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Working Paper/Document de travail 2011-7 Money and Costly Credit by Mei Dong Bank of Canada Working Paper 2011-7 February 2011 Money and Costly Credit by Mei Dong Currency Department Bank of Canada Ottawa, Ontario, Canada K1A 0G9 [email protected] Bank of Canada working papers are theoretical or empirical works-in-progress on subjects in economics and finance. The views expressed in this paper are those of the author. No responsibility for them should be attributed to the Bank of Canada. ISSN 1701-9397 © 2011 Bank of Canada 2 Acknowledgements This paper was completed as one chapter in my thesis when I was at Simon Fraser University. I would like to thank David Andolfatto and Fernando Martin for many discussions and comments. I also thank Jonathan Chiu, Geoffrey Dunbar, Ben Fung, Hua Jiang, Alex Karaivanov, Daniel Monte, Irina Telyukova, Chris Waller, Randy Wright and seminar participants at Simon Fraser University, the Federal Reserve Bank of Cleveland, the Bank of Canada, Shanghai University of Finance and Economics, the 2008 Summer Workshop on Money, Banking and Payments at the Federal Reserve Bank of Chicago, the 2009 Midwest Macro Meetings, and the 2009 Society of Economics Dynamics Annual Meetings in Istanbul for helpful suggestions. ii Abstract I study an economy in which money and credit coexist as means of payment and the settlement of credit requires money. The model extends recent developments in microfounded monetary theory to address the choice of payment methods and the effects of inflation. Whether a buyer uses money or credit depends on the fixed cost of credit and the inflation rate. In particular, inflation not only makes money less valuable, but also makes credit more expensive because of delayed settlement. Based on quantitative analysis, the model suggests that the relationship between inflation and credit exhibits an inverse U-shape which is broadly consistent with anecdotal evidence. Compared to an economy without credit, allowing credit as a means of payment has three implications: [1] it lowers money demand at low to moderate inflation rates; [2] it improves society’s welfare when the inflation rate exceeds a specific threshold; and [3] it can raise the welfare cost of inflation for some reasonable values of the credit cost parameter. JEL classification: E41, E50 Bank classification: Credit and credit aggregates; Inflation: costs and benefits Résumé L’auteure étudie une économie où la monnaie et le crédit coexistent comme moyens de paiement et où le remboursement du crédit requiert de la monnaie. Son modèle tire parti, en les prolongeant, des progrès récents réalisés dans l’élaboration des fondements microéconomiques de la théorie monétaire pour analyser le choix de la méthode de paiement et les effets de l’inflation. La méthode retenue pour le règlement des achats dépend du coût fixe du crédit et du taux d’inflation. En particulier, l’inflation non seulement érode la valeur de la monnaie, mais elle renchérit le crédit du fait que celui-ci est remboursé avec un délai. Les résultats de la résolution numérique du modèle indiquent que la relation entre l’inflation et le crédit a la forme d’une bosse, ce qui cadre dans l’ensemble avec les observations. Par rapport à une économie sans crédit, l’inclusion du crédit parmi les modes de paiement a trois conséquences : 1) elle entraîne un recul de la demande de monnaie quand l’inflation est faible ou modérée; 2) elle a pour effet d’améliorer le bien-être de la société lorsque l’inflation dépasse un seuil déterminé; 3) elle peut accroître le coût de l’inflation sur le plan du bien-être pour certaines valeurs raisonnables du paramètre relatif au coût du crédit. Classification JEL : E41, E50 Classification de la Banque : Crédit et agrégats du crédit; Inflation : coûts et avantages iii 1 Introduction This paper develops a model of money and credit in order to study issues in monetary economics, in particular, the choice of payment methods and the e¤ects of in‡ation. Not too long ago, consumers typically paid for their purchases using either cash or checks. In recent decades, the payment instruments available to consumers have expanded to include debit and credit cards as well as other electronic payments. In a credit transaction, credit is usually o¤ered by a third party, and this third party is willing to postpone debt settlement to the inde…nite future – at high rates of interest. A prominent di¤erence between credit and other means of payment is that acquisition of money in advance is not necessary to make a purchase using credit. In other words, credit transactions are settled "later".1 Credit has become increasingly popular as a means of payment in many countries. Based on the Payment System Statistics from the BIS, the percentage of total number of transactions using cards with a credit function in 2005 was 24.8% in Canada, 23.4% in the United States, and 13% in the United Kingdom. Table 1 contains the ratio of transaction value by cards with credit function to GDP in Canada and the United States. Figure 1 shows the number of card with credit function issued from 1990 to 2008 in selected countries. These statistics suggest that more credit cards have been issued and more transactions have been paid for with credit cards. Canada the United States 2000 11.68 12.54 2002 13.26 12.66 2004 14.04 13.6 2006 16.1 14.58 2008 18.29 14.95 Table 1: Ratio: Value of Transaction by Card with Credit Function to GDP (units: 10 5) If consumers can use both money and credit to pay for their purchases, how will money and credit interact in such an environment? In particular, what is the e¤ect of monetary policy or in‡ation on credit? How does the introduction of credit a¤ect money demand, social welfare and the e¤ects of monetary policy? These questions are interesting and important for the conduct of monetary policy in an economy in which money and credit coexist as means of payment. 1 Checks can be viewed as a short-term form credit. However, since the time until settlement is typically very short for checks, checks are not considered as "settled later" in this paper. 2 United States 1400 70 1200 # of credit card issued # of credit card issued Canada 80 60 50 40 1000 30 20 1990 800 600 400 1992 1994 1996 1998 2000 year 2002 2004 2006 200 1990 2008 1992 1994 1996 United Kingdom 2004 2006 2008 2002 2004 2006 2008 4 3.5 # of credit card issued 60 # of credit card issued 2002 Sweden 70 50 40 30 20 1990 1998 2000 year 3 2.5 2 1.5 1992 1994 1996 1998 2000 year 2002 2004 2006 1 1990 2008 1992 1994 1996 1998 2000 year Figure 1: Number of Card Issued with Credit Function (in Millions) Technological advances in electronic record-keeping is one of the main factors that have facilitated the use of consumer credit.2 Specialized intermediaries are now able to o¤er consumer credit with limits that can vary with each person’s credit history. These technological advances allow credit to substitute for money and other means of payment in transactions. For example, based on U.S. household-level data, Duca and Whitesell (1995) estimate that for every 10% increase in the probability of owning a credit card, checking balances are reduced by 9%. However, how in‡ation a¤ects the money-credit margin in practice is less clear. Empirical investigation here is hampered to some extent by data limitations and the recent improvements in credit card technology. Nevertheless, evidence from countries that su¤ered high in‡ation episodes tends to suggest that credit card usage increased after in‡ation was brought under control. For example, credit cards gained widespread popularity in Brazil following the successful reduction of in‡ation to sustainable levels, and the number of cards issued grew by 88% between 2000 and 2004. High in‡ation episodes also delayed the adoption and widespread use of credit cards in Turkey. Even in Australia, households’debts increased dramatically when there were lower in‡ation rates and thus lower cost of borrowing during the 1990s.3 Reducing in‡ation seems to be consistent with 2 Financial deregulation is another important factor that explains the increase in credit usage. The in‡ation rate was on average around 8% in Australia in the 1980s and was reduced to around 3% in the 1990s. 3 3 the increasing use of credit. Evidence based on total private credit to GDP ratio, a broader measure of credit, suggests