## wp1714

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---

### Overview and objectives
- Develop a theory of money and credit as competing payment instruments using a New Monetarist approach that explicitly models exchange/payment processes.
- Integrate Burdett-Judd "noisy" search (sellers post prices; each buyer sees a random number of them) into the Lagos and Wright (2005) alternating centralized (AD) and decentralized (BJ) markets framework.
- Introduce credit transaction costs C(d_t) = 1(d_t) + d_t (fixed cost  and proportional cost ) and show these specifications perform differently; the proportional (variable) cost specification outperforms the fixed-cost specification in theory and data.
- Aim to jointly match micro facts on payment methods and macro facts on money demand together with price-change statistics; apply the model to sticky prices, welfare, participation, and dynamics.

### Key mechanisms and equilibrium concepts
- Cash faces an inflation tax; credit requires paying C(d_t) to promise d_t dollars in the next AD.
- Noisy search and price posting imply a distribution of nominal prices F(p); any p in nondegenerate support yields same seller profit.
- Increasing money supply M shifts F(p) so real distribution stays the same; overlapping supports produce apparent nominal price stickiness.
- Costly credit generates a unique money demand and resolves indeterminacy present in indivisible-good/BJ posting models.
- Stationary equilibrium classes:
  - NME (nonmonetary equilibrium): z = 0 (all BJ trades use credit).
  - MME (mixed monetary equilibrium): 0 < z < q̄ (cash for q ≤ z, credit for q > z).
  - PME (pure monetary equilibrium): z ≥ q̄ (all BJ trades use cash).

### Firm problem and price distribution (Section 3.1) — exact expressions preserved
- Seller profit posting p:
  - Πt(p) = bt ht β1 + 2β2 ^Ft(pt) i (p − ct)  (notation preserved as in source)
- Equilibrium Ft(p) for p ∈ Ft = [pt; p̄t]:
  - Ft(p) = 1 − (β1 / (1 + 2β2)) ( (ct p̄t − ct p) / (p̄t − p) ).
- Lower bound from Ft(pt) = 0:
  - pt = (β1 / (1 + 2β2)) p̄t + (2β2 / (1 + 2β2)) ct.
- Real-price numeraire qt = ct pt; real distribution Gt(q) associated as in source.
- Lemma 3 characterization of q̄:
  - NME: z = 0 and q̄ = (m̄ − ε) / (1 + ς).
  - MME: z ∈ (0; m̄ − ε) and q̄ = (m̄ − ε + ς z) / (1 + ς).
  - PME: q̄ = z ≥ m̄ − ε.

### Household value functions and cash/credit decision
- AD value: W(A) = max_{x,ℓ,z} { U(x) − ℓ + β V(z) s.t. x = A + ℓ − (1 + π) z }.
- With U′(x*) = 1, W(A) = A + U(x*) − x* + β max_{z} Oi(z), where Oi(z) ≡ V(z) − (1 + i) z.
- Lemma 2: W′(A) = 1 and z choice does not depend on A.
- BJ value:
  - V(z) = W(A) + (β1 + β2) [ m̄ − E_H q − ε ^H(z) − ς E_H max(0; q − z) ]
- H(q) defined as:
  - H(q) = [ β1 G(q) + β2 (1 − ^G(q))/2 ] / (β1 + β2).

### Money demand: fixed-cost (ε > 0, ς = 0) — closed-form and equilibrium existence
- Objective Oi(z) linear on bounds; first-order condition for interior MME:
  - (β1 + β2) ε H′(z_i) = i.
- Proposition 1 (fixed-cost, βn = 0 for n ≥ 3):
  - (i) NME exists for high i.
  - (ii) MME exists iff ε < ε̄ and i ∈ (i; ī) (threshold expressions preserved in source).
  - (iii) PME exists under specified ranges of ε and i (thresholds given in source).
- Fixed-cost money demand (as given in source):
  - ẑ_i = m̄ + [ (β2 / β1) ε (m̄ − ε − ct)^2 ]^{1/3} i^{−1/3}  (expression formatting preserved from source)

### Money demand: variable-cost (ς > 0, ε = 0) — uniqueness and closed-form
- Oi(z) differentiable everywhere; q̄ and q expressions:
  - q̄ = (m̄ + z ς) / (1 + ς)
  - q = [ β1 (m̄ + z ς) + 2 β2 ct (1 + ς) ] / [ (β1 + 2β2) (1 + ς) ]
- Equilibrium regimes:
  - If i > (β1 + β2) ς, unique NME.
  - If i < (β1 + β2) ς, unique MME.
  - PME cannot exist for i > 0.
- Proposition 2 (variable-cost, βn = 0 for n ≥ 3):
  - (i) NME iff ς ≤ m̄ / ct − 1.
  - (ii) MME iff i < min{ ς (β1 + β2), ĩ }.
  - (iii) No PME for i > 0.
- Closed-form money demand in MME preserved as in source (equation (12)).

### Generalized sampling and parametric examples
- General firm profit for sampling up to N:
  - Πt(p) = bt (p − ct) Σ_{n=1}^N βn (n choose 1) ^Ft(p)^{n−1}
- First-order conditions for MME:
  - Fixed-cost: Σ_{n=1}^N βn ε H′(z_i) = i.
  - Variable-cost: Σ_{n=1}^N βn ς [1 − H(z_i)] = i.
- Poisson n (βn = e^{−λ} λ^n / n!):
  - Ft(p) = 1 − (1/λ) [ log( ct p̄t − ct ) − log( ct p − ct ) ] (as in source).
  - Fixed-cost money demand: ^z_i = m̄ + [ e^{−λ} ε (m̄ − ε − ct) ]^{1/2} i^{−1/2}
  - Variable-cost money demand: ^z_i = m̄ + (m̄ − ct) ς e^{−λ} / ( (1 + ς) i + ς e^{−λ} )
- Logarithmic n (βn = −θ^n / (n log(1 − θ))):
  - Ft(p) and closed-form money demand expressions as preserved: fixed-cost ^z_i = m̄ − ε / ( i log(1 − θ) ), variable-cost ^z_i = m̄ + (m̄ − ct) (1 − θ) i^{1/ς} / ( 1 + ς − ς (1 − θ) i^{1/ς} ).

