Dealer Pricing of Consumer Credit Giuseppe Bertola,a,b Stefan Hochguertel,a,c Winfried Koenigera,d This draft: January 2003.∗

Abstract

We study how monopolistic price discrimination may induce dealers to bear the financial cost of their customers’ credit purchases, focusing in particular on the role of financial market imperfections in allowing sellers to segment their customers. Our empirical analysis takes advantage of a set of installment-credit, personal-loan, and regional interest rate data. We characterize consumers’ intertemporal consumption decisions when their borrowing and lending rates are different not only from each other, but also from the internal rate of return of financing terms for a specific durable good purchase. Individual choices depend on the spread between the borrowing and lending rates, which therefore bear on the pricing of cash and credit purchases by sellers endowed with monopoly power. Our data include detailed information as to the incidence of dealer-subsidized credit, and offer considerable evidence in support of our theoretical frameworks’ assumptions and implications.

Keywords: Price discrimination, Financial market development, Liquidity constraints. JEL Nos: D10, D42, G2

a Finance and Consumption in the EU Chair, E.U.I., via Badia dei Roccettini 9, I-50016 San Domenico di Fiesole FI. http://www.iue.it/FinConsEU b Universit` a di Torino; c Free University Amsterdam; d IZA, Bonn.

∗

For helpful comments on earlier drafts we thank Fumio Hayashi, Guglielmo Weber, and other participants at the first Economics of Consumer Credit workshop, at the 2001 EEA Congress and 2002 Royal Economic Society Annual Congress, and at a Bonn University seminar. We are also grateful to Findomestic Banca for making available the data set analyzed in this paper and for sponsoring, with CETELEM, the Finance and Consumption in the EU Chair research program.

1

Introduction

Installment-payment terms for durable good purchases are typically quite attractive from a financial point of view. In fact, favorable credit terms are often extended to consumers by sellers of durable goods rather than by lending institutions (banks): the customer’s flow of installment payments is then lower than what would be required by the bank’s cost of funds, processing costs, and assessment of repayment probabilities, and the amount credited by the bank to the seller’s account is less than the cash price. In principle, this phenomenon may be explained by the same incentives to engage in monopolistic price discrimination that explain the lower prices charged to consumers who own particularly old trade-ins, or take the time to clip coupons. In real-life financial markets, borrowing rates are higher than lending rates. Hence, different groups of consumers are attracted by cash and credit purchases, and sellers can set cash and credit terms so as to effectively charge different prices to cash-rich and liquidity-constrained customers. Demand by consumers who find credit purchases attractive can be more or less price-sensitive than that by consumers who can readily pay cash, depending on the characteristics of the population of potential customers facing a seller. So, this intuitive mechanism can explain not only the existence of dealer-subsidized consumer credit, but also its different prevalence across different regional and sectorial markets, and the possibility that dealers charge finders’ fees to lending institutions–effectively increasing the borrowing rate for liquidityconstrained customers above the interest rate that lending institutions would be willing to extend for loans tied to durable purchases.

1.1

Related literature

We are certainly not the first to analyze seller-financed credit from a monopolistic price discrimination perspective. Brennan, Maksimovic, and Zechner (1988) study incentives for sellers of investment goods to finance their ‘poor’ (liquidity constrained) customers’ purchases, and an extensive literature studies more general forms of trade credit. In a business environment, suppliers rather than banks may provide credit when they are in a better position to screen, select, and discipline the lender, or to repossess and use the loan’s collateral, as well as for price discrimination purposes. Many contributions (surveyed by Petersen and Rajan, 1997) analyze credit extended to producers, and focus in particular on asymmetric information channels, which play a crucial role when firms’ inputs or outputs may serve as loan collateral. As regards credit extended to consumers, credit is generally cheaper when explicitly tied to the purchase of a good or service than in the case of a general consumption loan. Relatively low interest

1

rates are not surprising if the good purchased serves as collateral, as in the case of houses, cars, and other valuable items. And while repossession is not an economically appealing option in the case of appliances and consumer electronics, borrowing in order to purchase items with low second-hand value is only sensible if the item is actually used. Hence, credit risk is lower than in the case of non-finalized loans: lenders need not worry about borrowers gambling away the loan, and purchase of household appliances may a better indication of the consumers’ unobservable inclination to repay than purchases of (say) fast motorcycles, guns, and other goods which are unlikely to feature such a correlation. More generally, Wertenbroch (in press) reviews possible reasons for credit to be packaged with sales to consumers, including facilitating impulse purchases and reducing transaction costs. However, the “zero” annual percentage rates credit advertised by car dealers and department stores are only possible when dealers choose to bear the financial cost of their customers’ borrowing, thereby charging different prices to ‘cash’ and ‘credit’ customers–a behavior that, as mentioned above, is most readily explained by price discrimination incentives. Some of the issues that motivate our work have been addressed in specific instances, such as the use of credit cards as means of payments, and pricing of installment-loan purchases which make the lender jointly liable in case of seller default on the obligation to deliver goods. Customers who expect to pay the balance in full should always use their credit card, to take advantage of the float, rather than pay cash. Since transaction fees are levied on merchants, and card issuers often require that customers pay the same price as for cash transactions, a seller’s choice of whether or not to accept credit (or debit) card in lieu of cash payment is less trivial. It depends on the quantity and quality (in willingness-to-pay terms) of additional sales generated by credit card acceptance (Murphy and Ott, 1977; Chakravorti and To, 1999). In the model of Iossa and Palumbo (2000), sellers also have incentives to stipulate joint-liability agreements with lenders if this increases their sales volume, because the additional financing costs entailed by the lender’s joint liability are more than offset, from the perspective of buyers, by the fact that the installment loan will not need to be repaid in case of dealer default. Little has been done, however, to model from first principles the consumers’ side of subsidized or ‘discriminating’ credit deals. Realistic interest-rate differentials across different consumer-loan instruments have not been subject to theoretical and empirical analysis as frequently and carefully as other forms of financial market imperfections, such as straightforward liquidity constraints independent of the borrower’s consumption bundle. Juster and Shay (1964) noted that interest rates are different on consumers’ assets, liabilities, and durable purchases, characterized qualitatively the implications of this state of affairs for consumer choices, and explored empirically survey data focusing in particular on the sensitivity of aggregate consumption to changes in macroeconomic monetary

2

conditions. Attanasio (1995) also focuses on the importance of cash outlays for liquidity constrained consumers, who are prepared to pay higher interest rates in exchange for longer loan duration. Empirical relationships between borrowing opportunities and durable-good purchases have been studied by Brugiavini and Weber (1994) and Alessie, Devereux, and Weber (1997), who propose and study models where borrowing limits depend on the existing stock of durable goods, rather than on new purchases as would be implied by the mechanisms outlined above.

1.2

Outline of the paper

Our value added on this small body of literature is both theoretical and empirical. While we do not address information and competition issues, as done by studies of similar phenomena in business environments, we explicitly model optimization by financially constrained consumers. Our analysis has to explicitly take into account discrete features in the consumer’s problem, which features discretely different borrowing rates upon purchase of durable goods, and of the dealer’s choice to subsidize credit. This makes it possible to link customers’ inclination to borrow (and sellers’ pricing strategies) to deep features of the economy under study, such as financial market structure, rather than to ad hoc segmentation of the population by liquidity constraints. Hence, we will be able to take advantage of data on incidence of phenomenon and of data on relevant (consumer-related) aspects to seek evidence of the structural mechanisms we study. After reviewing the data in Section 2, in Section 3 we propose and solve a simple but fully specified model of purchase decisions by consumers endowed with different, and differently timed, amounts of purchasing power. Consumers are faced with different interest rates on positive assets and negative assets (consumption loans). The rate of return implicit in the comparison of cash and credit prices for the good under consideration is larger than the interest rate on the consumer’s assets, but lower than that charged on his negative assets. In Section 4 we study how interest rates influence consumers’ choices and sellers’ credit subsidy incentives, and in Section 5 we bring empirical analysis of a credit contract database to bear on the real-life relevance of the model’s simple theoretical insights, finding that theoretical predictions are confirmed in our data set.