that in‡ation tends to have a negative impact on credit at high rates, but not at low rates.4 Using a sample of 97 countries, Boyd et al. (2001) conclude that in‡ation has a negative impact on credit. See also Boyd and Champ (2003). Later, Khan et al. (2006) also use a large cross-country sample, but they …nd that there is a threshold e¤ect of in‡ation on credit. In‡ation has a negative impact on credit when it exceeds a threshold. In this paper, I construct a model that is able to replicate the observed relationship between in‡ation, money demand and credit. The model is built on Lagos and Wright (2005). In monetary theory, frictions that render money essential make credit arrangements impossible. In order for credit to exist, I assume that there exist competitive …nancial intermediaries that can identify agents and have access to a record-keeping technology. There are two frictions associated with credit arrangements. First, arranging credit is costly. In a bilateral trade, if a buyer wants to use credit, he must incur a …xed utility cost in order to make the seller and himself identi…ed to a …nancial intermediary. When buyers have heterogeneous preferences, the …xed cost of credit will endogenously determine the fraction of buyers using credit. Second, the settlement of credit is available only at a particular time in each period, during which the …nancial intermediaries accept repayment of credit and settle debts. Due to the timing structure of the model, settlement is "delayed" and money becomes the only means of settlement. These two features of the model allow some interesting interactions between money and credit. In‡ation tends to increase the fraction of buyers using credit at low in‡ation rates, but decrease the fraction of buyers using credit at high in‡ation rates. That is, the relationship between in‡ation and credit exhibits an inverse U-shape. Compared to an economy without credit, the model has three implications: [1] the real demand for money is lower at low to moderate in‡ation rates; [2] social welfare is higher when the in‡ation rate exceeds a speci…c threshold; and [3] for a given in‡ation rate, the welfare cost of in‡ation can be higher for some reasonable values of the credit cost parameter. 4 Total private credit may be too broad in comparison to consumer credit. Given that [1] it is di¢ cult to obtain data on consumer credit for a large sample of countries over an adequately long time span; and [2] di¤erent measures of credit tend to highly comove, as can be seen from the U.S. data, it seems useful and reasonable to review this evidence. 4 This paper contributes to the literature by developing a model which incorporates the following three features observed in an economy with money and credit: …rst, money and credit coexist as means of payment; second, the choice of using money or credit is endogenous; and third, the settlement of credit requires money. Several recent papers have attempted to construct models with both money and credit.5 However, they are not able to capture all three features. In Berentsen et al. (2007), banks can record …nancial history, but they cannot record goods transaction history. In this case, credit is available through bank loans in the form of money. Therefore, money is the only means of payment. Chiu and Meh (forthcoming) extend Berentsen et al. (2007) to study how credit/…nancial intermediation a¤ects allocations and welfare in an economy where ideas (or projects) are traded among investors and heterogeneous entrepreneurs. Again, credit does not serve as a means of payment. See also Camera and Ruscitti (2008) for a paper on in‡ation and …nancial intermediation. Telyukova and Wright (2008) build a model where agents can use money and credit to explain the credit card debt puzzle. Their market structure determines that agents do not use money and credit simultaneously. Sanches and Williamson (2010) adopt the notion of limited participation in the sense that only an exogenous subset of agents can use credit. The choice of using credit is not endogenous. See also Monnet and Roberds (2008). Another contribution of the paper is that the model has a novel prediction of the relationship between in‡ation and credit. To generate the inverse U-shape relationship, both frictions associated with credit arrangement are important. Indeed, the two frictions imply opposite e¤ects of in‡ation on credit. Costly credit has been used in the cash-in-advance model or in the OLG model.6 With the presence of the …xed cost, in‡ation always decreases money demand and increases credit demand. I label the e¤ect of in‡ation on credit through the …xed cost channel as the …xed cost e¤ect. "Delayed" settlement is used in Ferraris (2006), where money and credit are complements. In fact, this idea can be traced back to Stockman (1981), where in‡ation reduces the capital stock if money and capital are complements. The delayed settlement e¤ect of in‡ation on credit is that in‡ation should reduce credit. Overall, it turns out that the …xed cost e¤ect dominates at low in‡ation rates 5 The literature on money and credit is vast and thus I do not intend to review all these papers. Rather, I will only mention a few papers that are more recent or are more directly related to this paper. For example, for recent attempts to rationalize the coexistence of inside and outside money, see Cavalcanti and Wallace (1998), Kocherlakota and Wallace (1998), Mills (2007), Sun (2007). A more recent paper by Lester et al. (2008) studies the coexistence of multiple assets which di¤er in their return and liquidity. 6 For cash-in-advance models, see Lacker and Shreft (1996), Aiyagari et al. (1998), and English (1999) for examples. For the OLG framework, see Freeman and Hu¤man (1991) for an example. 5 and the delayed settlement e¤ect dominates at high in‡ation rates. This prediction is consistent with the anecdotal evidence. Azariadis and Smith (1996) obtain a similar result. However, the key friction driving their result is asymmetric information associated with credit. The rest of the paper is organized as follows. Section 2 lays out the physical environment. Section 3 solves for the equilibrium and analyzes the equilibrium when the repayment of credit can be enforced. I numerically study the model in Section 4. In Section 5, I consider monetary equilibrium when the repayment of credit cannot be enforced. Finally, Section 6 concludes. All proofs are provided in the Appendix. 2 Environment Time is discrete and runs forever. In each period, there are three submarkets that open sequentially. The …rst submarket is characterized by bilateral trades and is labelled as market 1. The second submarket is characterized by a centrally located competitive spot market and is labelled as market 2. No trade occurs in the third submarket and it is labelled as market 3. The activity in the third submarket will be described in detail later. There are two permanent types of agents –buyers and sellers, each with measure 1. Buyers are those who want to consume in market 1 and sellers are those who produce in market 1. All agents are anonymous and lack commitment. Each buyer receives a preference shock " at the beginning of each period, which determines the buyer’s preference in market 1.7 The preference shock " is drawn from a c:d:f: G("). The preference shocks are iid across buyers and across time. The realization of these preference shocks is public information. There are two types of goods. Goods that are produced and consumed in market 1(2) are called good 1(2). All goods are nonstorable. In market 1, buyers and sellers are matched randomly according to a matching technology. The probability that a buyer (seller) meets a seller (buyer) is with 0 < 1. Given that a buyer and a seller meet, the terms of trade are determined by the buyer’s take-it-or-leave-it o¤er. After exiting market 1, all agents enter market 2. Buyers supply labor for production and consume good 2. Sellers only consume good 2. The production technology in market 2 is assumed to be linear and 1 unit labor can be converted into 1 unit of good 2. 7 There are a variety of ways to model the heterogeneity in this model. For example, one can model heterogeneous sellers that have di¤erent cost functions or model heterogeneity as match speci…c shocks. 