### Repricing rule and apparent nominal stickiness
- Period-to-period density shifts right with π > 0; tie-breaking repricing rule (Head et al. 2012 style) preserved:
  - If pt ∈ F_{t+1} then p_{t+1}(pt) = p̂; else p_{t+1}(pt) = { pt with prob. ξ; p̂ with prob. 1 − ξ }.
- Calibration uses ξ = 0:90 as baseline (indifferent sellers change prices only 10% of the time).
- Unique symmetric equilibrium in repricing rule; repricing distribution Rt+1(p) given by equation (19) in source.

### Calibration targets and parameter anchors (quantitative section)
- Data period for price-change observations: 1988-2004; model period = month.
- Money demand target (Lucas and Nicolini (2012) M1J quarterly):
  - Average annualized nominal rate: E i = 0.041
  - Implied L_{Ei} = 0.277
  - Implied elasticity η_{Ei} = −0.116
  - Alternate longer sample: E i = 0.038, L_{Ei} = 0.279, η_{Ei} = −0.149
- Markup targets:
  - Low-end gross margins: between 1:17 and 1:21.
  - High-end gross margins: between 1:42 and 1:44.
  - Target gross margin: 1:3 → implies markup 1:39 (Bethune et al. 2014).
  - Aggregate markup target: 1:1 (Basu and Fernald 1997); with BJ markup 1:39 and AD markup 1:0, BJ market ≈ 25% of total output.
- Money vs. credit shares in payments:
  - Empirical credit-card share target ≈ 20% (evidence: Klee 2008 grocery data 12%; Boston Fed 2012 survey 22% and diary 17%; Bank of Canada 19%).
- Price-change facts (Klenow and Kryvtsov 2008):
  - Average duration of price spells: 8:6 (months).
  - Average absolute price change: 11:3 percent.
- Utility specification: U(x) = log(x) → x* = 1.
- Average duration between price changes used to pin down ξ.

### Basic findings and calibration comparison (Section 5.2)
- Fixed-cost calibration ("Fix" row preserved parameters):
  - Hits all macro targets except fraction of credit transactions given MME constraint.
  - Using smallest ε consistent with MME at maximum observed i = 0:103 yields 11:9% credit transactions.
  - This 11:9% ≈ 4:7% of BJ utility parameter  (mainly from matching average real balances).
  - BJ utility and parameters (preserved numbers): baseline probabilities pr(n=1) = 0:013, pr(n=2) = 0:081; tie-breaking ξ = 0:90.
  - Baseline calibration row labeled "Fix": 8:62 2:91 0:40 40:013 0:081 0:90 (values preserved from source).
- Variable-cost calibration ("Var" row preserved parameters):
  - Approximates all targets well including 20% BJ credit share.
  - Trade surplus   lower than fixed-cost; E_H q ≈ 4:21; average transaction cost E_H max(0;q z) ≈ 0:017.
  - Scaled: E_H max(0;q z)== 0:0029 (less than average credit card fees 1:5-2%).
  - Tie-breaking ξ constant at 0:90.
  - Baseline calibration row labeled "Var": 5:93 3:14 0:20 20:034 0:048 0:90.

### Money demand fit
- Both fixed- and variable-cost specifications fit money demand well; differences emerge at low i.
- Variable-cost specification matches money demand and micro payment data simultaneously; fixed-cost struggles under calibration constraints.

### Sticky prices — model fit to price-change statistics
- Average absolute price change (Klenow-Kryvtsov data vs models):
  - Data: 11:3%
  - Fixed-cost model: 20:3%
  - Variable-cost model: 12:3%
- Fraction of small changes (<5%):
  - Data: 44%
  - Fixed-cost model: 28%
  - Variable-cost model: 31%
- Fraction of big changes (>20%):
  - Data: 16%
  - Fixed-cost model: 34%
  - Variable-cost model: 21%
- Fraction of negative changes:
  - Data: 37%
  - Both models: 43%
- Hazard rate pattern:
  - Variable-cost model hazard falls with duration for a while (consistent with some empirical studies) but lacks enough action at low durations; hazard becomes U-shaped over very long horizons (turns up ≈ 4 years).
  - Even at 10 years model hazard reaches only 12:35%.
- Sensitivity to tie-breaking ξ:
  - ξ = 0:90 baseline; higher ξ increases expected duration.
  - Example: ξ = 0:95 gives expected duration 16 months and fits better than benchmark 8:6 months; extreme ξ = 0:9999 reduces fraction of negative changes to 1:5%.
- Sensitivity to inflation :
  - Increasing  reduces fraction of negative adjustments and increases frequency and size of price changes.
- If money-demand fit is abandoned, model fits price-change data much better; the paper emphasizes simultaneous fit to monetary and price-change facts.

### Welfare effects of inflation (Section 6.2) — preserved magnitudes and mechanisms
- Welfare metric: percent change in consumption equivalent to changing  to 0 (equation (20) preserved in source).
- Welfare cost monotone in inflation; MME exists for  from Friedman rule  = 1 up to ≈9% (fixed-cost) or 20% (variable-cost).
- Magnitude:
  - Eliminating  = 0:10 annual inflation is worth only 0:23% of consumption in baseline (variable-cost).
- Intuition: welfare effects are smaller than in some prior models because inflation mainly affects the cash–credit margin (usage of credit) rather than intensive or extensive margins of trade.
- Markups and dispersion:
  - Fixed-cost:  does not affect G(q), markups, or dispersion.
  - Variable-cost: higher  makes G(q) first-order stochastically smaller; average markup and coefficient of variation decline with ; both q and q fall with  but q falls faster.

### Endogenous participation (Section 6.3) — entry, money demand, and welfare
- Buyers choose to incur entry cost k; W(A) = max{W1(A); W0(A)} and partial-entry equilibrium simplifies to 	 = k with expected surplus expression preserved.
- RB (money demand) and FE (free entry) curves:
  - RB decreasing and convex; FE concave; unique MME at intersection.
  - Higher  shifts both curves toward origin → fewer buyers enter, lower BJ output.
- Monetary surprise neutrality differs from constrained-price models: a surprise jump in M does not change G(q) or b in this search model.
- Welfare: endogenizing participation roughly doubles the welfare cost of inflation relative to benchmark because inflation both raises credit resource costs and reduces participation.
- Effects on markups, dispersion, payment methods similar to variable-cost baseline: BJ markup and coefficient of variation decline with ; credit usage rises with .

