition 2 in Appendix A. - Intuition: - Households can opt out before being charged monopolistic loan rates; absent credible precommitment not to exploit data, integration of data and credit leads to zero DC take-up and zero lending profit. ### Exclusive data access (4.2) - Under exclusive access (only one lender buys access): - A lender with exclusive data access optimally charges all revealed households the equilibrium loan rate for the unrevealed, R_u (given by (9)). - Revealed households are cornered at R_u and have no attractive alternative, so DC use yields the same loan rate as cash but with a privacy cost. - Foreseeing this, no household uses DC and the DC issuer earns zero profit. --- ### 5 Policy analysis (preview) - Section 5 outlines regulatory implications: - Section 5.1 considers regulation to constrain the DC’s intrusiveness. - Section 5.2 analyzes whether a second policy instrument (tax or subsidy on DC use) can improve welfare. - Detailed policy analysis and welfare implications are explored in Section 5. *Source: wpiea2023121-print-pdf - 3.1 Equilibrium with few bad borrowers* ### 5.1 Welfare analysis: regulation ### 5.1 Welfare analysis: regulation ### Setup and timing - The policy maker determines regulation at the beginning of the game; socially optimal DC design is denoted by  and the DC issuer faces the constraint  ∈ [0; ]. - DC policy and DC design are set long-term, before the aggregate state is known. An uncertain state variable y realizes after policy and design: - With probability ∈ [0;1], y = y_h (the good state of Proposition 1). - With probability (1 ), y = y_l (the bad state as in Proposition 2). - y_h represents "High y" with y_h > 2c + . - y_l represents "Low y" with y_l ∈ (24/3 c; 4/3 c] (notation preserved from source). - Welfare is the sum of expected payoffs of households and total profits of lenders and the DC issuer; welfare depends only on: - The value added qyc of funded projects. - The privacy costs experienced by DC users. ### Welfare expression - Aggregate welfare W is expressed as: - W = ∫_{borrowers} (qyc) f(q | "borrowing") dq ∫_{DC users} ( ') g(' | "DC use") d' - Policy maker maximizes expected welfare E[W] with respect to : - max_{} E[W] = max_{} { W_h + (1 ) W_l } (equation (12) as given) - Closed-form expressions given in the source: - W_h = 3/4 y_h c 2  [ 1  1 + 4 /(3c) ] E[q | u] (equation (13)) - W_l =   (equation (14)) - E[q | u] closed form is referenced as (26);  is a collection of constants with  > 0 shown in (40). ### Socially optimal regulation (main insights) - Behavior of W_h and W_l as a function of : - In the bad state, welfare W_l increases linearly with . Higher  helps households reveal types and there are no disclosure externalities when the loan market for unrevealed households is inoperative. - In the good state, all households receive loans so total value added is independent of ; DC impacts aggregate welfare via privacy costs and disclosure cascades. W_h decreases more than linearly as  increases (direct effect via privacy costs and indirect effect via E[q | u]). - Proposition 5 (paraphrased from source): - If the good state is certain ( = 1), privacy costs dominate and the policy maker bans the DC:  = 0. - If the bad state is certain ( = 0), credit inclusion from type differentiation dominates and laissez-faire is optimal:  = 1. - For ∈ (0;1), for some parameterizations the policy maker chooses interior regulation  ∈ (0;1), solved in (41). ### Distributional effects - Bad state: - DC users’ disclosure is a Pareto improvement: creates value to DC users, no externalities to others, and generates corporate profits. - Good state: - Some (high q, low ') households gain from lower customized loan rates; their gains are exactly offset by losses of unrevealed households facing a higher loan rate. - Total corporate profits do not depend on  in the good state; higher  transfers profits within the corporate sector (DC issuer gains at lenders’ expense). - Net impact on W_h derives from privacy loss experienced by DC users swept into disclosure cascades. ### Comparative statics (three highlighted mechanisms) - Society’s care for privacy ( ): - Regulation plays a role ( < 1) when the two means of payment each have enough users (Corollary 1). - For → 0, privacy ceases to matter and expected