6 The preference of a buyer with a preference shock " is "u(q)+v(x) h; where "u(q) is the buyer’s utility from consuming q units of good 1. The utility function u(q) satis…es u(0) = 0, u0 (0) = 1 and u00 (q) < 0 < u0 (q). In market 2, the buyer’s utility from consuming x units of good 2 is v(x), where limx!0 0 (x) = 1 and preference of a seller is 00 (x) <0< 0 (x). The buyer’s disutility from working is h. The c(q) + y; where c(q) is the seller’s disutility from producing q units of good 1 with c(0) = 0, c0 (0) = 0, c0 (q) > 0 and c00 (q) 0. The seller has a linear utility in market 2, where y is the amount of consumption of good 2. All agents discount between market 3 and the next market 1. The discount rate is . In this economy, money is essential because agents are anonymous and lack commitment. I assume that there is a monetary authority that controls the supply of money. Let M denote the aggregate money supply at any given date. It grows at a gross rate the hat denotes the variable in the next period. I will consider > ^ = M . Here > 0; i.e., M and ! from above. New money is injected (or withdrawn) via a lump-sum transfer (or tax) to each buyer at the beginning of each period and the transfer is ^ = ( 1)M: Besides the monetary authority, there exist competitive …nancial intermediaries. These …nancial intermediaries possess a record-keeping technology, which allows them to identify agents and keep track of goods market transaction history. Clearly the availability of the record-keeping technology makes credit arrangements through the …nancial intermediaries possible in this economy. To sustain the essentiality of money, I assume two frictions associated with the record-keeping technology. The …rst friction is that the record-keeping technology or the …nancial intermediaries are not available in market 2. This restriction implies that agents may arrange credit transactions in market 1, but cannot settle their debts in market 2. As the …nancial intermediaries are available in market 3, buyers who have used credit in market 1 repay their debts and sellers who have extended credit get repayment in market 3. One can think of market 3 as an overnight market for settlement. Without such a restriction, agents would settle their debts in market 2. In some sense, the settlement of debts is delayed. Money becomes the only means of settlement because goods are nonstorable. The second friction associated with the record-keeping technology is that it is costly. As all agents are anonymous, the buyer in a match in market 1 can incur a …xed utility cost k to make the pair identi…able to a …nancial intermediary so that the seller can extend credit to the buyer.8 8 One may argue that in reality, sellers actually pay the cost of using credit. The model can be modi…ed to have 7 Without incurring the …xed cost, the buyer and the seller remain anonymous and cannot make credit arrangements. Figure 2 provides a timeline of events.9 Financial Intermediaries Financial Intermediaries IOU IOU Buyer1 $ Seller1 $ Buyer1 Seller1 Buyer2 Seller2 Buyers: ( x, h,$) q Buyer2 IOU $ $ IOU Sellers: ( y,$) Seller2 q t Market 1: V1 Market 2: V2 Market 3: V3 t +1 Figure 2: Timeline of Events Now I consider a planner’s problem as the benchmark allocation. Suppose that the planner weights all agents equally and is subject to the random matching technology. I restrict the attention to stationary allocations in what follows. In market 1 of each period, given a buyer’s preference shock " and the buyer meeting a seller, the planner instructs the seller to produce q(") for the buyer. Those agents who do not …nd a match consume and produce nothing. In market 2 of each period, the planner assigns the consumption of good 2 x; y and the labor supply h subject to the resource constraint. Formally, the planner’s problem is max q(");x;y;h Z ["u(q(")) c(q("))]dG(") + (x) h+y (1) s.t. x + y = h: 0 (x) The optimal (x ; q (")) are characterized by = 1 and "u0 (q(")) = c0 (q(")) for all ". The optimal q(") is increasing in ". In fact, the optimal allocation features a slight indeterminacy. That is, given the quasi-linear preference structure, h and y are indeterminate as long as h y=x . the seller pay the …xed cost in a match. All the main results hold. 9 Note that the IOU in Figure 2 refers to the record kept by the …nancial intermediaries. Financial intermediaries do not issue physical IOUs. I also rule out the existence of private claims on credit. 8 3 Monetary Equilibrium with Enforcement In this section, I assume that there is enforcement in the economy. It implies that the …nancial intermediaries can enforce the repayment of credit, so there is no credit limit for buyers. It also implies that the monetary authority can impose lump-sum taxes, i.e., 3.1 < 1 is feasible. Buyers To facilitate the analysis, I begin with buyers in market 2. Suppose that in nominal terms, a buyer carries money balance m and debt ` at the beginning of market 2. Let V2b (m; `) and V3b (z; `) be the value functions for a buyer in market 2 and 3, respectively. Notice that the buyer cannot pay o¤ his debt in market 2 because the …nancial intermediaries are not available. However, the buyer can accumulate money balance. Let z denote the money balance that the buyer carries to market 3. The buyer’s choice problem is n V2b (m; `) = max v(x) x;h;z o h + V3b (z; `) (2) s.t. x + z = m + h; where is the inverse of the price level (or the value of money). Substituting h from the buyer’s budget constraint into (2), the unconstrained problem is V2b (m; `) = m + maxfv(x) x;z x z + V3b (z; `)g: The …rst order conditions for interior solutions are v 0 (x) = 1 and @V3b (z; `) = : @z (3) As is standard, the choice of z does not depend on m; however, it depends on `. Intuitively, if the buyer incurs more debt in market 1, he must accumulate more money in market 2 for repayment 9 in market 3. The envelope conditions imply @V2b (m; `) @m @V2b (m; `) @` = ; (4) @V3b (z; `) : @` = (5) Note that V2b (m; `) is linear in m. For the buyer entering market 3, the value function is V3b (z; `) Z = V^1b (z ` + ^; 0; ")dG("): The only activity for the buyer in market 3 is to repay his debt. Due to the quasilinear structure of the buyer’s preference, I focus on one period contract without loss of generality. That is, if the buyer has any debt, he repays in the current market 3. It will become clear later that the buyer should be indi¤erent between repaying the debt in the current market 3 or in any future market 3. Let m ^ =z ` + ^ be the buyer’s money holding at the beginning of the next period. For a buyer receiving a preference shock " at the beginning of the next period, let V^1b (m; ^ 0; ") be the buyer’s value function. As V^1b (m; ^ 0; ") depends on ", I take the expected value for the buyer in market 3 and discount it by . The envelope conditions yield @V3b (z; `) @z b @V3 (z; `) @` = = Z @ V^1b (m; ^ 0; ") dG("); @m ^ Z ^b ^ 0; ") @ V1 (m; dG("): @m ^ (6) (7) Combining (3) and (5) with (6) and (7), @V3b (z; `) = @z Z @ V^1b (m; ^ 0; ") dG(") = @m ^ From (8), V2b (m; `) is linear in ` and V2b (m; `) = m @V3b (z; `) = @` @V2b (m; `) = : @` (8) ` + V2b (0; 0). After exiting market 3, each buyer realizes a preference shock ". For a buyer with ", the value function in market 1 is V1b (m; 0; ") = ["u(q) k I(a) + V2b (m 10 d; a I(a))] + (1 )V2b (m; 0); where (q; d; a) are the terms of trade. With probability , the buyer spends d units of money and uses a units of credit in nominal terms in exchange for q units of good 1 from the seller. An indicator function I(a) is such that I(a) = 1 if a > 0 and I(a) = 0 if a = 0. With probability 1 , the buyer is not matched and carries his money to market 2. 3.2 Sellers Let V2s (m; `) and V3s (z; `) be a seller’s value functions in market 2 and 3, respectively. Since the seller is the creditor, ` should be either 0 or negative. The seller’s value function in market 2 is V2s (m; `) = max fy + V3s (z; `)g y;z (9) s.t. y + z = m: By substituting y from the constraint into (9), the …rst order condition of the unconstrained problem is @V3s (z; `) @z , and z = 0 if @V3s (z; `) < : @z (10) The envelope conditions yield @V2s (m; `) @m @V2s (m; `) @` = = ; (11) @V3s (z; `) : @` (12) Again, V2s (m; `) is linear in m. For the seller in market 3, the value function is V3s (z; `) = Z V^1s (z `; 0; ")dG("): If the seller has extended any credit in the previous market 1, the seller will receive repayment from the …nancial intermediary in market 3. I take the expected value function of the seller because the seller anticipates that a potential buyer he will meet in the next market 1 may have a preference shock " drawn from G("). Let m ^ =z ` denote the seller’s money holding at the beginning of the 11 next market 1. The envelope conditions are @V3s (z; `) @z s @V3 (z; `) @` = = Z @ V^1s (m; ^ 0; ") dG("); @m ^ Z ^s @ V1 (m; ^ 0; ") dG("): @m ^ (13) (14) Now combining (12) with (13) and (14), I obtain @V3s (z; `) = @z Z @ V^1s (m; ^ 0; ") dG(") = @m ^ @V3s (z; `) = @` @V2s (m; `) : @` (15) For the seller who potentially meets a buyer with ", the value function in market 1 is V1s (m; 0; ") = [ c(q) + V2s (m + d; a)] + (1 )V2s (m; 0): (16) If the seller meets a buyer, the seller sells q units of good 1, receives d units of money and extends credit with the nominal value a if the buyer chooses to use credit. 