### Dynamics (Section 6.4) — nonstationarity, cycles, chaos, sunspots
- Money provides liquidity and can generate nonstationary equilibria due to self-referential liquidity valuation.
- Interpretations of ζ (return on m):
  - ζ > 0: m as share in a dividend-bearing technology.
  - ζ < 0: m as storage cost.
  - ζ = 0: fiat currency.
- Euler equation (22) (notation preserved):
  - ϵt = ϵt+1 + ζ/(1 + r) [1 + (β1/(1 + β2)) ς Ĥ(^z)]
  - Liquidity premium (β1 + β2) ς Ĥ(^z) augments asset-pricing.
- Phase dynamics and multiplicity:
  - Calibrated example with ζ = 0: unique steady-state MME at ϕ* = 4:4; monetary steady state unstable; nonmonetary steady state stable; equilibria exist where ϕ → 0.
  - Reducing β1 from 0:034 to 0:0001 yields unique steady state MME with ϕ* = 3:14; B′ < −1 at monetary steady state generates a period-2 cycle (ϕ oscillates between ϕL and ϕH) as a self-fulfilling prophecy.
- Higher-order cycles and chaos:
  - Third iterate B^3(ϕ) can exhibit multiple intersections → existence of 3-cycles.
  - By Sarkovskii and Li–Yorke results, existence of a 3-cycle implies existence of N-cycles for 8N and chaotic dynamics.
- Stochastic (sunspot) equilibria:
  - Introducing an extrinsic two-state Markov process s with transitional probability "s gives stochastic equilibria; continuity argument from deterministic cycles yields equilibria for "s < 1.
- Role of ζ:
  - ζ > 0 can eliminate ϕ = 0 equilibrium; qualitative dynamics persist for ζ not too large.
  - ζ < 0 can generate two positive steady states with lower one stable, allowing sunspot equilibria about that steady state.
- Main takeaway: costly credit suffices to generate cyclic, chaotic, and stochastic dynamics; fluctuations in price distributions F(p) and payment method usage can be driven by endogenous liquidity and credit-cost expectations without relying on fiat money alone.

### Appendix B — calibration formulae (variable- and fixed-cost models)
- Variable-cost expressions (preserved formatting and variables):
  - G(q)   =   1 

  1
  2
  2
   q+(^z
  i
  q)
  (1 +) (q )
  - H(q)   =   1 

  2
  1
  [ q+(^z
  i
  q)] [+^z
  i
  + (q 2) (1 +)]
  4
  2
  (
  1
  +
  2
  ) (1 +)
  2
  (q )
  2
  - Fraction of monetary transactions and markup expressions as given (H(^z_i), E_G q, L_i, _i) with explicit dependence on ^z_i and parameters.
- Fixed-cost expressions (preserved formatting and variables):
  - G(q)   =   1 

  1
  2
  2
    q
  q 
  - H(q)   =   1 

  2
  1
  (  q) ( +q 2)
  4
  2
  (
  1
  +
  2
  ) (q )
  2
  - Fraction of monetary transactions and markup expressions and L_i, _i formulas preserved as in source.
- Key calibration sensitivities and derivative notation preserved (e.g., @^z_i/@i ^z_i=i).

*Source: wp1714*

### 1. Introduction ........................................................................................................

### 1. Introduction

### Overview and objectives
- Develop a theory of money and credit as competing payment instruments using a New Monetarist approach that explicitly models exchange/payment processes.
- Integrate Burdett-Judd "noisy" search (sellers post prices; each buyer sees a random number of them) into the Lagos and Wright (2005) alternating centralized (AD) and decentralized (BJ) markets framework.
- Introduce both fixed and proportional transaction costs for credit: C(d_t) = 1(d_t) + d_t, and show these specifications perform differently; an unanticipated finding is that the proportional (variable) cost specification outperforms the fixed-cost specification in theory and data.
- Aim to match micro facts on payment methods and macro facts on money demand together with price-change statistics and to use the model in several applications (sticky prices, welfare, participation, dynamics).

### Key mechanisms
- Cash is subject to an inflation tax; credit involves transaction costs (fixed and/or proportional).
- Buyers in BJ must use cash at point of sale unless they access credit technology by incurring cost C(d_t), which allows them to purchase by promising d_t dollars in the next AD.
- Noisy search and price posting imply a distribution of prices F(p); any p in the nondegenerate support yields the same profit for a seller because lower per-unit margin is offset by larger volume.
- Increasing money supply M shifts F(p) so that the real distribution stays the same; overlapping supports imply some firms can keep the same nominal p as M rises, producing apparent nominal price stickiness despite costless price adjustment.
- Costly credit generates a simple money demand and resolves an indeterminacy present in similar models with indivisible goods and posting (continuum of stationary equilibria).

### Quantitative findings and calibration targets
- Eliminating  =  10% inflation is worth only 0:23% of consumption in the baseline setting where welfare effects operate mainly through the cash–credit margin.
- The fixed-cost specification can match standard money demand observations but not the simultaneous facts on money and credit shares in payment data; the proportional-cost specification can match both.
- The model can match salient price-change facts qualitatively and quantitatively:
  - long durations,
  - large average changes,
  - many small changes,
  - many negative changes,
  - a decreasing hazard,
  - repricing behavior that depends on inflation.
- When freed from discipline of matching money demand, the model can fit price-change data almost perfectly, but the paper emphasizes simultaneous fit to monetary facts.