welfare E[W] = 3/4 y_h c + (1 )  , which increases with  implying  = 1. - For very high , cash use dominates and for some parameterizations  = 1 can also be observed. - When society cares an intermediate amount about privacy, cascades can affect many households and regulation can be interior or even 0. - Market power in lending (): - Relationship centers on the bad state: ∂/∂ < 0. - A higher  raises loan rates for DC users in the bad state (cash users remain without credit), pushing more households into cash and reducing the social benefits of larger  in bad times; thus  decreases. - In the good state,  does not enter W_h in (13); therefore  does not affect W_h from the social planner’s perspective. - Type of economy (developing vs advanced): - When is smaller (bad state more likely in expectation), ∂/∂(1 ) > 0: the policy maker attaches more weight to gains from type differentiation and optimal regulation may be looser in expected terms for a developing economy. - Representing greater economic volatility via a "mean preserving spread" (increasing y_h and decreasing y_l) implies a lower  and tighter DC regulation. ### Second instrument: tax or subsidy on DC use - A second policy instrument T is introduced, where: - T > 0 is a tax on DC use. - T < 0 is a subsidy on DC use. - T affects the extensive margin (who opts into DC use) but not the intensive margin (how much information per DC user is collected). - Proposition 6: Combining regulation with a tax or subsidy on DC use can attain higher expected welfare than regulation alone. - T = 0 replicates the regulation-only outcome, so two instruments weakly improve welfare. - For some parameterizations two instruments do not raise welfare; for others they do (numerical examples provided in source Figures 6 and 7). - Numerical examples (parameterization noted in source): - y_h = 2:2, y_l = 1:3, c = 1, = 0:05,  = 0:1 while is varied. - Examples of how instruments are used: - When the good state is very likely ( = 0:95), regulation alone sets  = 0; a tax can allow the policy maker to exit this corner and permit limited DC use to sustain some credit provision in bad states. - When the states are equally likely ( = 0:5), a subsidy can increase DC uptake in the bad state where households disclose less than socially optimal, pairing the subsidy with looser regulation. - Instruments can act as substitutes (Figure 6 example) or complements (Figure 7 example). ### Lender consortium alternative - If lenders own the DC issuer (consortium) and collected DC data is freely available to all lenders: - In the good state, each lender’s total profit equals 3/4  and does not depend on ; hence the consortium is indifferent and privately optimal  is indeterminate. - In the bad state, lender profits increase with  because a higher  raises the mass of revealed households who receive credit; the consortium sets  = 1. - Proposition 7: Unlike the independent DC issuer, the lender consortium does not actively exploit disclosure externalities in the good state; unlike a social planner, the consortium does not lean against such externalities either—optimal  is indeterminate in the good state. ### Principal policy takeaways (concise) - The DC monopolist optimally makes data collection as intrusive as possible to amplify disclosure cascades and favorable information asymmetries for profit. - Socially optimal regulation trades off: - Preserving credit provision in downturns (bad state) via higher intrusiveness that enables type differentiation. - Shielding households from privacy losses and disclosure cascades in upturns (good state). - Optimal regulation is most interventionist when: - Market power in lending () is strong. - Both the monopolist’s payment system and cash have solid take-up (so disclosure cascades can affect many). - A tax or subsidy on DC use complements regulation by shifting the extensive margin; combining instruments can improve expected welfare and can be used as substitutes or complements depending on parameters. *Source: IMF Working Paper — Section 5.1 "Welfare analysis: regulation" (excerpts and equations as presented in the supplied content).