2

Dealer credit subsidies in Italian data

The data set available for our analysis includes a random sample of more than 200,000 credit applications by more than 120,000 individuals, over the period 1995—1999, to the leading supplier of consumer credit in Italy. Not all applications are accepted, but we refrain from modelling theoretically and empirically possible relationships between concession of credit by the bank and dealer

3

subsidization of financial terms.1 In the Italian markets from which our data set is drawn, dealers can choose to absorb interest payments and offer attractive financial terms to those among their customers who are inclined to purchase on credit. Since most items are small, there is no negotiation of financing terms, which are simply posted in the store along with the item’s price. The installment-purchase data set contains information on the characteristics of both the individual and of the good purchased. Of course, the data do not offer any information about customers who pay cash, little information about credit customers, and no information about dealers (only the item purchased is observed). However, they are better suited than any we are aware of to offer relevant empirical evidence, both because the heterogeneity of Italian regions’ financial development offers a useful source of relevant variation, and because each observation features an indicator of whether the dealer pays interest wholly, in part, or not at all (in the latter case, financial charges can be computed as the internal rate of return of the consumers’ repayment obligations). In practice, and presumably for simplicity, sellers typically pay all interest if they do subsidize their customers’ credit.2 This is the familiar case of ‘zero’ percentage rate, though of course even full subsidization of interest rates does not make credit completely costless for consumers (who bear transaction costs on installment payments, and may also be required to pay processing and insurance fees). The incidence of such subsidized credit purchases is high and variable across goods, as well as across regions of Italy and over time. Tables 1 and 2 below show percentages of complete dealer subsidy in region-quarter-item cells, averaged over time for the quarters 1996/I and 1999/I (our data set includes observations for earlier and later dates, but is most representative for the 1996-99 period; we focus on the first quarter to mitigate seasonal variation in the raw data). Table 1 takes averages within regions across items, and Table 2 takes averages within items across regions. The incidence of subsidized credit declines slightly over time, from 28.3% to 27.0%. Low rates of complete dealer subsidization are observed in Val d’Aosta and Molise, but these small regions are not representative of Northern and Southern Italy, respectively. High rates were found in, for instance, Friuli—VeneziaGiulia (North) or Sicily (South). In 1996 the regional discrepancies were larger than three years later. We also find large differences of dealer subsidization across items bought. Relatively low subsidization rates are observed for cars, while household appliances and electronics (“white” and “brown” goods) are relatively frequently subsidized in Table 2. There is also substantial variation over time 1 Alessie, Hochguertel, and Weber (2001) offer a detailed description of the data set, and empirical analysis of the time-series and cross-sectional impact of introduction of a law on usury rates in 1997. 2 Very few (less than 1.5%) of the contracts in the sample see dealers paying only a portion of the bank’s interest charges, and most (74%) of such contracts relate to motorcycle purchases, perhaps as a reflection of the high rates charged on these relatively expensive items. We decided to drop all partially subsidized installment plans from the sample.

4

Table 1: Dealer Subsidization by Region region

1996/I

1999/I

Piemonte Valle d’Aosta Lombardia Trentino/Alto Adige Veneto Friuli-Venezia-Giulia Liguria Emilia Romagna CENTER Toscana Umbria Marche Lazio SOUTH & ISLANDS Abruzzo Molise Campania Puglia Basilicata Calabria Sicilia Sardegna Total

24.5 14.3 26.0 41.7 37.8 51.4 20.3 40.5

22.5 20.0 30.0 31.6 32.0 41.6 23.5 28.8

37.5 24.3 50.3 26.5

25.9 23.9 33.7 22.6

21.9 5.3 18.9 21.7 31.0 10.7 41.9 19.8 28.3

21.1 15.0 27.2 18.3 3.7 38.3 37.6 43.1 27.0

NORTH

Table 2: Dealer Subsidization by Item Bought item financed white goods hh appliances brown goods computers furniture telephony motorcycles (used) cars new cars Total

1996/I 42.8 37.0 32.4 21.0 25.6 20.7 52.1 6.0 14.3 28.0

5

1999/I 50.4 39.8 41.5 15.7 23.2 24.1 39.9 6.9 4.0 27.3

Table 3: Distributions of Quarterly Regional Interest Rates quarter 1996/4 1998/4 1996/4 1998/4 1996/4 1998/4 1996/4 1998/4

variable level spread bank borrowing rate bank lending rate

mean 9.08 5.24 6.33 5.47 5.92 2.50 12.24 7.97

std.dev. 0.66 0.58 1.51 1.16 0.23 0.20 1.40 1.15

coeff.var. min max 0.07 8.10 10.44 0.11 4.38 6.18 0.24 4.47 9.56 0.21 3.58 7.26 0.04 5.40 6.27 0.08 2.14 2.93 0.11 10.33 15.22 0.14 6.17 9.77 source: Bank of Italy

of subsidization rates across goods. The subsidization rate for white goods, for instance, increased from 42.8% to 50.4%, while it fell for new cars from 14.3% to 4.0%. Italy is a country of stark regional differences in terms of financial market development. To document these, Table 3 shows nominal bank lending and borrowing rates, the rate spread (difference between lending and borrowing rates), and the level of these rates (average of borrowing and lending rates). These statistics are based on quarterly regional measures of financial rates applicable to relationships between the banking sector and the private sector. The available data, collected and published by the Bank of Italy, refer to loans exceeding LIT 150m (approximately euro 75,000) and deposits exceeding LIT 20m (approximately euro 10,000). Clearly, these rates are not directly relevant to consumers’ financial environment, but their level and the spread between active and passive rates are related, over time and across regions, to those applicable to smaller loans and deposits. The table shows that interest rates declined over time from 9.1% to 5.2%. The rate spread also declined over time, from on average 63 basis points to 55. The interregional differences of the spread measured by the coefficient of variation declined. In particular, lending rates declined sharply. In terms of regional segmentation we find higher levels and larger spreads in the southern regions, conforming to popular wisdom of regional financial development in Italy. Our modeling approach indicates that such financial market variables should be relevant to dealers’ incentives to offer subsidized credit. To see whether and how the incidence of dealer subsidization covaries with financial market structure, we next show patterns of temporal correlation of dealer subsidization with interest rate spreads controlling for the rate level. We partition available data in three macro regions (North, South, and Center, as defined in Table 1). To control for interest rate level variability, we further partition each subsample in the five quintiles of the mean bid/ask interest rate distribution. Within each of the 15 subsamples, we analyze the role of interest rate

6

0-20% 60-80%

20-40% 80-100%

40-60%

dealer subsidization (fraction)

.5

.4

.3

.2

.1 4

3

North

spread

5

6

Figure 1: Relationship between dealer subsidization and interest rate spreads, average within the five quintiles of the rate level distribution: Northern Italy.

0-20% 60-80%

20-40% 80-100%

40-60%

dealer subsidization (fraction)

.8

.6

.4

.2 4

Center

5 spread

6

Figure 2: As Figure 1, for Central Italy.

0-20% 60-80%

20-40% 80-100%

40-60%

dealer subsidization (fraction)

.4

.3

.2

.1 4

6

South

spread

8

Figure 3: As Figure 1, for Southern Italy.

7

10

bid-ask spreads (a measure of financial market imperfection) by regressing the region- and quarterspecific incidence of seller-subsidized credit on a third degree polynomial in the applicable bid-ask spread. The resulting descriptive evidence is displayed graphically in Figures 1 through 3. A hump shape is apparent in most cases: in the following sections we show that this type of non-monotonic relationship is indeed implied by our theoretical perspective, and we provide additional empirical evidence for it in a controlled-regression framework.