### Relations to existing literature and empirical facts
- Resolves a technical indeterminacy that appears in Burdett-Judd price-posting embedded in monetary economies with indivisible goods by introducing costly credit: holding more cash reduces expected costly credit use and yields a well-behaved money demand and unique equilibrium.
- Contrasts with approaches that assume divisible goods or linear menus to avoid indeterminacy.
- Connects to empirical findings on price durations and changes:
  - Campbell and Eden (2014): average duration between price changes in grocery-store data of 10 weeks.
  - Bils and Klenow (2004): at least half of prices last less than 4:3 months, or 5:5 months excluding sales.
  - Klenow and Kryvtsov (2008): durations from 6:8 to 10:4 months.
  - Nakamura and Steinsson (2008): report 8 to 11 months, excluding substitutions and sales.
  - Eichenbaum et al. (2011): duration of 11 months for reference prices.
  - Cecchetti (1986): magazine price durations from 1:8 months to 14 years.
  - Carlton (1986): wholesale price durations from 5:9 to 19:2 months.
- Empirical stylized facts the model is consistent with (as summarized in discussions):
  1. Prices change slowly, but exact durations vary across studies.
  2. Frequency and size of changes vary across goods.
  3. Two sellers changing at the same time do not typically pick the same ^p.
  4. Many changes are negative.
  5. Hazards decline slightly with duration.
  6. Many small (below 5%) and many big (above 20%) changes.
  7. Frequency and size of changes, and fraction of negative changes, vary with inflation.
  8. There is price dispersion even at low inflation.

### Welfare, markups, and money-demand connections
- Welfare effects of inflation in this model are smaller than in some prior models (e.g., Lagos and Wright (2005) or some search-and-bargaining models); example: eliminating  = 0:10 yields a welfare gain of 0:23% of consumption in baseline.
- Inflation decreases markups by directly affecting the cash–credit choice; model predictions on markups and price dispersion are consistent with some empirical evidence.
- The model yields closed-form money-demand solutions reminiscent of Baumol-Tobin: opportunity cost of holding cash versus cost of tapping financial services, but in a general-equilibrium setting that allows study of monetary phenomena like inflation as a self-fulfilling prophecy.

### Environment and model setup (summary)
- Time is discrete t = 1,2,...; each period has two subperiods:
  - BJ (decentralized Burdett-Judd market): trade indivisible goods y_t, sellers post nominal prices, buyers sample n firms (random sampling with probabilities _n); buyers use cash or credit subject to cost C(d_t).
  - AD (frictionless Arrow-Debreu centralized market): agents consume divisible good x_t, supply labor `t, adjust cash balances, and settle accounts.
- Sets and measures:
  - Measure 1 of firms (retailers).
  - Measure  of households (buyers).
- Costs and preferences:
  - BJ good produced at unit cost  ≥ 0.
  - Household per-period utility U(x_t) + 1(y_t) − `t, with U'(x_t) > 0 > U''(x_t),  >  + .
  - Discount factor between AD today and BJ tomorrow is  = 1/(1 + r), r > 0.
- Credit transaction cost: C(d_t) = 1(d_t) + d_t, with fixed cost  and proportional cost ; 1(d_t) is indicator of d_t > 0.
- Buyers in BJ sample n firms: baseline assumes _0, _1, _2 > 0 and _n = 0 for n ≥ 3; Section 4.3 generalizes this.
- Money supply per buyer evolves M_{t+1} = (1 + ) M_t; AD implements changes via lump-sum taxes if  > 0 or transfers if  < 0; AD price of money in numeraire is _t.
- Stationary equilibrium:
  -  is the inflation rate.
  - Nominal interest rate satisfies 1 + i = (1 + ) (1 + r).
  - Model growth rate satisfies  > ^{-1} − 1, and the Friedman rule corresponds to  → ^{-1} − 1.

*Source: wp1714 - 1. Introduction*

### 3.1  Firm Problem

### wp1714 - 3.1  Firm Problem

### Firm problem and equilibrium price distribution
- Profit for a firm posting p is
  - Πt(p) = bt ht β1 + 2β2 ^Ft(pt) i (p − ct)
  - where ^Ft(p) ≡ 1 − Ft(p), bt denotes the measure of participating households in the BJ market (tightness), and for now bt = b.
  - Net revenue per unit is p − ct; the number of units sold comes from sampling probabilities explained in text (probability a household contacts this firm alone is β1, contacts this firm plus another is 2β2, and if two contacts the firm wins with probability ^Ft(p)). All multiplied by bt.
- Profit maximization implies every p ∈ Ft yields same profit. Let Ft = [pt; p̄t] and assume F is continuous.
- Profit from posting the highest price p̄t is
  - Πt(p̄t) = bt β1 (p̄t − ct).
- Equating profits yields the equilibrium price distribution:
  - Lemma 1: for p ∈ Ft = [pt; p̄t],
    - Ft(p) = 1 − (β1 / (1 + 2β2)) ( (ct p̄t − ct p) / (p̄t − p) ).
- Additional checks:
  - F′t(p) > 0 and F′′t(p) < 0.
  - Using Ft(pt) = 0 gives
    - pt = (β1 / (1 + 2β2)) p̄t + (2β2 / (1 + 2β2)) ct.
- Real price numeraire: qt = ct pt, real price distribution
  - Gt(q) = 1 − (β1 / (1 + 2β2)) ( (q̄t − q) / (q̄t − q) )
  - Support Gt = [qt; q̄t]; ^Gt(qt) ≡ 1 − Gt(qt).

### Household problem and value functions
- Stationary equilibrium: real variables constant, nominal variables grow at rate π.
- AD state variable (net worth): A = m̂ − d − C(d) + I, where m̂ and d are real money balances and real debt carried into BJ, C(d) transaction cost of using credit, and I other income (includes transfers net of taxes plus profit).
- BJ state variable: real balances z.
- AD and BJ value functions W(A) and V(z) satisfy:
  - W(A) = max_{x,ℓ,z} { U(x) − ℓ + β V(z) s.t. x = A + ℓ − (1 + π) z }.
  - Eliminating ℓ and letting x* solve U′(x*) = 1, get
    - W(A) = A + U(x*) − x* + β max_{z} Oi(z)
    - where Oi(z) ≡ V(z) − (1 + i) z and i given by the Fisher equation.
- Lemma 2:
  - W′(A) = 1 and the choice of z does not depend on A.
- BJ value function:
  - V(z) = W(A) + (β1 + β2) [ m̄ − E_H q − ε ^H(z) − ς E_H max(0; q − z) ]
  - where ^H(q) ≡ 1 − H(q), H(q) is the CDF of transaction prices:
    - H(q) = [ β1 G(q) + β2 (1 − ^G(q))/2 ] / (β1 + β2)
  - Interpretation: benefit of higher z is reducing expected cost of tapping credit; costs ε and ς(q − z) are paid if q > z.