* ### References ### wpiea2023121-print-pdf - References ### Key propositions and equilibrium characterization (summary of Appendix A proofs) - The proofs apply backward induction to solve for the Subgame Perfect Nash Equilibrium with symmetric pure strategies among other lenders. - Limit-pricing arguments treat R_k(q) = c/q + " and R_u = c/E[q|u] + " with "!0 to model captive pricing and breakeven preference. - Home lender differentiated rates: - R_kh(q) and R_ki(q) denote differentiated loan rates charged to revealed households by the home lender and other lenders, respectively. - R_uh and R_ui denote loan rates charged to unrevealed households by home and other lenders. - Stage outcomes (high level): - Stage 6: Unrevealed households borrow from home lender if y - R_uh > y -  R_ui; revealed households borrow from home lender if y - R_kh(q) > y -  R_ki(q). - Stage 5: Breakeven loan rates are c/q for revealed households and c E[q|u] for unrevealed households. Bertrand competition among other lenders implies R_ki(q) = c/q. - Stage 4: Other lenders obtain household data only if the DC issuer offers free access; optimal strategy for DC issuer is to provide data free to other lenders and charge the home lender a fee equal to home lender profits on revealed households (m E[q|dc] , with "!0). - Stage 2: A household prefers DC over cash iff ' < q(R_u - R_k(q)), which given (8) and (9) reduces to sorting condition in (7). - Stage 1: DC issuer sets access fee m E[q|dc]  to the home lender and chooses  (probability of revelation) by maximizing m E[q|dc] ; closed-form solutions imply  = 1 is privately optimal for the DC issuer in the baseline. ### Deviation analysis and sufficient conditions - Deviation case: If home lender does not purchase data and instead charges R_dev, it must set R_dev < c E[q|u] +  to attract revealed borrowers; adverse selection implies such deviation is suboptimal under sufficient bounds on . - A sufficient condition for non-deviation (ensuring @Π_dev/@R_dev > 0 for R_dev ≤ c E[q|u] + ) is  < c/3. - Expression for deviating home lender profit Π_dev (equation (15)): - Π_dev = ∫_{c/(R_dev -  E[q|u])}^{c} (q R_dev - c) f(q) dq + (1 - m) (E[q|u] R_dev - c) - Density used on revealed range: f(q) = c/ (q/E[q|u] - 1). ### Closed-form solutions and comparative statics - Closed-form expressions derived for E[q|u], m(), E[q|dc], and the DC issuer optimal fee b̄: - E[q|u] given by equation (26): - E[q|u] = c/ [3c(2 - 1) + sqrt{3 sqrt{3c^2 (1 - ) + 4 ^2  (4 - 3)} (6 c^2 + 8 ^2)}] / (6 c^2 + 8 ^2) - (negative root discarded; E[q|u] ∈ [1/2, 3/4]) - m() given by equation (27): - m() = [3c(c+ 3 (1 - 2)) - 8 ^2  - ( +c) sqrt{3 sqrt{3c^2 (1 - ) + 4 ^2  (4 - 3)}}] / (3 c^2 - 4 ^2) - DC issuer fee b̄ = m() E[q|dc]  with closed-form expression in (28); @b̄/@ > 0. - Loan rate for unrevealed households R_u() (equation (29)): - R_u() = [6 c^2 + 8 ^2] / [3 c(2 - 1) + sqrt{3 sqrt{3c^2 (1 - ) + 4 ^2  (4 - 3)}] +  - Comparative statics for R_u() (equation (30)): - @R_u()/@ > 0 - @^2 R_u()/@^2 > 0 - @R_u()/@ < 0 - @R_u()/@c > 1 - @R_u()/@ = 1 - Marginal effects intuition: - Higher  worsens expected quality of unrevealed pool, raising R_u() and causing convex cascade effects. - Higher (privacy valuation) improves unrevealed pool quality, reducing R_u(). - Higher c (funding cost) increases loan rates. - Higher home lender market power  raises loan rates. ### Propositions under alternative parameter ranges (Proposition 2, 4, 5 summaries) - Proposition 2 (when y < 4/3 c): - No breakeven R_u exists; lenders unwilling to lend to unrevealed households. - For q < c/y, projects have negative NPV and no lending occurs. - For q ∈ (c/y, c y - ], only home lender can make offers and sets R_kh(q) = y - " with "!0 (monopoly appropriation). - DC issuer fee expression differs and is given by equation (36); b̄ increases linearly as  rises;  = 1 is privately optimal. - Welfare terms W_h and W_l derivations: - W_h (equation (37)): - W_h = 3/4 y_h - c - 2  [1 -  (1 + 4 /(3 c))] E[q|u] - W_l expressions depend on whether y_l < 2 + c +  or y_l > 2 + c + ; see equations (38)–(40). - Proposition 5 (social planner choice of ): - W_h maximized at  = 0 (privacy cost term eliminated). - W_l maximized at  = 1. - Interior solution * ∈ (0,1) solves @E[W]/@ = 0 and is given in closed form by equation (41). ### Extensions (B) - B.1 Alternative sources of credit quality data: - Pre-existing public information (e.g., credit registries, FICO) can be modeled