3.3 3.3.1 Equilibrium Take-it-or-Leave-it O¤er Before deriving the equilibrium conditions, I solve for the terms of trade in market 1. The terms of trade in a match are determined by the buyer’s take-it-or-leave-it o¤er.10 There are two types of trades in market 1, depending on whether the buyer in a match uses credit or not. Suppose that a buyer with " uses only money. Recall that V1b and V1s are linear in m. The buyer’s problem is max["u(q) q;d d] s.t. c(q) = d and d where m is the buyer’s money holding. Let 1 and 10 2 m; be the Lagrangian multipliers associated It will be interesting to generalize the buyer’s bargaining power from 1 to less than 1. It will also be interesting to study other pricing mechanisms that have been used in the literature such as competitive pricing and price posting. In this paper, I focus only on the buyer’s take-it-or-leave-it o¤er to get the main intuition from the model and leave those extensions for future work. 12 with the two constraints. L = max ["u(q) q;d; d] + 1; 2 1[ d c(q)] + 2 (m d): It is straightforward that the solution is the following. 8 > > < > > : 2 = 0 : (q; d) are given by "u0 (q) = c0 (q) and d = c(q); 2 > 0 : (q; d) are given by d = m and c(q) = d: Suppose that the buyer with " uses credit. From (8), V2b is also linear in `. However, it is not clear that V2s must be linear in ` at this stage. So I de…ne the buyer’s problem as max["u(q) q;d;a k d s.t. c(q) = d + V2s (0; a) a] V2s (0; 0) and d m: It is obvious that the seller’s money holding does not appear in the above problem. Therefore, the terms of trade with credit do not depend on the seller’s money holding. In addition, recall that the terms of trade without credit do not depend on the seller’s money holding. It follows from (16) that ^ 0; ") @ V^1s (m; = ^ + (1 @m ^ )^ = ^: (17) From (15) and (17), @V3s (z; `) = @z @V2s (m; `) = ^: @` (18) Two results follow from (18). First, V2s is linear in ` and V2s (m; `) = m ^ ` + V s (0; 0). Second, 2 s . As I only consider > and @V3s (z; `) = @` ! from above, the second result is derived from (10) and (18). Using (18), the Lagrangian is L= max ["u(q) q;d;a; 1; 2 k d a] + 1[ d + ^a c(q)] + 2 (m d): It turns out that the inequality constraint is always binding, and thus d = m. The solutions for 13 (q; a) are "u0 (q) = c0 (q); ^ a = c(q) (19) d: (20) It is interesting to note that q depends on . In this economy, if a buyer uses credit in market 1, he will accumulate money for debt repayment in market 2. However, for the seller who extends the credit in the match, he will not be paid in the same market 2. Instead, the seller must wait to get settled in market 3. After receiving the money, the seller carries the money to the next market 1, but he cannot spend it since he does not want to consume. Hence, the seller actually spends the money one period after the buyer accumulates the money. There is an asymmetry between the time at which the buyer accumulates the money for repayment (production) and the time at which the seller can spend the money from repayment (consumption). The buyer must compensate the seller for the loss in the value of money. From (20), the buyer essentially borrows a in nominal terms. The nominal interest rate of credit is 1 + i = ^ = . As ^ a and repays is higher, credit is more costly in nominal terms. For any given ", q is decreasing in . Credit transactions are subject to in‡ation distortion. As the structure of the model implies that money is the only means to settle credit, one may think that it is natural in‡ation a¤ects a credit transaction’s terms of trade. However, this is not necessarily true. The key feature that makes credit subject to in‡ation distortion is the time lag between buyer’s production for repayment and seller’s consumption from repayment.11 Imagine an environment in which …nancial intermediaries exist in market 2 and the settlement of credit requires money. Sellers receive the repayment in the form of money and can spend it in the same market 2. It is clear that a credit transaction’s terms of trade do not depend on the in‡ation rate in this scenario although money is imposed as the only means of settlement. 11 If sellers can deposit the money from repayment in a bank and earn interest as in Berentsen et al. (2007), then they would be fully compensated and the terms of trade would not be a¤ected by in‡ation. However, a full general equilibrium analysis that incorporates bank deposits involves more careful consideration. For example, since …nancial intermediaries are not available in market 2, sellers cannot withdraw their deposits (if there is any) in market 2, which impedes the use of deposits. It is also not obvious who would borrow those deposits from the …nancial intermediary in the current model. 14 3.3.2 Money versus Credit Having solved the terms of trade, I proceed to …nd the condition that determines whether a buyer uses credit or not. For a buyer with " in market 1, if he uses only money, V1b (m; 0; ") = ["u(q) c(q)] + m + V2b (0; 0): If the buyer uses credit, V1b (m; 0; ") = ["u(q) c(q) + ( 1) m k] + m + V2b (0; 0): Let T (") be the net bene…t of using credit for the buyer, where T (") = ["u(q c ) c(q c ) + ( 1) m k] ["u(q m ) c(q m )]: (21) I use q c to denote the quantity traded with credit and q m to denote the quantity traded without credit. For the rest of the paper, I assume that " is uniformly distributed, " U (0; 1]. Lemma 1 For any given in‡ation rate , there exist two threshold values of "; "0 and "1 such that 8 > > > > 0 " "0 , the buyer spends d < m; a = 0 and consumes q where "u0 (q ) = c0 (q ); > > > < "0 " "1 ; the buyer spends d = m; a = 0 and consumes q where c(q) = m; > > > > > > > : "1 " 1; the buyer spends d = m; a > 0 and consumes q c where "u0 (q c ) = c0 (q c ): Lemma 1 is very intuitive. If a buyer receives a very low ", he has enough money to a¤ord q , which is the optimal consumption for him. Here "0 is the threshold that determines whether a buyer is liquidity constrained. For a buyer who receives an intermediate ", the money may not be enough to a¤ord his q . The buyer is liquidity constrained. Using credit can relax the buyer’s liquidity constraint, but this is costly. Therefore, buyers with intermediate "s …nd it optimal not to use credit, because the bene…t from using credit is not enough to cover the …xed cost. For those buyers who have large "s, paying the …xed cost to relax their liquidity constraints becomes optimal. The threshold "1 determines whether a buyer uses credit. 15 The decision to use credit is endogenous in this environment. Buyers use credit for large purchases. This result is in accordance with the evidence on consumers’choices of payment methods. Empirically, the mean value of cash purchases is smaller than the mean value of credit purchases. In English (1999), the mean values of credit card purchases and cash purchases are $54 and $11, respectively. Klee (2008) documents that these respective mean values are $30.85 and $14.2. 