### Equilibrium definitions and characterization of q̄
- Definition 1: stationary equilibrium is a list {G(q); z} such that: given G(q), z solves households' problem; given z, G(q) solves firms' problem with q̄ determined as in Lemma 3.
- Definition 2: equilibria classes
  - NME (nonmonetary equilibrium): z = 0 (all BJ trades use credit).
  - MME (mixed monetary equilibrium): 0 < z < q̄ (cash for q ≤ z, credit for q > z).
  - PME (pure monetary equilibrium): z ≥ q̄ (all BJ trades use cash).
- Lemma 3 (characterization of q̄):
  - In NME: z = 0 and q̄ = (m̄ − ε) / (1 + ς).
  - In MME: z ∈ (0; m̄ − ε) and q̄ = (m̄ − ε + ς z) / (1 + ς).
  - In PME: q̄ = z ≥ m̄ − ε.
- Lemma 4:
  - In MME, Oi(z) is continuous, smooth and strictly concave in z ∈ (q; q̄), and linear at z = q or z = q̄.

### Fixed-cost (ε > 0, ς = 0) equilibrium and money demand
- Fixed-cost setup: ε > 0, ς = 0. If ε < m̄ − ct there is an NME where all transactions use credit (similar to Burdett-Judd with deferred settlement).
- Objective Oi(z) is linear for z ∈ [q; q̄] with slope O′i(z) = −i < 0; O′′i(z) < 0 for z ∈ (q; q̄).
- First-order condition for interior MME (z_i ∈ (q; q̄)):
  - (β1 + β2) ε H′(z_i) = i.
- Global maximizer ^z_i conditions described (left/right derivatives); this leads to:
- Proposition 1 (fixed-cost, βn = 0 for n ≥ 3):
  - (i) NME exists for high i.
  - (ii) MME exists iff ε < ε̄ and i ∈ (i; ī).
  - (iii) PME exists under specified ranges of ε and i (thresholds satisfy q̄2(i;1), i = ε β2 / (β1 2β2) (m̄ − ε − ct) and ε̄ = (m̄ − ct) (2β2^2 + 2β1 β2) / (2β2^2 + 2β1 β2 − β2 β1).
  - (text preserves the threshold expressions as in source)
- Money demand in MME (closed form):
  - ẑ_i = m̄ + [ (β2 / β1) ε (m̄ − ε − ct)^2 ]^{1/3} i^{−1/3}
  - expressed in source as:
    - ^z_i = m̄ + 2 β2 1 ε (m̄ − ε − ct)^2 = 2 β2 ^{1/3} i^{−1/3} (equation (11) formatting preserved from source).

### Variable-cost (ς > 0, ε = 0) equilibrium and money demand
- Variable-cost setup: ς > 0, ε = 0. Oi(z) is differentiable everywhere including at bounds, simplifying analysis and avoiding lotteries.
- Price distribution results:
  - q̄ = (m̄ + z ς) / (1 + ς)
  - q = [ β1 (m̄ + z ς) + 2 β2 ct (1 + ς) ] / [ (β1 + 2β2) (1 + ς) ]
- Equilibrium regimes:
  - If i > (β1 + β2) ς, unique NME.
  - If i < (β1 + β2) ς, unique MME.
  - PME cannot exist.
- Proposition 2 (variable-cost, βn = 0 for n ≥ 3):
  - (i) NME iff ς ≤ m̄ / ct − 1.
  - (ii) MME iff i < min{ ς (β1 + β2), ĩ }.
  - (iii) No PME for i > 0.
  - ĩ(ĩ(ς)) is the nominal rate that drives buyers' payoff to 0.
- Closed-form money demand in MME (equation (12) from source):
  - ^z_i = m̄ + (m̄ − ct) [ ς + (1 + ς) sqrt( 1 + 4 β2 / β1 (1 + ς) ) ]^{...} (preserve exact expression as in source)
  - (see source equation (12) for exact preserved formula).

### Generalized sampling distributions (N up to arbitrary)
- General profit expression for a firm posting p, sampling up to N prices:
  - Πt(p) = bt (p − ct) Σ_{n=1}^N βn (n choose 1) ^Ft(p)^{n−1}
- Profit equality with p̄t leads to general Ft(p) via equation (14) in source.
- First-order conditions for MME in fixed- and variable-cost models:
  - Σ_{n=1}^N βn ε H′(z_i) = i (fixed-cost).
  - Σ_{n=1}^N βn ς [1 − H(z_i)] = i (variable-cost).
- Parametric examples:
  - Poisson for n: βn = e^{−λ} λ^n / n!
    - Profit simplifies: Πt(p) = bt (p − ct) λ e^{−λ Ft(p)}
    - Ft(p) = 1 − (1/λ) [ log( ct p̄t − ct ) − log( ct p − ct ) ]
    - Propositions 3 and 4 give analogs to Propositions 1 and 2 with thresholds and closed-form money demand:
      - Fixed-cost money demand: ^z_i = m̄ + [ e^{−λ} ε (m̄ − ε − ct) ]^{1/2} i^{−1/2}
      - Variable-cost money demand: ^z_i = m̄ + (m̄ − ct) ς e^{−λ} / ( (1 + ς) i + ς e^{−λ} )
  - Logarithmic for n: βn = −θ^n / (n log(1 − θ))
    - Ft(p) = 1 − ( ct (p̄t − p) )^θ / ( ct (p̄t − ct) )
    - Propositions 5 and 6 give analogs; closed-form money demand:
      - Fixed-cost: ^z_i = m̄ − ε / ( i log(1 − θ) )
      - Variable-cost: ^z_i = m̄ + (m̄ − ct) (1 − θ) i^{1/ς} / ( 1 + ς − ς (1 − θ) i^{1/ς} )
- For tractability in applications, revert to N = 2.