as a revelation draw before the game; revealed households receive R_k(q) offers; renormalization by factor 1/ if pre-revelation probability is  ∈ (0,1). - Alternative payment technology with revelation probability _alt < 1 and _dc > _alt: household choice reduces to same condition ' < q(R_u - R_k(q)); DC issuer optimality unaffected, with _dc = 1 privately optimal (_alt appears as a constant). - B.2 Correlation of credit quality and privacy preferences: - Perfect positive correlation: ' = 4 q - 2 (so ' and q collapse to one dimension); solving yields E[q|u] = sqrt{c} / (sqrt{c} + 16 - c) / (8 ) (closed-form summarized in text as E[q|u] = sqrt{c} sqrt{c + 16 - c} / (8 ) — see equation text). - Under perfect positive correlation,  = 1 remains privately optimal; W_h decreases and W_l increases with ; W_h is linear in  so interior * is either 0 or 1. - Perfect negative correlation leads to full unraveling (Lemons’ Market); no E[q|u] ∈ [1/2, 3/4] solves fixed point. - B.3 Quadratic privacy costs: - Replacing linear privacy cost with quadratic form u(q,') = q max{y -  I - R, 0} - '^{2}/2 (equation (42)) increases privacy cost convexity; main Stage 3–7 results for Proposition 1 unchanged in outline though quantitative thresholds differ. ### Key numeric thresholds and parameter bounds (preserved exactly) - Breakeven and feasibility conditions and bounds used throughout: - "!0 (limit pricing) - f(q) support q ∈ [1/2, 1] - E[q] = 3/4 in baseline when all unrevealed - Sufficient condition for non-deviation:  < c/3 - Parameter restrictions used: < 1/6 c,  ∈ (0, 1/3 c] in various derivations - R_u() derivatives: @R_u()/@ > 0, @^2 R_u()/@^2 > 0, @R_u()/@ < 0, @R_u()/@c > 1, @R_u()/@ = 1 - Welfare and fee closed forms referenced: equations (15), (18)–(29), (36), (37)–(41) *Italic: Source: wpiea2023121-print-pdf - References (Appendix A and B proofs and extensions).* ### Section 2.5. In Stage 2, the derived indifference frontier in (7) now becomes ### Section 2.5. In Stage 2, the derived indifference frontier in (7) now becomes ### Indifference frontier with quadratic privacy costs - The derived indifference frontier becomes: - ' c  q E[qju]1 (equation (43)) - For  = 1, (7) and (43) are equivalent;  = 1 remains optimal for the DC issuer (as in Proposition 1). - Calculations referenced are available in a Mathematica file (available on request). ### Implications for Proposition 2 and mass functions - Equation (31) becomes: - ' < q(y)c (equation (44)) - The resulting expressions for() and m() are: - () = - 2  1 c+  y  if  < yc 2 - 1 2   (yc) 2 y  if  > yc 2 - m() = - 2  1 c+  y  if  < yc 2 - 1 2  (yc) 2 y  if  > yc 2 (equation (45)) - The two expressions for each function coincide at  = yc 2 , producing kinks but no discrete jumps in masses of DC users and revealed borrowers. - The quadratic privacy-cost functional form is less well suited to extended DC design analysis but provides additional mechanism insight. ### Responses of masses to  (low-y case) - In the low y case (Proposition 2) with quadratic privacy costs: - @() @ <0. Explanation: larger marginal privacy costs repel more households from DC use as  rises than are attracted by higher unrevealed loan rates. - For  > yc 2 : - @m() @ = 0 — decline in mass of DC users is exactly offset by higher revelation probability per DC user; mass of revealed borrowers unchanged. - For  < yc 2 : - @m() @ >0 (same sign as in the baseline). - Overall,  = 1 remains optimal for the DC issuer in the low-y case. - Even when @m() @ = 0, optimal data access fees (m()E[qjdc]) rise as  increases because @E[qjdc] @ >0. ### B.4 Fees or subsidies on DC use - Introduce fee F set at Stage 1 and paid at Stage 2: - F > 0: positive fee; F < 0: subsidy; F = 0: baseline. - Main insight: - Incentive for DC issuer to charge a positive fee increases as  decreases (weaker lender market power). - In limit  ! 