3

A consumer’s problem

We illustrate more general theoretical insights with an admittedly simple, but far from trivial formal model. In general, a consumer’s decision to purchase a durable good depends on tastes, on prices, and on the funds available for that and other purposes. When financial markets are imperfect, current and future funds are not perfectly substitutable: hence, optimal choices depend not only on the total amount but also on the timing of the consumer’s resources, which interacts in interesting and complex ways with the relationship between the cash and credit prices of the good and with the intertemporal rates of transformation applicable to borrowing and lending contracts. In order to model such issues in the simplest possible way, we consider a two-period representation of the consumer’s preferences. Let C denote nondurable consumption in the current period, when the choice of whether to buy a specific durable good is made, and let A denote the funds available for future purchases. Utility is assumed to be increasing and concave in both current consumption and future resources and, of course, given levels of C and A should be associated with higher levels of utility when the durable good is available for use in the current and future periods. We choose to illustrate the character of the solution in a simple special case where closed-form solutions are available, namely:3 max log(C) + log(A) + δk, C,δ

(2)

where the additive constant k representing the positive impact of the durable’s services on utility. This parameter summarizes a host of possible reasons why the durable is attractive (including, in a 3 More general representations of intertemporally separable preferences would take the form U (C, δ) + V (A, δ), with δ = 1 if the durable good is available and δ = 0 otherwise, such that

U (C, 1) + V (A, 1) ∂U (·) ∂ C

>

U(C, 0) + V (A, 0),

>

∂V (·) ∂ 2 U(·) ∂ 2 V (·) 0, > 0, < 0, (1 + rb ) W . When the intertemporal pattern of the consumer’s resources falls in the cone from the origin between these two lines, the optimal constrained consumption pattern simply coincides with available resources and, if the durable is not purchased, utility only accrues from C = W and A = Y . The other lines plotted in the figure, whose slopes also coincide with the intertemporal rates of transformation associated with positive and negative assets, identify cones originating from P1 on the vertical axis and from P0 on the horizontal axis. If the consumer purchases the durable good on a credit basis, then the levels of C and A coincide with the amount of current and residual future funds if these lie within the cone originating from P1 , and reflect optimal borrowing or lending choices if they lie outside of it. It is similarly easy to characterize the implications for C and A of a decision to purchase the durable on a cash basis, referring to the cone originating from P0 on the horizontal axis of the figure. In order to establish optimality of cash, credit or no purchase, the utility levels achieved in those cases need to be compared with each other. The three utility levels depend univocally on the parameters of the problem (resources and prices) through the intertemporal allocation choices determined by slack Euler conditions like (5). Hence, such comparisons are conceptually easy and, 10

•N •M

•Q

Figure 4: A graphical representation of the Euler conditions for choice of C and A, conditional on whether the good is purchased with cash, on an installment loan basis, or not at all. as mentioned, they could be performed numerically for much more general preference specifications than that proposed in (2). One might for example allow for discounting of future utility, or for non-homothetic utility. Such generalizations would imply more complex relationships than those represented in Figure 4, where the borrowing, lending, and liquidity-constrained ranges would in general be delimited by non-linear upward-sloping lines. The qualitative character of the solution, however, does not depend on such details and, because of the discrete nature of the optimization problem, formal quantitative representations are already quite complex and intriguing for the simple objective function.

3.1

Character of the solution

We proceed to characterize the solution of the problem (2) with respect to durable purchase choices, in terms of indifference conditions between purchase and no-purchase choices. As is intuitive, and apparent in figures we discuss below, the current and future funds endowments that make the consumer indifferent to purchasing or not purchasing the durable identify a (weakly) downwardsloping locus in a figure like Figure 4: since the marginal utility of C and A is decreasing, the consumer needs to be rich enough (in terms of current and/or future funds) before the choice of diverting some of his or her purchasing power from C and A to the durable (which affords a given

11

utility level k) becomes attractive. The endowment structures that make consumers indifferent to purchasing the good can never lie on an upward-sloping locus in the (W, Y ) space.4 Also intuitively, among consumers who do purchase the durable indifference between cash or credit purchase (use of current or future funds) is depicted by an upward-sloping locus in a figure like Figure 4: consumers need to be relatively well endowed with future funds for a credit purchase to be preferable. The general character of the solution is illustrated in Figure 5 below, but the shape of the relevant indifference loci depends in interesting ways on the relationship between the borrowing and lending rates on the one hand, and on the cash and credit price of the good on the other. To organize the derivation of the indifference loci and to offer some intuition for their shape, it will be helpful to refer to the regions of the three points M , N and Q in Figure 4. In what follows, we present the solution for these three regions; the Appendix discusses the complete solution of all the ten regions defined by the Euler conditions in the (W, Y ) space, within which the possible lending, borrowing, and purchase choices are restricted in different ways.

3.2

Indifference to purchase

If the consumer’s endowment is such that optimal assets are positive regardless of whether the durable good is purchased on a cash basis, or on credit, or not at all (as is the case at point Q in Figure 4), it is straightforward to characterize the optimal purchase decision. If the durable good is not purchased, then current consumption is arg max C

[log(C) + log (Y + (W − C)(1 + ra ))] µ ¶ Y 1 W+ = ≡ CN,a , 2 1 + ra

while a cash purchase reduces current consumption to arg max C

[log(C) + log (Y + (W − C − P0 )(1 + ra ))] µ ¶ 1 Y W − P0 + = ≡ CD,a 2 1 + ra

and affords additional utility k. The choice is a matter of indifference for the consumer when log(CD,a ) + log (Y + (W − CD,a − P0 )(1 + ra )) + k

(6)

= log(CN,a ) + log (Y + (W − CN,a )(1 + ra )) , 4 Suppose instead the purchase-indifference locus were positively sloped. Consider a point (W ∗ , Y ∗ ) on that locus and a point (W ∗ + x, Y ∗ ) to its right, where x > 0. The consumer would purchase the durable at (W ∗ , Y ∗ ), but not at (W ∗ + x, Y ∗ ) despite the fact that overall resources have increased. This cannot be optimal.

12

a quadratic equation in P0 and W + Y / (1 + ra ), the present value of funds discounted at the lending rate. The left-hand side of (6) is larger than the right-hand side (to imply that cash purchase is preferable to no purchase) if

5 1

W+

Y e2k > 1k P0 , 1 + ra e2 − 1

(7)

namely if the consumer’s endowment of current and future resources,at a point such as Q in Figure 4, lies to the north-east of a downward sloping line. It remains to be checked whether credit purchase might in turn be preferable to cash or no purchase. Quite intuitively, however, credit purchase cannot be strictly optimal for the consumer if the endowment, as is the case at point Q, is such that assets are positive regardless of whether and how the durable is purchased. In fact, when assets are positive, use of future rather than current funds can never increase the amount of future purchasing power A for any choice of C, and hence utility, since Y + (W − C)(1 + ra ) − P1 ≤ Y + (W − C − P0 )(1 + ra ) as long as P1 ≥ (1 + ra ) P0 as assumed in (4). Symmetric reasoning is applicable when the endowment is such that assets are negative regardless of whether and how the durable is purchased (see point M in Figure 4). In that case, credit purchase is always at least weakly preferable to cash purchase, and preferable to no purchase if 1

Y e2k P1 W+ > 1k , 1 + rb 1 + rb 2 e −1 again a downward-sloping line in the plane depicted by the figures. The two indifference lines would coincide if the borrowing and lending rates were equal to each other and, by (4), to the internal rate of return of the durable’s installment-credit plan. In such a perfect-capital-markets case, the solution would be easy and uninteresting: all the cones would collapse to lines in the figure, consumers would always be indifferent between cash and credit purchase, and only the present value of intertemporal resources would affect their choice of whether to purchase or not. When ra < rb , conversely, the indifference locus is a more steeply declining line in the region of point M than in the region of point Q, and becomes interestingly nonlinear when the consumer’s endowment lies outside of those regions. 5 The

Y left-hand side is also larger if W + 1+r < a

1k

e2

P0 , 1k e 2 +1

but such low wealth levels would imply negative current

consumption in the relevant region: Ã Ã ! ! 1 ¶ µ 1 1 1 1 e2k Y = − < 0. − P0 < − P P P W+ CD,a = 0 0 0 2 1 + ra 2 e 12 k + 1 2 e 12 k + 1

13

Characterization of the purchase decision is also quite straightforward when the (W, Y ) endowment of purchasing power lies in the region of point N , i.e., within all three of the cones plotted in Figure 4. When the consumer has zero assets regardless of whether and how the good is purchased, a cash purchase is preferable to no purchase if log(W ) + log (Y ) < log(W − P0 ) + log (Y ) + k, i.e. if W > P0

ek ; ek − 1

(8)

and credit purchase is preferable to no purchase if log(W ) + log (Y ) < log(W ) + log (Y − P1 ) + k, i.e. if Y > P1

ek . ek − 1

(9)

The parameters may be such that these conditions are automatically satisfied for all endowment patterns in the region of point N . The horizontal and vertical coordinates of all points in that region, in fact, are larger than those of the intersection point of the flatter boundary of the cone originating from P1 on the vertical axis with the steeper boundary of the cone originating from P0 on the horizontal axis, i.e., the solution of P1 + (1 + ra ) W = (W − P0 ) (1 + rb ), W =

P1 + P0 (1 + rb ) , rb − ra

(10)

on the horizontal axis and Y = (P1 + P0 (1 + ra ))

1 + rb . rb − ra

(11)