### Repricing behavior and price stickiness (repricing rules)
- With positive inflation π > 0, period-to-period density shifts right: F′t+1 lies to right of F′t.
- Firms with p < pt+1 must reprice; firms with p > pt+1 may remain unchanged if still profit-maximizing.
- Tie-breaking repricing rule (Head et al. 2012 style):
  - If pt ∈ F_{t+1} then p_{t+1}(pt) = p̂ (new price); if pt ∉ F_{t+1} then:
    - p_{t+1}(pt) = { pt with prob. ξ; p̂ with prob. 1 − ξ }
  - Tie-breaking parameter ξ set in calibration; example uses ξ = 0.90 so 10% of indifferent firms change price.
- Unique symmetric equilibrium in repricing rule; repricing distribution Rt+1(p) given by equation (19) in source.
- Implication: model can match average repricing behavior without Calvo or menu-cost assumptions; once ξ is pinned down, precise predictions follow.

### Quantitative results: targets, data, and calibration
- Period: focus on 1988-2004 for price-change observations; model period = month (price-change data monthly), money demand aggregated to quarterly.
- Money demand target from Lucas and Nicolini (2012) M1J quarterly series:
  - Average annualized nominal rate: E i = 0.041
  - Implied L_{Ei} = 0.277
  - Implied elasticity η_{Ei} = −0.116
  - (Alternate longer annual sample: E i = 0.038, L_{Ei} = 0.279, η_{Ei} = −0.149; similar results)
- Markup targets from U.S. Census Annual Retail Trade Report 1992-2008:
  - Low-end gross margins: between 1:17 and 1:21.
  - High-end gross margins: between 1:42 and 1:44.
  - Target gross margin: 1:3 → implies markup 1:39 (Bethune et al. 2014).
  - Aggregate markup target: 1:1 (Basu and Fernald 1997). With BJ markup 1:39 and AD markup 1:0, BJ market ≈ 25% of total output.
- Money vs. credit shares in payments:
  - Interpret monetary transactions as cash, check, debit (liquid demand deposits).
  - Empirical targets: credit-card share ≈ 20% (evidence: Klee 2008 grocery data 12%; Boston Fed 2012 survey 22% and diary 17%; Bank of Canada 19%).
- Price-change facts (Klenow and Kryvtsov 2008):
  - Average duration of price spells: 8:6 (months).
  - Average absolute price change: 11:3 percent.
- Calibration notes:
  - Utility U(x) = log(x) → x* = 1.
  - Model fits money demand L_i = ẑ_i / Y, with Y = x* + (β1 + β2) E_H q.
  - Average duration between price changes pins down ξ in tie-breaking rule.

*Italicized source: wp1714 - 3.1  Firm Problem (source PDF: wp1714 - 3.1  Firm Problem).*

### 5.2  Basic Findings

### 5.2  Basic Findings

### Calibration: fixed-cost versus variable-cost specifications
- Fixed-cost model:
  - Hits all targets except the fraction of credit transactions because the parameter search is constrained to stay within the region where MME exists.
  - Trying to get 20% BJ credit transactions forces parameters into a region where MME does not exist for some sample values.
  - Uses the smallest value consistent with MME at the maximum observed i =  0:103, which yields 11:9% credit transactions.
  - This 11:9% is about 4:7% of the BJ utility parameter , which comes primarily from matching average real balances.
  - The value of , about one-third of , comes primarily from the BJ markup.
  - The probability of sampling one price (two prices) in BJ is 
    1
    = 0:013 (
    2
    = 0:081).
  - Tie-breaking parameter  = 0:90, implying indifferent sellers change prices only 10% of the time.
  - Baseline calibration table row labeled "Fix": 8:62 2:91 0:40 40:013 0:081 0:90 (corresponding to BJ utility, BJ cost, credit cost, pr(n=1), pr(n=2), tie breaker).

- Variable-cost model:
  - Approximates all targets very well, including 20% for BJ credit.
  - Trade surplus   is lower than in the fixed-cost case, so BJ goods are less important relative to AD goods.
  - With E
    H
    q around 4:21, the average transaction cost E
    H
    max (0;q z) is about 0:017.
  - Scaled by BJ utility, E
    H
    max (0;q z)==  0:0029, which is less than average credit card fees, which are 1:5-2% without counting small fixed costs per transaction.
  - A constant across specifications is tie-breaking parameter  = 0:90.
  - Baseline calibration table row labeled "Var": 5:93 3:14 0:20 20:034 0:048 0:90.

### Money demand (Figure 6)
- Money demand is plotted with:
  - Solid curve from the fixed-cost model.
  - Dashed curve from the variable-cost model.
- The fit is good in both cases, with differences at low values of i; these differences matter for other issues but not materially for the applications in the paper.
- Conclusion: The variable-cost specification can match both money demand and micro payment data well; the fixed-cost model struggles with micro payment data given the calibration method.

### Sticky prices (Section 6.1) — price-change distribution and hazards
- Price-change distribution (Klenow-Kryvtsov data versus model predictions):
  - Average absolute change:
    - Data: 11:3%
    - Fixed-cost model: 20:3%
    - Variable-cost model: 12:3%
  - Fraction of small changes (below 5% absolute):
    - Data: 44%
    - Fixed-cost model: 28%
    - Variable-cost model: 31%
  - Fraction of big changes (above 20% absolute):
    - Data: 16%
    - Fixed-cost model: 34%
    - Variable-cost model: 21%
  - Fraction of negative changes:
    - Data: 37%
    - Both models: 43%
- Hazard rate (probability of changing p as a function of time since last change):
  - Variable-cost model hazard slopes downward for a while (consistent with Nakamura and Steinsson (2008)), but does not generate enough action at low durations.
  - Hazard turns up at around 4 years; the hazard is U-shaped over a longer horizon because continuing inflation means any p eventually falls out of the equilibrium support.
  - Even at 10 years, model hazard is only up to 12:35%.
- Sensitivity to duration/tie-breaking parameter  (Figures 9 and 10):
  - Left panel: '0 and expected duration 1 month.
  - Right panel:  = 0:95 and expected duration 16 months.
  - Right panel fits better than benchmark duration of 8:6 months.
  - With too much stickiness (e.g.,  = 0:9999) the fraction of negative changes drops to 1:5%.
- Sensitivity to inflation  (Figure 10):
  - Left panel sets  to 0; right panel to 20%.
  - As  increases:
    - Fraction of negative adjustments falls.
    - Frequency and size of price changes increase.
  - These implications align with evidence in Klenow and Kryvtsov (2008) and are hard to explain with simple Calvo models.
- Ignoring money demand (Figure 11):
  - If calibration gives up matching money demand, the fit to price-change statistics improves markedly.
  - Conclusion: Search models can capture sluggish nominal prices quantitatively; imposing macro and micro discipline reduces fit but models still capture broad facts.