0, only fees can make DC issuer profit; optimal fee maximizes F · m(;F) where m(;F) is mass of DC users as function of  and F. - When lending market power is larger, information rents rise and DC issuer has incentive to cross-subsidize information acquisition with F. - Households choose DC over cash if: - In Proposition 1: '+ F   c  q E[qju]1  - In Proposition 2: q '+c+F y -  = 1 remains optimal for the DC issuer with fees. - Properties: - @m(;F) @F < 0. - First-order condition for optimal fee: - @F m(;F) @F + @m(;F)E[qjdc] @F = 0 - Can be rearranged (as given) and evaluated at F = 0 to determine whether fee or subsidy will be offered. ### B.5 Monetary policy - Consider policy maker setting c after state realization; allow arbitrarily negative policy rates (c 2 (0;1) possible). - Monetary policy can prevent the bad (low-y) state equilibrium with constrained credit by setting c low enough so y > 2c+ (given y >  and in fact y > 3 by (4) and (5)). - Monetary policy cannot overcome data externalities that dominate welfare in the good state. - With monetary policy leaning against the bad state: - Disclosure loses social value because credit provision is assured and social benefits of type differentiation vanish. - Monetary policy influences aggregate credit access but cannot prevent disclosure cascades. - First-best can be attained by combination:  = 0 (ban DC) and state-dependent monetary policy preventing limited-credit equilibrium. ### B.6 Portable data (household-owned data; quadratic privacy costs) - Setup: households choose own revelation probability k in [0,1]; payment service provider verifies q and reveals with chosen probability. - Embedded in quadratic privacy cost case to yield interior solution; 4-stage game with households choosing k at Stage 1. - Household optimization: - max k 2[0;1] { k q maxf(yIRk(q));0g + (1k) q maxf(yIRu);0g '2k } (equation (46)) - Focusing on high-y case, loan market equilibrium as in (8) and (9); optimization becomes: - max k 2[0;1] { k q  y c q  + (1k) q  y c E[qju]  '2k } (equation (47)) - Solution: - k = - min { 1 2  c '  q E[qju]1  ;1 } if q > E[qju] - 0 if q < E[qju] (equation (48)) - Comparator model (quadratic extension equilibrium) revelation probability: - = - 1 0 if c '  q E[qju]1  1 - otherwise (equation (49)) - Intuitions comparing data porting (free portability) vs data monopoly: - Households with q < E[qju]: zero disclosure privately optimal in both models. - Households with q > E[qju]: - In data porting: can choose partial disclosure k in (0,1). - In comparator (monopoly) model: forced choice between full revelation and zero (cash) leads to more overdisclosure via cascades; DC issuer exploits cascades to commercialize overdisclosure. - From the good-state social perspective,  = 0 (ban DC and data porting) is socially optimal; free data porting leads to overdisclosure, and private data monopoly induces even more overdisclosure. ### B.7 Intermediate y in the baseline model - Proposition 8 (Intermediate y: y 2 4 3 c;2c+ ): - Equilibrium can be as in Proposition 1 or Proposition 2 depending on parameters. - There can exist threshold value e  such that: - For  < e : Proposition 1 prevails. - For  > e : Proposition 2 prevails, with DC issuer optimally setting  = 1. - Reasoning: - For y = 4 3 c the unrevealed loan market is shut for  = 0; for y = 4 3 c + " it remains shut when  > 0 because @E[qju] @ < 0. - e  in (0,1) exists if at  = 1 the unrevealed do not obtain credit (possible because @E[qju] @ < 0 from (26)). - At threshold crossing, mass of DC users may increase or decrease; condition for mass to be greater after crossing is c E[qju] < y with E[qju] 2  1 2 ; 3 4  and y 2 4 3 c;2c+ . ### B.8 Expressions for the social planner with two instruments - Households’ payoff: - u(q;') = q maxf(yIR);0g ' + JT + transfers (equation (50)) - J = 1 if household uses DC; 0 if cash. T > 0 tax (T < 0 subsidy); transfers are lump-sum redistribution of total revenues/costs. - Equilibrium sorting conditions (DC over cash) with tax/subsidy: - Modified forms of Proposition 1 and Proposition 2 sorting inequalities: - ' 1  c  q E[qju]1  T  and q 1 y '+c+ T  . - Expressions for E[qju] and E[W] = W h + (1 ) W l are given (explicit algebraic forms in source): - E[qju] = - 3c(21) + p 3 q  (43) 3T 2 + 6 T+ 4 2  2  + 3c 2 (1)  (6c 2 8 2) 2 6T(T+ 2 ) - W h = - 3 4 y h c 2   (4 + 3c) + 3T 3c E[qju] - W l = - Two-piece expression depending on y l relative to 2 + c +  + T (full forms in source). - Note: These expressions are recorded without derivations; derivations available on request. *Bank Competition and Household Privacy in a Digital Payment Monopoly — Working Paper No. WP/2023/121* --- _Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023121-print-pdf.pdf_