If the right-hand sides of (8) and (9) are larger than those of (10) and (11), then purchase of the durable is always optimal in the region of point N . Otherwise, the purchase-indifference locus goes through that region, where it consists of a vertical and horizontal segment. To complete the characterization of optimal choices in the region of point N , note that for a consumer who always has zero assets credit purchase is preferable to cash purchase if log(W − P0 ) + log (Y ) + k < log(W ) + log (Y − P1 ) + k, i.e., if Y /W > P1 /P0 . Thus, any portion of the upward-sloping cash/credit purchase indifference locus that lies in the region of point N is a straight line segment, with slope given by the installment plan’s internal interest factor. We have already shown that all points in the region of point M (Q) lie above (below) that locus; we will derive the shape in other regions of the (W, Y ) plane in the Appendix. Whenever the purchase-indifference locus goes through the region of point N , it has a horizontal and a vertical segment. In general, the nonlinearity of the purchase-indifference locus becomes more 14

pronounced as the consumer’s imperfect access to borrowing and lending opportunities becomes more relevant to the purchase decision. Intuitively, the indifference locus tends to become (and is, in the region of point N ) horizontal when the consumer is liquidity constrained, because when future resources are too low to make a credit purchase appealing (and financial markets are not accessed) then an increase of current resources does not make such a purchase any more desirable. For a given degree of financial market imperfection, as represented by the difference between rb and ra in the model, the relevance of financial market access for durable purchase decisions depends on k. Indifference conditions in the other regions, where purchasing the durable good is always associated with a discrete change in the consumer’s asset position, can be derived by much the same reasoning as that applied to the regions–considered above–where assets are positive, negative, or zero regardless of purchase decisions. In all cases, indifference to purchase is characterized by comparisons of utility reflecting the optimal intertemporal pattern of consumption if the consumer smooths it by either borrowing or lending, or just the intertemporal pattern of resources if the consumer is liquidity constrained. We discuss solution methods in the Appendix, where we characterize in closed form the subset of the (W, Y ) space where cash purchase is preferable to no purchase, and that where credit purchase is preferable to no purchase. The intersection of these sets includes all intertemporal endowment patterns that make purchase optimal. To complete characterization of the solution, the set of (W, Y ) points for which purchase is preferable to no-purchase can be further partitioned according to whether cash or credit payment is optimal. The boundary of these subsets is an (upward-sloping) locus of points such that the choice of cash or credit terms is a matter of indifference. We have already characterized this locus in the region of point N , where the consumer has no assets in all cases and, quite intuitively, prefers credit to cash purchase when Y /W > P1 /P0 , i.e. when the financing terms of the durable purchase offer an attractive intertemporal rate of transformation for the assumed logarithmic specification of preferences. As mentioned above, in other regions (where the purchase is associated with a change in the consumer’s asset position) complex considerations are relevant to the choice of whether to purchase on a cash or credit basis. We report in the Appendix the exact solution for the cash/credit indifference locus in all regions, which is continuous with slope lower than 1 + rb and larger than 1 + ra . Intuitively, in all regions where the purchase implies a qualitative change in the consumer’s asset position (which can be positive, negative, or zero) the slope of the cash-credit indifference locus is a weighted average of the applicable intertemporal marginal rates of substitution, which all lie between 1 + ra and 1 + rb . We illustrate the qualitative nature of the solution in Figure 5. The purchase-indifference locus is the lower envelope of the loci which define when the consumer is indifferent between cash purchase

15

Figure 5: The solution for the purchase indifference and cash-credit indifference frontier and no purchase or credit purchase and no purchase, respectively. The shape of the purchase indifference locus is intuitive. For example, the locus defining indifference between credit purchase and no purchase becomes flatter in the region of the (W, Y ) plane in which the consumers’ endowment is more tilted towards the present. Starting from an endowment at which the consumer does not purchase the durable, the consumer needs relatively more additional current resources W to buy the durable on credit if his initial endowment is relatively more tilted towards current resources. In the figure, and below, we denote with ω the level of current wealth such that the purchase-indifference and the cash-credit indifference locus intersect.

4

Consumer choices and dealer pricing

We proceed to illustrate graphically the properties of the solution. Our functional form assumptions imply that all conditions of indifference to purchase define quadratic equations in the (W, Y ) plane. Any modification would substantially complicate analytic solution. For example, applying a discount factor β 6= 1 to utility accruing from future purchasing power A would require solution of equations of order 1 + β, rather than of quadratic equations. This and other extensions, however, would not alter the qualitative character of the consumer’s choices. More complex models could in principle be studied by numerical methods, but even in our simple case derivation and description of the 16

solution are considerably complicated by the need to consider all the possible interactions between the discrete decision to purchase the durable good and the (also discrete) changes in the budget set implied by the possibility of switching from positive to negative asset positions, or vice versa, when a credit or cash purchase alters the intertemporal pattern of residual funds. This feature captures important elements of reality, since not only the rate of interest but also the structure of installments is severely limited in most durable-good purchases, but has complex implications for optimal behavior. For example, the discreteness of purchase and financing choices implies that it may be optimal for consumers to buy the durable on credit and carry a positive (though less than in the case of no durable purchase) amount of positive assets into the next period. This phenomenon is reminiscent of the puzzling coexistence of credit card debt and liquid assets in many consumers’ portfolios (identified and studied by Bertaut and Haliassos, 2002, and their references), which may also be explained at least in part by transaction-cost discreteness. In our context, holding positive assets and borrowing against a durable good purchase is optimal when the intertemporal endowment structure is such that the (discrete) first-period consumption compression implied by a cash purchase would yield lower utility than a smaller consumption decline in the first period, smoothed intertemporally by a larger positive asset position, and payment of the good on a credit basis. Crucially for our purposes, the nonlinearity of the purchase indifference frontier depends on the relationship between rb and ra . This is illustrated in Figure 6, where we plot purchase-indifference loci for increasingly large values of rb , keeping k, ra , and P0 constant, and varying P1 so that the internal rate of return of the installment-payment plan is always equal to the average of the borrowing and lending rates in the financial market (P1 /P0 = 1 + (ra + rb )/2). The straight downward sloping line in the figure represents the case of perfect financial markets, where ra equals rb and the internal rate of return of the installment plan. We do not plot the cones defining liquidity-constrained regions in the figure. It is not difficult to see, however, that regions where assets are zero (contingent on one or more of the possible choices open to the consumer) become larger as financial market access becomes more difficult. In the figure, financing terms for durable-good purchase become relatively more favorable compared to unconditional loan rates, and the nonlinear character of the consumer’s choices becomes more and more pronounced. Additional qualitative features of the solution are illustrated in Figure 7, where for a given k we keep P0 , ra and rb constant but vary P1 . Not surprisingly, as P1 increases the purchase-indifference locus shifts outward, i.e., the consumer needs to be richer in order to find it optimal to purchase the durable good. Of course, this is more pronounced for configurations of the endowment that are relatively tilted towards the future: any change of the delayed-payment terms for the good (as long

17

Figure 6: Consumer purchase decisions: implications of different borrowing rates; internal rate of return set equal to the average of lending and borrowing rate. as the internal rate is larger than ra ) is irrelevant for consumers who have positive assets and buy the durable with cash, while consumers who are liquidity constrained or borrow can be induced to buy the good by a better financing deal.

4.1

A dealer’s optimal pricing of financial terms

The simple qualitative insights outlined in the introduction, and the more precise quantitative perspective offered by our modelling approach, can be brought to bear on various aspects of reality and of the data available to us. In particular, we explore how incentives by dealers to subsidize their consumers’ credit purchases depend on the structure of financial markets. We consider the case of a monopolist dealer who has a specific durable good available for sale. The basic insights generalize to other models with imperfect competition as long as there is scope for price discrimination. Normalizing marginal cost to zero with no loss of generality, the objective of the dealer’s pricing decisions is defined as follows: R(P0, P1 ) = P0 D0 (P0 , P1 ) +

P1 D1 (P0 , P1 ), 1 + rf

(12)

where D0 (P0 , P1 ) is the quantity purchased on a cash basis at price P0 , and D1 (P0 , P1 ) is the quantity purchased on a credit basis. For each customer who purchases the good on credit, the dealer receives 18

Figure 7: Consumer purchase decisions: implications of different installment-credit interest rates. from the bank the customer’s installment payment P1 , discounted at rate rf . In the model, as in reality, the bank handles the financial side of the durable purchase transaction, and the dealer faces no default risk. Suppose the dealer is faced by a heterogeneous population of potential customers that behave like the one characterized in the previous section. The parameter k, of course, is in general different across consumers as well as across goods. No additional insight would be afforded by making that heterogeneity explicit, however, over and above the implications of consumer heterogeneity in terms of current and future financial resources. Formally, we suppose that purchasing power is distributed across consumers according to the bivariate density function f (W, Y ). Let Y = χ(W ; P0 , P1 , k, ra , rb ) denote the upward-sloping cash-credit indifference locus of Figure 5, and let Y = π(W ; P0 , P1 , k, ra , rb ) denote the downward-sloping line of indifference between purchasing and not purchasing the good in that and the other figures of the previous section. As shown above, the future-resource level Y identified by each of these loci depends on current resources W , as well as on the cash and credit prices and on the specification of tastes (parameterized by k in the model) and of the financial market environment (parameterized by ra and rb ). If ω denotes the current-resource coordinate of the two schedules’ intersection point (see Figure 5), implicitly defined by χ(ω; P0 , P1 , k, ra , rb ) = π(ω; P0 , P1 , k, ra , rb ), the quantity sold on a cash

19

basis can be expressed as an integral over the appropriate region of the previous section’s figures: # Z "Z D0 =

∞

ω

χ(W ;...)

f (W, Y )dY dW.