3.3.3 Monetary Equilibrium With di¤erent groups of buyers in terms of their choices of payment methods, I can now characterize the equilibrium. I de…ne (q0 ; q1 ) such that "0 u0 (q0 ) = c0 (q0 ); (22) c0 (q1 ): "1 u0 (q1 ) = (23) Notice that c(q0 ) = m represents the transaction demand for money. In market 3, the expected marginal bene…t of 1 unit money is Z Z @ V^1b (m; ^ 0; ") dG(") = ^ @m ^ "0 dG(") + Z "1 "0 0 u0 (q0 ) [ " 0 + (1 c (q0 ) )]dG(") + Z 1 [ + (1 )]dG(") : "1 From (8), the marginal cost of 1 unit money is . The optimal q0 is determined by "0 + 1 u0 (q0 ) 2 (" 2 c0 (q0 ) 1 "20 ) + (1 "1 ) = 1 + : (24) The last condition that completes the characterization of the equilibrium is derived from T (") = 0, "1 u(q1 ) Lemma 2 When is close to c(q1 ) k = "1 u(q0 ) c(q0 ): (25) or approaches 1, "1 = 1. Following Lemma 2, it is possible that no buyer wants to use credit. When is close to the rate of return of money is high enough so that there is no need to use credit. As the terms of trade with credit become worse. When , is higher, approaches 1, the gain from using credit cannot cover the …xed utility cost k. In (21), T (") is negative. Depending on the parameter values 16 of ( ; k; ), there are two types of monetary equilibrium. De…nition 1 When repayment of credit can be enforced, a monetary equilibrium with credit is characterized by ("0 ; "1 ; q0 ; q1 ) satisfying (22), (23), (24) and (25). A monetary equilibrium without credit is characterized by "1 = 1 and ("0 ; q0 ) satisfying (22) and (24). Proposition 1 For any in‡ation rate above the Friedman rule ( > ), monetary equilibrium exists. The optimal monetary policy is the Friedman rule ( ! ). Proposition 1 establishes the existence of monetary equilibrium. In general, monetary equilibrium without credit is unique. The uniqueness of monetary equilibrium with credit is less clear. However, one can show that monetary equilibrium with credit is unique for generic values of following the proof in Wright (2010). It is not surprising that the Friedman rule is the optimal monetary policy. If the monetary authority can run the Friedman rule, there is no cost to hold money.12 Buyers would hold enough money to a¤ord q . Credit is driven out as a means of payment. In the model, the …xed cost k of using credit a¤ects a buyer’s choice of payment methods. A lower k can be viewed as an improvement in credit transaction technology, which is likely to promote the use of credit and lower the transaction demand for money. The e¤ects of k are examined in Proposition 2. Proposition 2 (The E¤ ects of the Fixed Cost) In a monetary equilibrium with credit, the thresholds are increasing in k, i.e., d"0 dk > 0 and d"1 dk > 0. Moreover, dq0 dk > 0 and dq1 dk > 0. If k is too big, no buyer will use credit because it is too costly. The economy would function as the one where money is the only means of payment. The other extreme case is where credit is not costly at all. When k = 0; "0 = "1 = q0 = 0. In equilibrium, credit becomes the only means of payment in market 1 and money is driven out by credit as a means of payment in market 1. The transaction demand for money is 0. However, the total demand for money is not 0 as money is needed for settlement. Monetary equilibrium still exists, but money functions mainly as a means of settlement. 12 If money is subject to theft, in‡ation lowers the rate of return of money and hence reduces theft. As a result, the optimal monetary policy may deviate from the Friedman rule. See Sanches and Willamson (2010) for an example. Similarly, introducing counterfeiting is another way that may generate the optimal monetary policy above the Friedman rule. 17 Given that both money and credit are the means of payment, it seems that they substitute each other. Money and credit are also complements because the settlement of credit requires money. In this economy, the demand for real money balance is M = c(q0 ) + Z 1 [c(q c (") c(q0 )]dG("); (26) "1 where c(q0 ) re‡ects the transaction demand for real money balance. From Proposition 2, the introduction of credit lowers q0 . It does not follow that the total real money balance must be lower as k decreases. Given that money is the only means of settlement, the second term in (26) represents the repayment demand for real money balance. It may increase as k decreases because a lower k makes "1 smaller and induces more buyers to use credit. Therefore, the overall e¤ect of k on the total real money balance is ambiguous. Another parameter of interest is the trading probability . A higher frictions in goods market. Proposition 3 addresses the e¤ects of implies less trading on the economy. Proposition 3 (The E¤ ects of the Trading Probability) In a monetary equilibrium with credit, the thresholds are increasing in ; i.e., d"0 d > 0 and d"1 d > 0. Moreover, dq0 d > 0 and dq1 d > 0. Money becomes more popular as a means of payment when the trading probability increases. The search friction in goods market promotes the use of credit. Recall that a key di¤erence between money and credit is that money has to be acquired before making a purchase. In the case of not …nding a trade, the value of money depreciates when the in‡ation rate is above the Friedman rule. Credit allows buyers to avoid such a distortion because the money required for repayment is accumulated after making a purchase. If it is easier to …nd a trade, holding money is less costly so that money is more desirable. However, even if = 1, credit may still be useful as a means of payment depending on the in‡ation rate. As the paper is motivated by the observations on in‡ation and credit, it is useful to study how a¤ects ("0 ; "1 ; q0 ; q1 ). One can show that d"0 d < 0 and dq0 d < 0 under some conditions, e.g., = 1 or < 2 in a monetary equilibrium with credit. Intuitively, in‡ation should have negative impacts on "0 and q0 because in‡ation is a tax on money. The e¤ects of in‡ation on "1 and q1 are less clear. The two frictions associated with using credit generate two channels through which 18 a¤ects "1 . A higher lowers the rate of return of money and makes more buyers liquidity constrained. As a result, more buyers may …nd that the gain from relaxing the liquidity constraint by using credit can cover the …xed cost. Through the …xed cost channel, decreases "1 . Indeed, this type of e¤ect has been predicted by many other models using the cash-in-advance framework or the OLG framework. The other friction associated with credit is the delayed settlement. From (19), a¤ects the marginal bene…t of using credit. When is higher, the terms of trade using credit become worse. Therefore, buyers have less incentive to use credit. Through the delayed settlement channel, increases "1 . Having analyzed these two channels, it would be interesting to know the e¤ect from which channel dominates. From Lemma 2, "1 hits the boundary 1 when either the total e¤ect of ! or ! 1. Thus, on "1 should be non-monotonic. In fact, it is likely that the e¤ect displays a U-shape. I will rely on numerical results in the next section to verify these conjectures. 3.4 Welfare In order to analyze the e¤ect of monetary policy on aggregate welfare, I de…ne aggregate welfare in this economy as W and (1 )W = ("0 ; "1 ; q0 ) + [v(x ) x ] ( 1 ) Z 1 [c(q c (") c(q0 )]dG("); (27) "1 where ("0 ; "1 ; q0 ) = Z 0 "0 Z ["u(q (")) c(q ("))]dG(")+ "1 "0 Z 1 ["u(q0 ) c(q0 )]dG(")+ ["u(q c (")) c(q c (")) k]dG("): "1 Note that aggregate welfare is also buyers’ aggregate welfare since sellers in this economy earn 0 surplus from trades and their aggregate welfare is 0. The …rst and second terms in the aggregate welfare function are standard. What’s new in (27) is the third term, which is the production distortion from using credit in the following sense. After a seller extends credit in market 1, he receives payment from the …nancial intermediary in market 3 and must wait until the next market 2 to spend the money. As discussed earlier, the buyer who uses credit should pay the nominal interests to compensate the seller. The actual extra payment the buyer has to accumulate by working is the 19 real interest rate because buyers receive monetary transfers from the monetary authority in each period. This part is re‡ected in the third term, which can be viewed as the production distortion from using credit. Without knowing how a¤ects ("0 ; "1 ), it is not obvious how aggregate welfare responds to . Again, I leave a more general analysis in the next section. 