### Welfare effects of inflation (Section 6.2)
- Welfare measure: percent change in consumption equivalent to changing  from a given level to 0, with welfare expressed as equation (20):
  - Y (
    1
    +
    2
    )
    
    [1 H
    
    (z
    
    )] +
    Z
    q
    z
    
    (q z
    
    )dH
    
    
    ;
  - Where Y = U(x
    
    ) x
    
    + (
    1
    +
    2
    ) ( ) adds the AD and BJ surpluses; remaining terms subtract resource costs of credit.
- Figure 12: Welfare cost monotonically increases with inflation .
  - Range of horizontal axis is range over which MME exists, from Friedman rule  = 1 up to about 9% with a fixed cost or 20% with a variable cost.
- Magnitude of welfare effects:
  - With a variable cost, eliminating 10% annual inflation is worth .0:23% of consumption.
  - This is less than estimates in Lucas (2000) and much less than Lagos and Wright (2005).
  - Intuition: Changes in  do not affect the intensive margin (good indivisibility) nor the extensive margin (fixed population of participants); welfare effects mainly arise because inflation increases usage of credit.
- Inflation, markups and price dispersion (Figure 13):
  - Fixed-cost:  does not affect G(q), markups, or dispersion.
  - Variable-cost:  implies G(q) decreases with  in the sense of first-order stochastic dominance.
    - Consequently average markup and dispersion (coefficient of variation) both decrease with .
    - Both q and q fall with , but q falls faster.
  - Empirical evidence on inflation and dispersion/markups is mixed; model can match findings in Reinsdorf (1994) and Benabou (1992).

### Endogenous participation (Section 6.3)
- Buyers choose whether to participate in the BJ market at cost k > 0; output depends directly on inflation.
- Value functions:
  - W1(A) and W0(A) are AD value functions for households that enter and do not enter the next BJ market, respectively; W(A) = max{W1(A); W0(A)}.
  - In partial-entry equilibrium W(A) = W1(A) = W0(A), which simplifies to 	 = k, where expected surplus from participation is:
    - 	(
      1
      +
      2
      ) [ E
      H
      q E
      H
      max (0;q z)] i^z
      i
      .
- Buyers' arrival rates depend on buyer-seller ratio b via market tightness:
  n
  =
  n
  (b
  t
  ).
- Entry condition and curves (Figure 14):
  - Buyers' money demand ^z
    i
    now depends on b (RB curve).
  - 	 = k is the FE (free entry) curve.
  - RB is decreasing and convex; FE is concave; unique MME where RB and FE intersect.
  - Higher inflation  shifts both curves toward the origin, reducing buyer entry and BJ output.
- Comparison with constrained-price models:
  - In constrained-price models a surprise jump in M has real effects (nominal distribution F(p) does not change enough).
  - In this search model, a surprise jump in M affects neither G(q) nor b; monetary surprises are neutral.
  - Policy implication: Observing a fraction of sellers adjusting prices each period is not sufficient to infer that monetary expansions have real effects—the reason for price stickiness matters.
- Welfare with entry (Figure 15):
  - Endogenizing participation approximately doubles the welfare cost of inflation compared to benchmark, because inflation both increases resources used for credit and decreases participation.
- Effects on markups, dispersion, payment methods with entry (Figure 16):
  - Endogenizing participation does not substantially change the impact of inflation on markup or price dispersion relative to the baseline variable-cost model.
  - Fewer buyers enter the BJ market at higher , but higher arrival rates for entrants attenuate their reduction in real balances.
  - Figures show:
    - BJ markup and coefficient of variation decline with .
    - Average money payment percentage and average credit payment percentage per transaction respond as in the variable-cost baseline, with credit usage rising with .

*Source: wp1714 - 5.2  Basic Findings*

### 6.4  Dynamics

### 6.4  Dynamics

### Nonstationarity, liquidity, and asset returns
- Money conveys liquidity and can generate nonstationary equilibria because liquidity is at least partially self-referential: "how much you are willing to give for an asset depends on how much you think others will give in the future."
- Interpretations of m's return (ζ):
  - If ζ > 0, m can be interpreted as a share in a technology or "tree" bearing a dividend of "fruit" in terms of the numeraire.
  - If ζ < 0, m can be interpreted as a storage cost, making m a poor saving vehicle but potentially valuable as a medium of exchange.
  - ζ = 0 corresponds to fiat currency.
- In this section the authors set k = 0 so b̄ = b, focus on the variable-cost model, and keep the supply M fixed.

### Asset-pricing dynamics and Euler equation
- Household problem recast with A = ζm + ϵm − d − C(d) + I including dividend income ζm; Oi(z) = V(z) − (1 + r)z.
- Euler equation (22) (nonstationarity of ϕt not imposed):
  - ϵt = ϵt+1 + ζ/(1 + r) [1 + (β1/(1 + β2)) ς Ĥ(^z)]
- Interpretation:
  - If β1 = β2 = 0 (i.e., no BJ market), (22) becomes a standard asset-pricing equation with unique equilibrium ϵt = ζ = r8t; alternative solutions are explosive and violate transversality.
  - More generally, RHS is augmented by a liquidity premium capturing expected reduction in credit costs, (β1 + β2) ς Ĥ(^z), which dramatically changes the equilibrium set.