Similarly, the quantity sold on credit is given by: # Z "Z D1 =

∞

ω

∞

(13)

π(W ;...)

f (W, Y )dY dW +

χ(W ;...)

Z

0

ω

"Z

∞ π(W ;...)

#

f (W, Y )dY dW.

(14)

The model features three distinct interest rates, ra < rf < rb . Since “the bank” (or financial sector) is the counterpart of the consumers’ deposit and borrowing relationships, the wedges rf −ra > 0 and rb − ra > 0 reflect intermediation costs and non-repayment risk, left unmodeled for this paper’s purposes. The wedge rb − rf > 0 between the interest rates charged by the bank on unrestrained consumer borrowing and on durable good installment payments reflects differential transaction costs and repayment behavior, through the selection effects outlined in the Introduction. Of course a complete model would need to account for these mechanisms explicitly, in particular in the consumers’ optimization problems. Whereas an extension of the model along these lines is interesting for further research, in this paper we focus on the incentive for price discrimination taking rf as given. If the cash and credit prices are related according to P1 = (1 + rf ) P0 , then the dealer’s revenues are not affected by the proportion of cash and credit sales. Besides choosing the overall level of the item’s price, however, the dealer can also choose to set the installment price as a different ratio of the cash price. In fact, it will be generally optimal for the dealer to do so, and exploit the opportunity to price-discriminate among customers. As long as 1 + ra
1 + r¯) then no consumer would accept such unfavorable terms, and those who 20

prefer to devote future resources to the durable purchase would simply borrow on the market and pay cash. In reality, of course, consumers’ borrowing and lending rates are different: qualitatively, it will not be surprising to find that the wider the spread between them, the more pronounced are incentives for sellers to price discriminate.6 We illustrate the model’s implications with numerical computations based on a simple parameterized model. We suppose that the population’s distribution of current and future resources is well approximated by a bivariate normal distribution over the relevant region (this distribution attaches positive probability to negative levels of W and Y , which however are irrelevant because they can never be associated with purchase decisions). Qualitatively, the point could be illustrated with simpler distributional assumptions. However, a more general distributional assumption provides a suitable underpinning for empirical work. The normal distribution’s variance, mean, and correlation parameters can be varied very flexibly, and extensive numerical experimentation confirmed the robustness of the characterization results discussed below. In the data we observe that the dealer either fully subsidizes or does not subsidize at all. Hence, we focus on the case P1 = P0 (1 + ra ) when the dealer subsidizes.7 We compare profits in this case with profits when dealers do not subsidize, in which case P1 = P0 (1 + rf ) . The optimal choice of whether to subsidize depends on the discrete comparison between maximum profits in the two cases. Within each case, maximum profits can be computed evaluating numerical counterparts to the demand expressions (13) and (14) for every pricing choice by the dealer. Our program then determines the optimal one by numerically integrating the dealer’s objective function (12) and searching (with the Newton-Raphson algorithm) over the relevant region of P0 and P1 for each set of parameter values. Figure 8 reports the difference between the profits when dealers “subsidize completely” and “pass on all finance charges” as a function of the spread between customers’ borrowing and lending rates. 6 The information needed to establish the optimal (discriminating) pricing structure is not qualitatively different from that needed to implement optimal decisions in other monopolistic pricing models (see Dana, 2001, and references therein). It does include information regarding the customer pool’s need of, and access to, credit: and it is not unreasonable to suppose that dealers would be able to base their decision on relevant aggregate statistics, as well as on trial-and-error methods. 7 The model is applicable to more general situations, where dealers set prices so that 1 + r ≤ P /P ≤ 1 + r . a 1 0 b The main insights remain qualitatively the same, but it is worth noting that dealers might find it optimal to set P1 /P0 > 1 + rf and charge a fee for intermediating the consumer credit transaction. This may be the case in some segments of the Italian data set we analyze. However, information on incidence of that phenomenon is not available to us.

21

Figure 8: The difference in profits when subsidizing and non-subsidizing as a function of the spread (in basis points). Parameter values: k = .35, the mean of future and current resources is E(Y ) = 4, E(W ) = 3.5, their standard deviation and correlation is σ Y = 2, σ W = 2, ρY W = .5. We increase the spread symmetrically around rf = ra = rb = .16 (with 21 grid points each for rf and the spread), for a few values of rf . The difference in profits is a inversely U-shaped function. When the spread is small, in fact, there is little scope for price discrimination. As the spread increases, subsidization makes more of a difference in terms of possibly desirable price discrimination. However, as ra falls relative to rf , subsidizing becomes more expensive. Thus, the effect of an increase in the spread on dealer subsidies becomes less pronounced, and eventually decreasing. The smaller effect is, of course, also due to the fact that at high levels of the spread already most of the customers are “liquidity constrained” and an additional increase of the spread makes little difference. It is also possible to characterize the implications of increasing rf starting from ra , going towards rb . This is also inverse-U shaped: in the Figure, a higher rf shifts the difference in profits upward for low levels of rf but it shifts the difference in profits downward for high levels of rf . For relatively small interest rate levels and spreads, dealer subsidies become relatively more profitable for a given spread: the cost of subsidizing credit is relatively small, and a higher rf increases the scope for price discrimination. Initially, subsidizing means charging almost exactly the same price pattern and it is not surprising that it makes little difference to profits (in fact, the dealer might like to charge a higher borrowing rate than rf ≈ ra since rb is much higher). As rf increases, the large spread gives scope for price discrimination, and once subsidizing becomes attractive a larger rf actually makes it more attractive (because it implies a more pronounced change in the intertemporal price pattern). 22

However, subsidizing also becomes more costly, because the dealer has to absorb all charges. At some point, the latter effect dominates, and the profit difference becomes a declining function of rf . In reality, of course, the character of the distribution of current and future resources across the customer population also affects the scope for price discrimination (in the limit case of no variation across the population in the relevant respects, the dealer could not possibly discriminate among them). However our data, which we analyze empirically next, offer no information regarding this aspect.

5

Empirical evidence

To model empirically the variation observed in the incidence of seller-subsidized credit in our data, we focus on the probability that the dealer subsidizes credit as a function of the rate spread and of the interest rate charged by banks on the credit they grant. Estimating the empirical model on micro data makes it possible to control for other characteristics of the observed transaction. In light of the structural mechanism discussed above, we expect interest rates to matter nonlinearly for the likelihood of subsidization. To assess the empirical relevance of borrowing/lending spreads in the financial market, we rely on the substantial variation across regions and over time of the interest rates reported by the Bank of Italy (and discussed in our Section 2 above). The interest rate charged by banks on durable good credit purchases, however, is not observed in our data when the dealer pays it. Since we do observe interest rates when the consumer pays the interest in full, we estimate a model of interest rate determination and use predicted values from that empirical model to proxy for the rate charged by the bank to the dealer: this is appropriate if the bank’s financing rate is (to a first approximation) unrelated to dealers’ price discrimination incentives. Specifically, we run separate regressions for various broad categories of items purchased. In preliminary regressions, not reported, we included all available contract and customer information as explanatory variables, finding that interest rate variability is mostly accounted for by time, region, and type of item purchased, plus a limited number of contract characteristics (such as loan size).8 In the data, some types of item purchases are financed by personal loans (not intermediated by the dealer) as well as through standard installment credit contracts, and the coefficient of a personal-loan dummy estimate the interest rate surcharge when customers intending to make similar purchases have to obtain funds from the bank, rather than accept delivery of the goods from the dealer. 8 Variables that relate to the individual loan applicant are much less relevant, and not surprisingly so since creditworthiness does not affect the interest rate charged by the bank. Hence, individual characteristics become known (and may only affect whether credit is granted) after the terms of financing are decided.