4 Quantitative Analysis In this section, I solve the model numerically and derive more implications from the model. For the numerical exercise, I adopt some speci…c functional forms for u(q); c(q) and v(x) that have been used in the literature. Let u(q) = 1 q , v(x) = B log x; and c(q) = q, where 0 < < 1. In market 1, the matching technology that I specify is the urn-ball matching function, where =1 e 1. There are four parameters ( ; B; ; k) to be determined. The period length in this model is set to be 1 year mainly to facilitate comparisons with past work on the welfare cost of in‡ation. The time preference parameter is set 1 = 1:04, so the implied annual real interest rate is 0:04. For the other parameters, I follow Lucas (2000) and Lagos and Wright (2005), and …t the model’s money demand to the U.S. money demand data by nonlinear least square. The data covers annual nominal interest rate and the "money demand" (or the inverse of the velocity of money) for the period 1900 where 2000.13 The "money demand" predicted by the model is R1 c c(q0 )]dG(") c(q0 ) + M "1 [c(q (") = L(i) = : R "0 R "1 R1 PY Yc + [ 0 c(q ("))dG(") + "0 c(q0 )dG(") + "1 c(q c ("))dG(")] Z Yc = x + [ "0 c(q ("))dG(") + 0 + (1 ) Z Z "1 c(q0 )dG(") + "0 1 [c(q c (") Z 1 c(q c ("))dG(")] "1 c(q0 )]dG("): "1 The parameters from the best …t are in Table 2.14 13 The data are originally from Craig and Rocheteau (2008). From the numerical exercise, multiple equilibria have not been found although there is no general proof of a unique monetary equilibrium with credit. 14 20 ρ 0.4732 Parameters Values B 1.4436 k 0.0739 Table 2: Parameter Values The values of ( ; B) are in the ballpark of existing studies. I use a consumption equivalence measure to evaluate the plausibility of the value of k. The utility cost k = 0:0739 is worth of 1% of consumption for buyers. Based on these parameter values, I numerically solve the model and show the results in Figure 3. threshold e1 credit to GDP ratio 1 0.07 0.95 0.06 0.9 0.05 credit to GDP ratio 0.85 e1 0.8 0.75 0.7 0.65 0.04 0.03 0.02 0.6 0.01 0.55 0.5 0 10 20 30 40 50 inflation rate 60 70 80 90 0 100 0 10 20 30 total money demand 40 50 inflation rate 60 70 80 90 100 60 70 80 90 100 welfare improvement 0.8 0.015 credit no credit 0.7 0.01 consumption equivalence money demand 0.6 0.5 0.4 0.3 0.005 0 0.2 -0.005 0.1 0 0 10 20 30 40 50 inflation rate 60 70 80 90 -0.01 100 0 10 20 30 40 50 inflation rate Figure 3: The E¤ect of In‡ation –Benchmark The upper-left and upper-right panels are the e¤ects of in‡ation on the threshold "1 and the credit to GDP ratio, respectively.15 The lower panels are the comparisons with a no-credit economy. 15 The predicted credit to GDP ratio from the model is R1 [c(q c (") c(q0 )]dG(") "1 : R "0 R" R1 Yc + [ 0 c(q ("))dG(") + "01 c(q0 )dG(") + "1 c(q c ("))dG(")] 21 The lower-left panel presents the total demand for real money balances in the credit economy and the no-credit economy. The lower-right panel shows the welfare improvement of having credit based on a consumption equivalence measure. That is, the number on the vertical axis is the fraction of consumption that a buyer is willing to give up to live in a credit economy instead of a no-credit economy. There are several interesting …ndings from Figure 3. It is clear that in‡ation induces more buyers to use credit at low in‡ation rates and less buyers to use credit at high in‡ation rates. Moreover, the credit to GDP ratio predicted by the model has an inverse U-shape against in‡ation. As discussed in the previous section, in‡ation has two e¤ects on "1 . The …xed cost e¤ect implies that in‡ation makes more buyers use credit. This is because high in‡ation causes more buyers liquidity constrained so that more buyers may …nd using credit bene…cial enough to cover the …xed cost. The delayed settlement e¤ect on the other hand lowers incentives for buyers to use credit because of a deterioration in the terms of trade. The numerical results suggest that the …xed cost e¤ect dominates the delayed settlement e¤ect at low in‡ation rates, but the delayed settlement e¤ect dominates the …xed cost e¤ect at high in‡ation rates. High in‡ation makes credit involve high repayment and hence unfavorable terms. This exactly describes the consumer credit market in Brazil during the late 1980s.16 Compared to a no-credit economy, credit lowers demand for real money balances at low to moderate in‡ation rates, but slightly increases demand for real money balances at high in‡ation rates. One can show that the transaction demand in a credit economy is always lower than in a no-credit economy. As the repayment of credit also requires money, the repayment demand may increase as the in‡ation rate increases. It seems that credit and money are substitutes at low to moderate in‡ation rates, but are complements at high in‡ation rates. The …rst half of the result can be supported by the empirical work using U.S. data, since the in‡ation rates in the U.S. have been low to moderate in recent decades. See Duca and Whitesell (1995) for an example. The latter half of the result, however, has not been veri…ed empirically. The lower-right panel reveals that having credit does not always bene…t the society in terms 16 Due to the long time delay in credit card charges clearing through the banking system, vendors have been documented to normally add on a 20 to 30 percent surcharge to the price of the purchased item. In this way, vendors can protect themselves from the depreciation of money during the time the vendors are waiting to be paid by the credit card companies. 22 of aggregate welfare. Knowing that individuals optimally choose to use money versus credit, it seems puzzling that credit can hurt the economy. From (27), credit improves welfare by relaxing the liquidity constraint for some buyers, but may hurt welfare because of the production distortion. Besides these direct e¤ects, credit a¤ects welfare through the general equilibrium e¤ect. As analyzed above, credit may lower the demand for real money balances and thus the value of money, which will generate a negative externality on agents who use money. On the other hand, credit may increase the demand for real money balances and the value of money, which will generate a positive externality on agents who use money. The general equilibrium e¤ect implies that credit may hurt welfare at low to moderate in‡ation rates, but improve welfare at high in‡ation rates. Similar results appear in Chiu and Meh (forthcoming). Overall, credit improves aggregate welfare when the in‡ation rate exceeds a threshold. In terms of the e¤ect of monetary policy, the model predicts that aggregate welfare and aggregate output are decreasing in the in‡ation rate. This is not surprising although the model does introduce a channel through which in‡ation may potentially increase output levels at low in‡ation rates by encouraging more buyers to use credit. However, the e¤ect from this channel does not appear to be strong. In Figure 3, the threshold value for credit to improve welfare is around 20% in‡ation rate, which is fairly high. The potential problem is that …tting ( ; B; k) together implicitly assumes that these parameters do not change over the hundred years. However, it is hard to believe that the cost of credit transactions stays constant over time. Nevertheless, since there is no direct data that measures how k evolves over time and the focus of the paper is not to match any moment in the data, I take a simple approach to evaluate the model‘s predictions by varying k and …xing ( ; B). To highlight the e¤ect of changing k, I show in Figure 4 the credit to GDP ratio and the welfare improvement when k = 0:01; 0:05 and 0:1: 23 credit to GDP ratio welfare improvement 0.12 0.035 k=0.01 k=0.05 k=0.1 k=0.01 k=0.05 k=0.1 0.03 0.1 0.025 0.02 consumption equivalence credit to GDP ratio 0.08 0.06 0.04 0.015 0.01 0.005 0 0.02 -0.005 0 0 10 20 30 40 50 inflation rate 60 70 80 90 -0.01 100 0 10 20 30 40 50 inflation rate 60 70 80 90 100 Figure 4: Comparative Statics –Varying k According to Figure 4, a lower cost of credit promotes the use of credit. Using the average in‡ation rate 7:387 from 1969 2000, the predicted credit to GDP ratio is 0:26% and the predicted "money demand" is 0:395 when k = 0:1. For k = 0:01, the predicted credit to GDP ratio is 10:02% and the predicted "money demand" is 0:201. The low cost credit regime is featured by more credit and less "money demand". It has been noted that there is a trend decline in "money demand" in the recent decade, which has been viewed as a shift of the money demand curve.17 Clearly, improvements in the credit transaction technology contribute to the trend decline in "money demand". In terms of welfare, more costly credit makes credit less bene…cial to the society. One can see that the threshold for credit to be welfare-improving is higher when k is higher. In practice, if sellers receive repayment in the form of money, they may deposit the money to avoid any in‡ation distortion, which makes credit more bene…cial. This type of argument can be built into the model by allowing a fraction of agents to settle in market 2 and the rest to settle in market 3. While this is a nice extension, the current model still serves as a benchmark for analyzing the e¤ect of in‡ation on credit in a world in which credit is not entirely free of in‡ation distortion. As a robustness check, I choose values of ( ; B) by varying the sample period and evaluate the 17 Considering an economy without credit, I can …nd the values of ( ; B) by …tting the money demand curve. Starting in the 1980s, the predicted money demand diverges from the data. If one is willing to assume a speci…c functional form of the trend of …nancial innovation and assume that each year is in a steady state, then …tting the money demand data can generate the values of ( ; B) and the trend of …nancial innovation. By doing such an exercise, I found that the predicted money demand in recent years is much closer to the money demand data. A similar exercise is in Faig and Jerez (2006). 