### Phase dynamics, cycles, and multiplicity
- The dynamical system maps ϕt in terms of ϕt+1: ϕt = B(ϕt+1).
- Calibrated examples and steady states:
  - With calibrated parameters including ζ = 0, a unique steady state MME at ϕ* = 4:4 (i.e., 4.4). The monetary steady state is unstable; the nonmonetary steady state is stable. There exist equilibria where ϕ → 0; inflation can operate as a self-fulfilling prophecy.
  - Reducing β1 from 0:034 to 0:0001 yields a unique steady state MME with ϕ* = 3:14 (i.e., 3.14). Because B′ < −1 at the monetary steady state, B and B^{-1} cross the 45° line at (ϕL; ϕH) and (ϕH; ϕL), producing a period-2 cycle where ϕ oscillates between ϕL and ϕH as a self-fulfilling prophecy.
- Intuition for period-2 cycle:
  - If ϕt+1 = ϕL is low then liquidity is scarce at t+1, making buyers want more of the asset at t and driving ϕt = ϕH high.
  - Cyclic equilibria are familiar in monetary/search models and differ conceptually from OLG explanations (e.g., backward-bending labor supply or savings functions).

### Higher-order cycles, chaos, and sunspots
- With the same parameters that generate a 2-cycle, the third iterate B^3(ϕ) can have 6 intersections with the 45° line in addition to the steady state, implying the existence of pairs of 3-cycles.
- Standard results imply:
  - Existence of a 3-cycle implies existence of N-cycles for 8N (Sarkovskii theorem) and chaotic dynamics (Li–Yorke theorem).
  - Thus a large set of perfect-foresight dynamics can be generated for parameter values fairly close to calibration.
- Stochastic (sunspot) equilibria:
  - For these parameters there are also stochastic (sunspot) equilibria with random fluctuations in ϕ, F(p) and other endogenous variables, illustrating how costly credit can generate excess volatility as a self-fulfilling prophecy.
  - Sketch of existence proof: introduce an extrinsic two-state Markov process s ∈ {s1, s2} with transition prob "s = prob(st+1 ≠ s | st = s). If "1 = "2 = 1 this reduces to a 2-cycle; by continuity there are equilibria for "s < 1.

### Role of ζ (return on asset), multiplicity, and stability
- Setting ζ = 0 might suggest multiplicity because two steady states exist (ϕ > 0 and ϕ = 0). However:
  - The equilibrium with ϕ = 0 can be eliminated by setting ζ > 0; for ζ not too big, by continuity qualitative results (cycles, etc.) persist.
  - Multiple steady states are not necessary for complicated dynamics; dynamics can be fluctuations around one steady state.
- If ζ < 0 (example ζ = −0:4), the model can have two steady states ϕ1 and ϕ2 with ϕ2 > ϕ1 > 0; the lower steady state ϕ1 can be stable, allowing construction of sunspot equilibria fluctuating around it.

### Economic interpretation and main takeaway
- Models with costly credit admit cyclic, chaotic and stochastic dynamics; price distributions F(p) and the use of money and credit vary over time due to "animal spirits."
- These phenomena do not require fiat money (they hold for ζ ≠ 0).
- Underlying mechanism: a trade-off between paying transaction costs on credit and accepting low returns on liquid assets.
  - An asset useful in exchange has price above the fundamental ζ = r (see (22)), and high asset prices imply low returns.
- Result: even when credit is always available but costly, liquidity considerations and credit costs can generate rich dynamics (cycles, chaos, sunspots), fluctuations in price distributions (not just levels), and excess volatility.

*Content unit: wp1714 - 6.4  Dynamics*

### Appendix B: Formulae for Calibration

### wp1714 - Appendix B: Formulae for Calibration

### Variable-cost model: expressions for G(q), H(q), fraction of monetary transactions, and markup
- G(q)   =   1 

1
2
2
 q+(^z
i
 q)
(1 +) (q )
- H(q)   =   1 

2
1
[ q+(^z
i
 q)] [+^z
i
+ (q 2) (1 +)]
4
2
(
1
+
2
) (1 +)
2
(q )
2

- The fraction of monetary transactions and the markup are therefore
  - H(^z
i
)   =   1 

2
1
( ^z
i
) [+^z
i
+ (^z
i
 2) (1 +)]
4
2
(
1
+
2
) (1 +)
2
(^z
i
 )
2
  - E
G
q

=   1 +

1
(+^z
i
 +) log (1 + 2
2
=
1
)
2
2
(1 +)

- From this we get
  - L
i
=
(1 +) ^z
i

1
(+ ^z
i
) + (1 +) (1 +
2
)
  - 
i
=

1
+ (1 +) (1 +
2
)

1
(+ ^z
i
) + (1 +) (1 +
2
)
  - @^z
i
/@i
^z
i
=i

- Note: ^z
i
is given in the text.

### Fixed-cost model: expressions for G(q), H(q), fraction of monetary transactions, and markup
- G(q)   =   1 

1
2
2
  q
q 
- H(q)   =   1 

2
1
(  q) ( +q 2)
4
2
(
1
+
2
) (q )
2

- The fraction of monetary transactions and the markup are
  - H(^z
i
)   =
[2
1

2
(  )=]
2=3
i
2=3
 
2
1
4
2
(
1
+
2
)
  - E
G
q

=   1 +

1
(  ) log (1 + 2
2
=
1
)
2
2

- From this we get
  - L
i
=
+


2
1
(  )
2
=2
2

1=3
i
 1=3
1 +
1
( ) +
2

  - 
i
=
 1
3 + 3


2
1
(  )
2
=2
2

 1=3
i
1=3

### Key mathematical relationships summarized
- Variable-cost model yields closed-form expressions for:
  - G(q), H(q), H(^z
i
), E[G/q], L
i
, and 
i
, all expressed in terms of parameters , , , 
1
, 
2
, and ^z
i
.
- Fixed-cost model yields closed-form expressions for:
  - G(q), H(q), H(^z
i
), E[G/q], L
i
, and 
i
, with dependence on parameters , , , 
1
, 
2
, and ^z
i
.
- Sensitivities include derivatives such as @^z
i
/@i explicitly noted as ^z
i
=i in the variable-cost calibration.

*Source: wp1714 - Appendix B: Formulae for Calibration*

---


_Source: https://www.imf.org/-/media/files/publications/wp/wp1714.pdf_