23

Table 4: Internal rates of return, installment credit and personal loans

item # observations Adj. R-squared

furniture 8238 0.5583

motorcycles 5553 0.6754

coeff. t-value contract characteristics log(loan size in LIT) duration in months insurance* pay by bank* dealer-bank communication fax phone other personal loan personal loan [PL] PL x log loan size PL x insurance PL x quarter dummy regions Piemonte Val d'Aosta Lombardia Trentino/Alto Adige Veneto Friuli-Venezia-Giulia Liguria Emilia Romagna Toscana Umbria Marche Abruzzo Molise Campania Puglia Basilicata Calabria Sicilia Sardegna intercept

-0.0209 -24.12 -0.0003 -4.77 0.0187 4.29 -0.0084 -6.94

(used) cars 5059 0.7435

coeff. t-value -0.0072 -0.0001 0.0413 -0.0029

-6.11 -0.73 10.39 -2.18

new cars 6883 0.6140

coeff. t-value -0.0166 -17.64 0.0000 0.43 -0.0087 -9.61

white goods 4419 0.4554

coeff. t-value

hh appliances 8460 0.3803

coeff. t-value

brown goods 16156 0.4415

coeff. t-value

computers 3722 0.4130

coeff. t-value

telephony 17982 0.5020

coeff. t-value

coeff. t-value

-0.0038 -0.0001 0.0222 -0.0030

-4.92 -2.45 4.03 -5.69

0.0062 -0.0014 0.0990 -0.0086

2.50 -4.19 13.95 -3.52

0.0007 -0.0001 0.0780 -0.0090

0.54 -0.95 15.66 -4.61

-0.0068 0.0002 0.1105 -0.0057

-7.25 1.32 36.00 -4.58

-0.0151 -0.0010 0.1160 -0.0073

-7.79 -5.78 10.24 -3.61

0.0024 -0.0008 0.1050 -0.0056

2.08 -5.24 26.84 -4.74

-0.0140 -0.0054 -0.0016

-8.79 -3.28 -0.56

0.0000 0.0071 0.0040

0.01 5.02 1.21

-0.0106 -0.0116 -0.0127

-3.16 -3.45 -3.36

0.0013 0.0027

1.34 2.17

0.0190 0.0127 0.0271

3.86 3.03 2.43

0.0055 0.0023 0.0146

1.61 0.83 1.65

0.0155 0.0105 0.0121

5.81 5.44 2.05

0.0029 0.0096 0.0181

0.97 2.72 2.84

0.0093 0.0068 0.0042

5.25 3.91 0.76

0.0958 -0.0061 -0.0235 -0.0014

1.98 -2.00 -4.43 -2.80

0.2598 -0.0166 0.0177 -0.0021

2.64 -2.65 1.10 -1.96

0.3668 -0.0200 -0.0180 -0.0022

10.31 -9.17 -3.54 -3.83

0.2968 -0.0164 -0.0334 -0.0027

8.98 -8.25 -6.80 -6.54

-

-

-

-

-

-

-

-

-

-

-0.0072 -0.0232 -0.0073 -0.0401 -0.0102 -0.0232 -0.0182 -0.0244 -0.0038 0.0038 -0.0237 -0.0084 -0.0129 0.0034 0.0072 -0.0012 0.0077 -0.0051 -0.0171

-1.25 -1.75 -1.33 -4.86 -1.73 -2.53 -2.89 -4.31 -1.62 0.59 -5.73 -1.35 -0.98 0.62 1.29 -0.12 1.33 -0.96 -3.04

-0.0224 -3.73 -0.0400 -1.48 -0.0081 -1.48 -0.0181 -1.69 -0.0242 -4.15 -0.0343 -5.19 -0.0079 -1.16 -0.0245 -4.12 -0.0344 -14.93 -0.0203 -3.80 -0.0317 -7.34 -0.0076 -1.39 -0.0281 -1.89 -0.0283 -6.47 -0.0303 -6.25 -0.0160 -2.36 -0.0195 -4.10 -0.0316 -7.32 -0.0317 -5.94

-0.0197 -0.0221 -0.0156 -0.0171 -0.0178 -0.0215 -0.0141 -0.0208 -0.0134 -0.0063 -0.0099 -0.0267 -0.0166 -0.0159 -0.0284 -0.0192 -0.0186 -0.0258 -0.0277

-3.61 -3.16 -2.90 -2.11 -3.06 -3.13 -2.28 -3.66 -6.69 -1.49 -2.85 -4.39 -1.22 -3.26 -5.57 -3.29 -3.78 -5.38 -5.42

0.0001 0.0019 0.0036 -0.0001 0.0011 -0.0027 -0.0061 0.0039 -0.0052 0.0004 -0.0052 -0.0138 -0.0187 -0.0269 -0.0289 -0.0215 -0.0244 -0.0260 -0.0243

0.02 0.34 0.84 -0.01 0.25 -0.50 -1.29 0.88 -3.98 0.11 -1.70 -3.60 -2.33 -8.20 -8.46 -4.88 -7.37 -8.05 -7.16

0.0257 0.0598 -0.0067 -0.0001 -0.0208 0.0218 -0.0234 -0.0269 -0.0178 -0.0070 -0.0204 -0.0070 -0.0222 -0.0056 0.0037 -0.0201 0.0012 -0.0098 0.0090

2.79 1.72 -0.76 -0.01 -2.29 2.09 -2.36 -2.91 -4.05 -0.61 -3.89 -0.77 -1.47 -0.67 0.45 -1.44 0.14 -1.20 1.00

0.0320 0.0055 -0.0021 -0.0025 -0.0209 0.0278 -0.0120 -0.0177 -0.0086 -0.0060 -0.0221 -0.0077 -0.0267 -0.0123 -0.0006 -0.0110 0.0033 -0.0159 0.0221

4.65 0.31 -0.32 -0.22 -3.04 3.48 -1.59 -2.58 -2.83 -0.81 -5.14 -1.03 -2.25 -1.96 -0.09 -0.92 0.49 -2.60 3.28

0.0418 0.0638 0.0161 0.0168 -0.0025 0.0325 0.0021 -0.0070 -0.0171 -0.0072 -0.0145 -0.0041 0.0008 0.0004 0.0040 0.0122 -0.0035 -0.0097 0.0127

8.91 4.72 3.49 2.52 -0.52 5.82 0.38 -1.47 -7.44 -1.46 -5.05 -0.81 0.09 0.08 0.89 1.87 -0.73 -2.18 2.58

-0.0074 -0.0816 -0.0142 -0.0121 -0.0232 -0.0088 -0.0142 -0.0217 -0.0262 -0.0078 -0.0067 -0.0170 -0.0305 -0.0028 -0.0097 -0.0109 -0.0044 -0.0162 -0.0161

-0.51 -2.25 -1.00 -0.72 -1.58 -0.57 -0.93 -1.50 -6.36 -0.94 -1.13 -1.09 -1.67 -0.19 -0.63 -0.63 -0.28 -1.06 -1.01

0.0394 0.0235 0.0104 0.0156 -0.0013 0.0034 0.0089 0.0078 -0.0082 0.0010 -0.0158 -0.0034 -0.0047 -0.0002 0.0089 -0.0022 -0.0026 -0.0136 0.0148

6.80 1.87 1.84 2.00 -0.22 0.54 1.34 1.36 -3.85 0.24 -5.25 -0.63 -0.43 -0.03 1.84 -0.36 -0.52 -2.84 2.82

0.6118

47.50

0.5442

37.80

0.2721

21.90

0.2629

7.98

0.3116

16.71

0.4135

31.60

0.5548

17.60

0.2906

18.40

0.4079

23.71

Source: Findomestic Banca, authors' calculations. Note: Time dummies, their interaction with insurance and the interaction of the region dummies with time and insurance, respectively, are included in the regression, but not reported. goods are classified as follows: furniture includes all sorts of home furnishing, among which living and bedrooms and (modular) kitchens; motorcycles includes motorcycles and scooters; (used) cars includes cars, motorhomes, and caravans, they may or may not be second-hand; new cars includes cars, motorhomes, and caravans that are classified as being new; white goods include fridges, freezers, washing machines, dishwashers; household appliances are those not classified as white or brown goods; brown goods are consumer electronics like TV sets, VCRs, radios, cameras, etc. excluding home computers; computers and telephony cannot further be subdivided. *: dummy variable -: not enough observations for personal loans to estimate the coefficients.