24 model’s predictions. In these experiments, the value of varies from 0:387 to 0:589; but the value of B does not change much, which is around 1:4. The model’s predictions emerging from Figure 3 are very robust.18 To study how introducing credit a¤ects the welfare cost of in‡ation, I compute the welfare cost of 10% in‡ation based on the parameters given in Table 2. The measure of the welfare cost follows the recent literature by using the consumption equivalence measure. The numbers reported in Table 3 is the fraction of consumption a buyer is willing to give up to have 0% in‡ation rather than 10% in‡ation. As a benchmark, I compute the welfare cost for a no-credit economy, i.e., k = 1 and hence "1 = 1. The welfare cost of 10% in‡ation is 1:12% in the benchmark economy, which is relatively small because the take-it-or-leave-it o¤er by buyers avoids the holdup problem.19 k Welfare Cost benchmark 1.12% 0.01 0.22% 0.05 0.85% 0.0739 1.32% 0.1 1.40% Table 3: Welfare Cost of 10% in‡ation I then compute the welfare cost of in‡ation for di¤erent values of k. The introduction of credit can raise the welfare cost when credit is costly enough. For low values of k, the cost for buyers to substitute credit for money is relatively low. Therefore, in‡ation does not generate a large welfare loss. On the contrary, if the cost for buyers to switch from money to credit is high, in‡ation can result in a higher welfare loss compared to the benchmark economy. Note that if k is too big, no buyer uses credit and the economy is essentially the benchmark economy. Dotsey and Ireland (1996) and Lacker and Shreft (1996) both emphasize that credit costs are quantitatively important as a component of the welfare cost of in‡ation. The results in Table 3 further con…rm their results. 18 Taking one year as the period length in the model implies that the settlement is delayed for one year, which seems to be too long. To address this issue, I can take one month or one quarter as the model’s period length and compute the model’s annual "money demand" by aggregating from monthly or quarterly "money demand". I can also take one quarter as the period length and use quarterly data on treasury bill rate per annum, M1 and GDP from IFS. The quarterly "money demand" data is computed by converting treasury bill rate and GDP into quarterly measure. All these methods do not change the qualitative predictions of the model including the inverse U-shape relationship between in‡ation and credit usage. However, the quantitative predictions depend on the speci…c method and the initial values assigned to ( ; B; k). 19 In Lagos and Wright (2005), the welfare cost of 10% in‡ation is 1:4% when buyers have all the bargaining power. The current result does not deviate from their estimate. 25 5 Monetary Equilibrium without Enforcement So far I have assumed that …nancial intermediaries can enforce the repayment of credit, which implies that buyers are not credit constrained. In this section, I relax this assumption. Financial intermediaries can identify agents and keep records of goods market transactions, but they cannot enforce the repayment of credit. As in Berentsen et al. (2007) and Sanches and Williamson (2010), the punishment for default is permanent exclusion from the …nancial system. That is, if a buyer defaults, the buyer will never be able to use credit in any future period. Given the punishment, the amount of credit extended to a buyer should be consistent with the buyer’s incentive to repay. In an environment without enforcement, the government (or the monetary authority) cannot enforce buyers to pay taxes either. It implies that 1. With this modi…cation, buyers and sellers face the same choice problems as before in market 2 and 3. Only in market 1, the buyer’s take-it-or-leave-it o¤er should be reformulated. max["u(q) d q;d;a s.t. c(q) = d + ^ a; d a] m and a a: Here a is the credit limit faced by the buyer. An individual buyer takes a as given. In equilibrium, a will be endogenously determined. The Lagrangian is L= max q;d;a; 1; 2; 3 ["u(q) d a] + If the credit constraint is not binding, i.e., d + ^a 1[ 3 d) + 3 (a c0 (q); ^ a = c(q) 3 2 (m = 0, then d = m and (q; a) are given by "u0 (q) = If the credit constraint is binding, i.e., c(q)] + d: > 0, I have d = m, a = a and q solving c(q) = d + ^ a: 26 a): Introducing the credit limit may cause some buyers to be credit constrained. If such buyers exist, there are potentially four groups of buyers. Lemma 3 For any given in‡ation rate , there exist three thresholds of "; "0 ; "1 and "2 such that 8 > > > 0 " "0 ; the buyer spends d < m; a = 0, and consumes q where "u0 (q ) = c0 (q ); > > > > > > > > < "0 " "1 ; the buyer spends d = m; a = 0 and consumes q where c(q) = m; > > > > "1 > > > > > > > : "2 " "2 ; the buyer spends d = m, a < a and consumes q c where "u0 (q c ) = c0 (q c ); " 1; the buyer spends d = m, a = a and consumes q where c(q) = m + ^ a: The expected marginal value of money in market 3 is Z @ V^1 (m; ^ 0; ") G(") = @m ^ ^ + Z Z "0 dG(") + 0 "2 [ Z "1 "u0 (q0 ) + (1 )]dG(") c0 (q0 ) "0 Z 1 " [ )]dG(") + + (1 )]dG(") ; "2 "2 + (1 "1 [ where q0 is de…ned in (22). Combining with (8), the optimal q0 is implicitly given by "0 + 1 u0 (q0 ) 2 (" 2 c0 (q0 ) 1 "20 ) + ("2 "1 ) + 1 1 ( 2 "2 "2 ) = 1 + : (28) De…ne q2 such that "2 u0 (q2 ) = c0 (q2 ) ; (29) and q2 satis…es c(q2 ) = c(q0 ) + ^ a: (30) Now suppose that "2 < 1. Consider a buyer with " carrying debt a into market 2. The buyer must have spent all of his money. If he repays the debt, the buyer should work to accumulate the money for repayment. His value function is V2b (0; a). If the buyer defaults, he does not need to work as much in market 2. Denote the payo¤ from default by V2bD (0; a) where V2bD (0; a) = v(x ) x + max zD 27 n o z D + V3bD (z D ; 0) : The superscript D represents variables associated with default. The real credit limit is the value of ^ a that solves V2b (0; a) = V2bD (0; a): After some algebra, the real credit limit is ^a = Z 2 (1 Z + + ) "2 "1 Z 1 "0 ["u(q (")) c(q ("))]dG(") + ["u(q c (")) k k c(q c (")) + ( c(q2 ) + ( "2 Z "D 0 ["u(q (") c(q ("))]dG(") ( ) (1 ) ["u(q0 ) c(q0 )]dG(") (31) 1)c(q0 )]dG(") 1)c(q0 )]dG(") Z 1 "D 0 0 + "1 "0 0 ["u(q2 ) Z [c(q0D ) ["u(q0D ) ) c(q0D )]dG(") c(q0 )]: When the repayment of credit cannot be enforced, a monetary equilibrium with constrained credit is de…ned by a list of ("0 ; "1 ; "2 ; q0 ; q1 ; q2 ; ^ a) characterized by (22), (23), (25), (28), (29), (30) and (31). It is possible that ("1 ; "2 ) hit the boundary 1 in equilibrium. In particular, there are two special cases: [1] "2 = 1 and [2] "1 = "2 = 1. In case [1], the equilibrium corresponds to the monetary equilibrium with credit in Section 3. In case [2], the equilibrium corresponds to the monetary equilibrium without credit. Notice that in these two cases, the endogenous credit limit still exists, but it is not binding. It becomes very complicated to derive any analytical results. Hence, I use a numerical example to show how endogenizing credit limit a¤ects equilibrium and welfare. Based on the numerical exercise in Section 4, I again set = 0:4732 and B = 1:4436. As for k, I choose k = 0:03. The left panel in Figure 5 shows the endogenous credit limit and the maximum amount of borrowing for various in‡ation rates. The main …nding is that buyers are credit constrained only at very low in‡ation rates, which is consistent with Berentsen et al. (2007). High in‡ation rates help to relax the credit constraint because punishment of default becomes more severe. In‡ation also lowers the surplus from credit trades in this model. The overall e¤ect of in‡ation on credit limit is that in‡ation …rst relaxes and then tightens the credit limit. At high in‡ation rates, because buyers borrow less, the credit constraint does not bind although the credit limit is lower. 