Table 4 reports the results of this type of regressions. We control for the size, duration, and personal or installment nature of the loan, for whether or not the customer has bought an insurance against health risk (to cover repayment obligations in case of severe sickness), for geographical variation (region), time (quarter), and interactions (see the notes to the table; the reference case is an installment payment contract issued in Lazio in the third quarter of 1995). We report the coefficients of regional dummy, and also of dummies that index the communication technology used to transfer application data to the bank: rather than fax, phone, or other unspecified means of communication, some dealers have online access to the bank’s computers. The dealer-bank communication dummy variables (with online technology as the reference case) are not included in the probit model for the incidence of seller-subsidized credit below: while the business relation with the bank may result in particularly favorable credit terms, there is little reason to expect that the seller’s relationship with the bank matters for price-discrimination incentives. This credible exclusion restriction helps achieving identification of our empirical modelling approach. the communication technology used to transfer application data to the bank: rather than fax, phone, or other unspecified means of communication, some dealers have online access to the bank’s computers. The dealer-bank communication dummy variables (with online technology as the reference case) are not included in the probit model for the incidence of seller-subsidized credit below: while the business relation with the bank may result in particularly favorable credit terms, there is little reason to expect that the seller’s relationship with the bank matters for price-discrimination incentives. This credible exclusion restriction helps achieving identification of our empirical modelling approach. While the data strongly reject pooling of observations across item categories, the pattern of coefficient signs is broadly similar across the columns of the Table. The error term ε does not have a structural interpretation, but for descriptive purposes it is comforting to find that the model yields very high R-squared values on the large cross-section of data we analyze. The intended use of money borrowed is reported for personal loans: all the good categories where both dealer-intermediated and direct bank lending is observable yield a positive (and significant in 3 of the 4 cases) personal-loan dummy coefficient. Hence, instalment credit is associated with lower borrowing rates, as in our modeling framework. In order to interpret the magnitude of the coefficients, note that the included interactions of the personal loan dummy with contract characteristics and time dummies are quite significant, indicating that (for example) fixed costs of credit provision play a different role across personal and installment loans as well as across different good categories. The resulting predicted values can be used in our models of dealer subsidization. The unobservables that affect the level of the interest rate and the dealer’s decision to subsidize may of course be related at the level of the individual loan application. To mitigate the impact of that possible

25

Table 5: Dealer Subsidization: Probit Estimates Number of obs Log likelihood Pseudo R2

96999 -43443.15 0.2975 marg.eff. std.err. t-val GOOD BOUGHT∗ (ref.: new cars) Wald test (12DF) = 4645.77; p-value = 0.0000 white good 0.4665 0.028 16.53 HH appliances 0.4030 0.029 14.79 brown good 0.4014 0.027 15.48 0.0354 0.022 1.68 computer furniture 0.4713 0.021 22.14 telephony 0.2735 0.029 10.40 0.0749 0.017 4.62 (used cars) motorcycles 0.5733 0.016 32.08 0.4223 0.031 13.80 leisure home maintenance 0.2593 0.045 6.41 air conditioner 0.6041 0.025 18.60 0.3534 0.031 12.26 other CUSTOMER CHARACTERISTICS Wald test (10DF) = 347.78; p-value = 0.0000 age -0.0007 0.000 -4.98 one child∗ 0.0089 0.004 2.02 two children∗ 0.0161 0.004 3.70 -0.0090 0.006 -1.53 three or more∗ couple∗ 0.0324 0.005 6.34 divorced∗ 0.0051 0.008 0.62 -0.0122 0.009 -1.39 widowed∗ own home, morg.∗ -0.0241 0.007 -3.13 tenant∗ -0.0418 0.003 -11.72 -0.0159 0.005 -3.31 live w relativs∗ LOAN CHARACTERISTICS log amount 0.1186 0.003 41.09 duration (mths) -0.0273 0.000 -75.28 insurance∗ -0.2987 0.002 -79.53 0.0791 0.004 22.41 pay via bank∗ RATE SPREAD Wald test (3DF) = ; p-value = 0.0000 spread (linear) -0.1443 0.072 -2.01 (squared) 0.0215 0.013 1.70 (cubic) -0.0014 0.001 -1.93 RATE LEVEL Wald test (3DF) = ; p-value = 0.0000 rf, level (lin) -2.1700 0.988 -2.20 13.2018 4.112 3.21 (squared) (cubic) -25.3787 5.355 -4.74 INTERACTION 0.0491 0.022 2.21 ∗ marg. effect is for discrete change of dummy variable from 0 to 1; region and quarter dummies included, but not displayed

26

correlation, we average the predicted rates over all observations within a given region—quarter—good cell. The dealer subsidization model is a probit of full subsidization (against the alternative of non— subsidization), that relates the deal-specific subsidization indicator to a vector of customer characteristics, contract characteristics, region dummies, quarter dummies, goods dummies, a polynomial in the measured interest rate spread in a given region—quarter cell, a polynomial in the predicted interest rate in a given region—quarter cell (as described just above), and an interaction term of rate spread and rate level to control for correlation. The spread varies only across region—quarter cells, and the level across region—quarter—good cells. Controlling for good, region and quarter dummies hence implies that identification of interest rate spread and level effects is achieved by the interaction of time and space. When running the probit models, we relegate to the (normal) error term of the specification all the sources of unobservable heterogeneity around the specification of subsidization: dealers’ decisions to subsidize may, for example, depend importantly on information available to them (but not to the econometrician) as to the distribution of resources and of willingness to pay, as indexed by k in the model, across potential consumers. Table 5 reports estimated marginal effects on the probability of subsidization when the corresponding regressor increases infinitesimally (or, in case of a dummy variable, changes from zero to one) keeping all other regressors at their mean values. We focus the discussion on the main variables of interest, namely the rate spread and the rate level. Recall that our theoretical perspective implies nonlinear relationships (resulting from two offsetting mechanisms of varying strength) between financial rates and dealers’ incentives to subsidize their customers’ credit purchases. Accordingly, the probit specification includes a third degree polynomial in both spread and level, which allows for reverse-U shaped relationships without imposing them. For both the level and the spread we find that the probability of complete dealer subsidization is a concave function, falling at an increasing rate over the relevant range of values observed in the data. This result is perfectly consistent with our theoretical results, and proved robust to a variety of different specifications: we have estimated the same model separately for each good category, as well as allowing for random rather than fixed regional effects, and these regressions (available upon request) offered qualitatively similar insights into the role of interest rate levels and spreads. Our theoretical perspective and formal solutions also predict a negative interaction between the spread and level effects. This is not the case in the empirical results. Of course, the imputed level of the interest rate is only a rough proxy for the rate that the financial institution will charge to the individual dealer, and it does not vary in the sample across dealers in a given region—quarter—good cell. The standard errors reported in the Table account for the resulting structure of error term covariances, but failure to capture structural within-cell variation may be one possible explanation

27

for why the estimated correlation exhibits the wrong sign.

6

Concluding comments

We have motivated, proposed, and solved a model of interrelated purchase and borrowing decisions when, on the one hand, wedges between borrowing and lending rates in the financial market make consumers’ willingness to pay depend on the relative as well as the absolute size of current and future funds; and, on the other hand, installment-payment plans towards purchase of a specific item is available at relatively favorable rates. Further, we have rationalized in terms of pricediscrimination incentives the behavior of dealers who choose to bear the financial cost of their customers’ credit purchases. Wedges between borrowing and lending rates segment the population of potential customers into groups who are more or less inclined to borrow. Hence, any relationship between such inclinations and the overall willingness to pay induces dealers endowed with monopoly power to charge different present-value prices to the two groups. While the model proposed is very simple, its closed-form solution and numerical experiments offer intriguing insights into subtle aspects of real-life market interactions. Empirical analysis of a rich set of installment-credit and personal-loan data offers considerable support for the assumptions and implications of the proposed theoretical perspective. In particular, we have offered evidence that heterogeneity across geographic, market-segment, and time dimensions of the structure of borrowing, lending, and installment-plan interest rates has non-trivial implications for the incidence of dealersubsidized credit in our data. Such heterogeneity is taken as exogenous in our work, but may be endogenous to higher-level economic interactions. In the reality we model theoretically and study empirically, imperfections of the market for unconditional credit increase the scope for consumer-credit dealer subsidies. Competition among dealers, and among banks and specialized credit institutions, can in principle play a very important role in determining the scope of financial market imperfections and their incidence on different consumer groups. Aggregate financial market development indicators might hide important interactions between more detailed imperfections in unconditional and conditional credit markets, and the modelling perspective we propose may also have interesting implications for the pattern of durable and nondurable expenditure by individual consumers. For example, consumers with fast-increasing income patterns may not find it optimal to borrow in order to finance current nondurable consumption (especially when such borrowing is expensive and/or severely constrained). They should, however, be inclined to tilt their consumption bundles towards the kinds of durable goods that feature favorable financing arrangements. The resulting interaction between durable

28

stocks and expenditure flows on the one hand, and financial market imperfections on the other, appears interestingly different from that featured in Alessie, Devereux, Weber (1997) and other related contributions. It might be possible in further work to study such phenomena combining detailed information on the characteristics of customer who do apply for credit (and may or may not be offered a subsidy by dealers) with the information on financial market access and durable expenditure patterns available in representative data sets, such as the Bank of Italy survey studied here and by Alessie, Hochguertel, Weber (2001) and Bertola, Guiso, Pistaferri (2001).