28 real credit limit and max. borrowing welfare comparison 1.4 0.38 credit limit max borrowing constrained credit unconstrained credit no credit 0.36 1.2 0.34 1 0.32 0.8 welfare real credit 0.3 0.28 0.6 0.26 0.4 0.24 0.2 0.22 0 0 10 20 30 40 50 inflation rate 60 70 80 90 0.2 100 0 10 20 30 40 50 inflation rate 60 70 80 90 100 Figure 5: The E¤ect of In‡ation –Endogenous Credit Limit In the right panel of Figure 5, I compare welfare in an economy with endogenous credit limit with welfare in an economy with either unconstrained credit or no credit. The implication is that having the endogenous credit limit can improve social welfare at very low in‡ation rates. It sounds counterintuitive that imposing a constraint would make the allocation better. However, money functions well as a means of payment at low in‡ation rates. It is more likely that buyers would choose to default because the punishment may make them better o¤. If the credit limit reaches zero, the economy converts into a pure monetary economy. At very low in‡ation rates, I demonstrated in Section 4 that a monetary equilibrium without credit is better in terms of aggregate welfare than a monetary equilibrium with credit. Having a zero credit limit is essentially welfare-enhancing. 6 Conclusion Both money and credit are widely used as means of payment. It is important to understand how the use of credit a¤ects money demand and hence the e¤ects of monetary policy. I constructed a model in which money and credit coexist as means of payment and money is the means of settlement. There are two frictions associated with using credit – a …xed utility cost and delayed settlement. In this environment, a buyer’s choice of payment methods is endogenous. Credit lowers money demand at low to moderate in‡ation, but it slightly increases money demand at high in‡ation rates. The relationship between in‡ation and the credit to GDP ratio exhibits an inverse U-shape which is broadly consistent with existing evidence. Costly credit does not always improve social 29 welfare. Depending on the …xed utility cost of credit, allowing credit as a means of payment may raise the welfare cost of in‡ation. In a modi…ed environment where enforcement is imperfect, the endogenous credit limit tends to increase at low in‡ation rates, but decrease at high in‡ation rates. However, credit constraints only bind at very low in‡ation rates. Interestingly, imperfect enforcement may improve social welfare because it avoids socially ine¢ cient borrowing. One testable implication from the model is that the relationship between in‡ation and the credit to GDP ratio has an inverse U-shape. There are many studies testing the long run e¤ect of in‡ation on credit. Most of them use the total private credit to GDP ratio as the measure of credit. In the future, it would be of interest to use the consumer credit to GDP ratio and test its long run relationship with in‡ation, particularly using cross-country data.20 A Appendix A.1 Proof. Proof of Lemma 1 As discussed in the paper, if a buyer uses credit, he must spend all his money. De…ne q as the solution to u0 (q) = c0 (q). One can show that m < c(q) as long as > . It implies that there exists a threshold "0 such that "0 u0 (q0 ) = c0 (q0 ) where q0 is from c(q0 ) = m. For buyers with " > "0 , T 0 (") = ["u0 (q c ) Notice that "u0 (q c ) c0 (q c )] dq c d" ["u0 (q m ) c0 (q c ) = 0 from (19) and dq m d" c0 (q m )] dq m + [u(q c ) d" u(q m )]: = 0. Because q c > q m , it follows that T 0 (") > 0 for " > "0 . De…ne "1 such that T ("1 ) = 0. If an interior "1 exists, then [1] buyers with "0 < " < "1 are liquidity constrained and do not use credit; and [2] buyers with " > "1 are liquidity constrained and use credit. 20 I tested the long run relationship between in‡ation and the private credit to GDP ratio following Bullard and Keating (1995). The results moderately support the inverse U-shape prediction. In addition, I used Canada and the U.S. consumer credit data to perform the same test. For these two low in‡ation countries, a permanent increase in in‡ation increases the consumer credit to GDP ratio, which is consistent with the left half of the inverse U-shape. However, due to data availability, I do not have any consumer credit data for high in‡ation countries. Therefore, it is hard to verify the right half of the U-shape at this stage. A detailed description of the estimation is available upon request. 30 A.2 Proof of Lemma 2 When Proof. is close to ["u(q c ) , "0 ! 1, T (") ! c(q c ) k] ["u(q m ) c(q m )] and q c ! q m . It implies that T (") < 0 for all " > "0 , which means "1 = 1. Recall that T 0 (") > 0 for " > "0 . For " > "0 , it is true that m = c(q m ). The maximum of T (") is achieved at " = 1 and T (1) = [u(q c ) T (1) ! A.3 Proof. c(q c ) k. It is clear that "1 = 1 when k] [u(q m ) c(q m )]. When approaches 1, ! 1. Proof of Proposition 1 Notice that (24) determines q0 , where "0 and "1 are functions of q0 . From (22), one can solve for "0 (q0 ) and d"0 dq0 > 0. To solve for "1 , both (23) and are used. The choice problem of q0 can be rewritten as 1 u0 (q0 ) 2 1 c0 (q0 ) + " (q0 ) 2 u0 (q0 ) 2 c0 (q0 ) 1 "1 (q0 ) = 1 + (1 ) : As I focus on q0 2 [0; q ] where q is given by u0 (q) = c0 (q), a solution of q0 must exist by the Theorem of Maximum. For any > , q0 > 0 for any monetary equilibrium with or without credit. It follows that monetary equilibrium must exist for > . In monetary equilibrium without credit, the second hold condition is negative so that q0 is unique and hence "0 is unique. In monetary equilibrium with credit, it is less clear that q0 must be unique. Following the proof in Wright (2010), one can show that equilibrium q0 is unique for generic values of . If q0 is unique, "0 is also unique from (22). In addition, implies that "1 is unique. Given that d"1 dq1 d"1 dq0 > 0 from (25). It > 0 from (23), q1 is unique. When the monetary authority implements ! , "0 = "1 = 1 and q(") = q (") for all ". The Friedman rule achieves the e¢ cient allocation. 31 A.4 Proof of Proposition 2 Proof. Total di¤erentiate (24), (22), (23) and (25) with respect to k: c0 (q0 )u00 (q0 ) u0 (q0 )c00 (q0 ) 2 dq0 u0 (q0 ) d"1 ("1 "20 ) + ["1 0 ] 0 2 2[c (q0 )] dk c (q0 ) dk 0 00 0 00 c (q0 )u (q0 ) u (q0 )c (q0 ) dq0 1 d"0 + 2 0 2 [c (q0 )] dk "0 dk 0 00 0 00 d"1 c (q1 )u (q1 ) u (q1 )c (q1 ) dq1 + 2 [c0 (q1 )]2 dk "1 dk d"1 dq0 [u(q1 ) u(q0 )] 1 ["1 u0 (q0 ) c0 (q0 )] dk dk = 0; = 0; = 0; = 0: 0 (q0 ) From (25), one can show that q1 > q0 . Together with (22) and (23), "1 uc0 (q 0) that d"0 dq0 d"0 dk ; dk ; dk and d"1 dk > 0. It follows are equal in sign. Moreover, c0 (q0 )u00 (q0 ) u0 (q0 )c00 (q0 ) 2 ("1 2[c0 (q0 )]2 dq0 "20 ) dk 0 + (q0 ) ["1 uc0 (q 0) ]["1 u0 (q0 ) [u(q1 ) c0 (q0 )] dq0 u(q0 )] dk = 1 u(q1 ) : u(q0 ) (32) De…ne f (q0 ; ) = "0 (q0 ) + 1 u0 (q0 ) 2 [" (q0 ) 2 c0 (q0 ) 1 "20 (q0 )] + [1 "1 (q0 )] 1 ; where "0 (q0 ) is from (22) and "1 (q0 ) is from (23) and (25). As the equilibrium q0 must be local maximum, the second order condition must hold. It must be true that f 0 (q0 ; ) < 0, so c0 (q0 )u00 (q0 ) u0 (q0 )c00 (q0 ) 2 ("1 2[c0 (q0 )]2 From (32), I have dq0 dk > 0 and hence, d"0 dk 0 "20 ) > 0; + (q0 ) ["1 uc0 (q 0) d"1 dk 32 ]["1 u0 (q0 ) [u(q1 ) > 0 and dq1 dk > 0. u(q0 )] c0 (q0 )] < 0: (33) A.5 Proof of Proposition 3 Proof. The proof is similar to the proof of Proposition 2. Total di¤erentiate (24), (22), (23) and (25) with respect to : c0 (q0 )u00 (q0 ) u0 (q0 )c00 (q0 ) 2 u0 (q0 ) d"1 2 dq0 (" " ) + [" ] 1 1 0 0 2 0 2[c (q0 )] d c (q0 ) d 0 00 0 00 c (q0 )u (q0 ) u (q0 )c (q0 ) dq0 1 d"0 + 2 [c0 (q0 )]2 d "0 d d"1 c0 (q1 )u00 (q1 ) u0 (q1 )c00 (q1 ) dq1 + 2 0 2 [c (q1 )] d "1 d dq0 d"1 ["1 u0 (q0 ) c0 (q0 )] [u(q1 ) u(q0 )] d d It is clear that d"1 dk > 0 and d"0 dq0 d"1 d ; d ; d dq1 dk and d"1 d have the same sign. Using (33), dq0 d = 2 ; = 0; = 0; = 0: > 0. Moreover, d"0 dk > 0; > 0. 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