29

Figure 9: Partition of the {current funds, future funds} plane according to whether the durable goods is purchased with cash, credit, or not at all, and to whether the resulting intertemporal choices entail positive assets, negative assets, or a binding liquidity contraint. Appendices In this Appendix we derive algebraic formulae defining the combinations of current and future resources that a) make a consumer indifferent between purchasing or not purchasing the durable good. b) make a consumer indifferent between purchasing the durable good with cash or credit. To organize the derivation of such indifference loci, it will be helpful to refer to Figure 9. This figure partitions the (W, Y ) plane according to whether its points lie inside or outside the three cones defined by the Euler conditions. Within each of the ten resulting regions, labelled with Roman numerals, the possible lending, borrowing, and purchase choices are restricted in different ways. Note that the regions of the points

M , N and Q mentioned in the paper correspond to regions I, IV and X, respectively.

A

Exact solution: purchase indifference

In this Appendix we list algebraic formulae defining the combinations of current and future resources that make a consumer indifferent between purchasing or not purchasing the durable good. We omit details of the derivation, which are in general quite similar to those discussed in the paper for regions I, IV, X and always lead to quadratic equations. The sign of the root in the solution is always uniquely determined by considering restrictions on the possible values of W and Y in each of the regions. Again taking such restrictions into account, the slope of the indifference locus can be shown to be weakly negative in all regions considered here. The indifference locus is continuous at the boundaries of the regions: values at all boundary

30

points are reported in Tables at the end of this Appendix section. It is also continuously differentiable, unless indifference between cash and credit purchase also happens to obtain when a boundary is crossed by the purchase-indifference locus.

A.1

Region I

From inspection of Figure 9 it follows that in this region the consumer has negative assets independent of the purchase decision. Credit purchase is always at least weakly preferable to cash purchase because

Y + (W − C)(1 + rb ) − P1 ≥ Y + (W − C − P0 )(1 + rb ) given that

P1 ≤ (1 + rb ) . P0 We compare the utility afforded by a credit purchase of the durable with the utility the consumer obtains if he does not buy the durable. The endowments that make the consumer indifferent between a credit purchase and no purchase lie on the locus:

Y =

A.2

Ã

1

e2k P1 −W 1 k 1 + rb 2 e −1

!

(1 + rb ) .

Region II

As is clear from inspection of Figure 9, in this region the consumer has no assets (is liquidity-constrained) if the durable good is purchased on credit, and has negative assets (borrows) otherwise. The endowments that make the consumer indifferent between a cash purchase and no purchase lie on the locus:

Y =

Ã

k

P0

e2 k

e2 − 1

−W

!

(1 + rb ).

The endowments that make the consumer indifferent between a credit purchase and no purchase lie on the locus:

A.3

q Y = W (1 + rb )(2ek − 1) − 2 W 2 (1 + rb )2 (e2k − ek ) − ek (1 + rb )W P1 .

Region III

The asset position of the consumer depending on whether the durable is purchased and, if so, on a cash or credit basis can again be inferred easily from Figure 9 in this and all other regions. The endowments that make the consumer indifferent between a cash purchase and no purchase lie on the locus:

µ ¶ q 2 2 Y = (1 + rb ) P0 + W ( k − 1) + k W (ek P0 + (1 − ek )W ) , e e

and those that generate indifference towards a credit purchase are defined by

Y = P1

ek . ek − 1

31

A.4

Region IV

From inspection of Figure 9 it follows that in this region the consumer does not hold assets independent of the purchase decision. The endowments that make the consumer indifferent between a cash purchase and no purchase lie on the locus:

W = P0

ek ek − 1

and those that generate indifference towards a credit purchase are defined as in Region III.

A.5

Region V

The endowments in this region that make the consumer indifferent to purchasing the good with cash are described by the same relationship as in Region I or II. Indifference to credit purchase occurs when

¡ ¢ ¢ ¡ kp (1 + ra )W 1 − ek + ek P1 (1 + rb ) + e 2 (1 + ra ) (1 + rb ) ((rb − ra )W + P1 ) Y = . ek (1 + rb ) − (1 + ra )

A.6

Region VI

The endowments in this region that make the consumer indifferent to purchasing the good with cash are described by the same relationship as in Region III. The endowments that make the consumer indifferent between a credit purchase and no purchase lie on the locus:

Y =

A.7

µ

2 2 P1 + W (1 + ra )( k − 1) + k e e

¶ q k k W (1 + ra ) (P1 e + (1 + ra )W (1 − e )) .

Region VII

The endowments in this region that make the consumer indifferent to purchasing the good with cash are described by the same relationship as in Region IV. The endowments in this region that make the consumer indifferent to purchasing the good with credit are described by the same relationship as in Region VI.

A.8

Region VIII

The endowments that make the consumer indifferent between a cash purchase and no purchase lie on the locus:

Y =

´ ´´ ³ ³³ ¢ ¡ kp a −rb W + P (1 + rb ) (1 + ra ) W − (W − P0 ) ek + e 2 (1 + ra ) (1 + rb ) r1+r 0 b (1 + ra ) ek − (1 + rb )

.

The endowments that make the consumer indifferent between a credit purchase and no purchase lie on the locus: k

Y = P1

e2 k

e2 − 1

− W (1 + ra ) .

32

A.9

Region IX

The endowments that make the consumer indifferent between a cash purchase and no purchase lie on the locus:

µ ¶ q Y = (1 + ra ) 2ek (W − P0 ) − W − 2 (W − P0 )ek ((W − P0 )ek − W ) . The endowments in this region that make the consumer indifferent to purchasing the good with credit are described by the same relationship as in Region VIII.

A.10

Region X

From inspection of Figure 9 it follows that in this region the consumer holds positive assets independent of the purchase decision. The endowments that make the consumer indifferent between a cash purchase and no purchase lie on the locus:

Y =

Ã

P0

k

e2 k

e2 − 1

−W

!

(1 + ra ) .

Cash purchase is always at least weakly preferable to credit purchase because

Y + (W − C)(1 + ra ) − P1 ≤ Y + (W − C − P0 )(1 + ra ) given that

P1 ≥ (1 + ra ) . P0

A.11

Continuity

The following tables report the values of the indifference locus at all points where it crosses the boundaries of two regions. In all cases, the value is the same whether it is computed with the above analytic expressions for either one of the regions. The first table reports the intersection points for the cash purchase-no purchase indifference locus. The second table reports the intersection points for the credit purchase-no purchase indifference locus.

33

Table: Continuity Purchase Indifference I

Intersection

Cash purchase - no purchase indifference: Values of W 1

Region I-II

1

k k 1 P0 e 2 (1+rb )−P1 (e 2 −1) 1 2 (e 2 k −1)(1+r ) 1

Region II-III

1 e2k 2 P0 e 12 k −1 1

Region II-V Region II-VI Region III-IV Region III-VI

b

1

P0 e 2 k (1+rb )−P1 (e 2 k −1) 1 (2+rb +ra )(e 2 k −1)

P1 P1 rb −ra only if k = 2 log 2 2P1 −P0 (rb −ra ) ek P0 ek −1 ek (2+rb +ra )(P0 (1+rb )−P1 )+2P1 (1+rb ) + k k 2 4(e q −1)(1+ra )(1+rb )+e (rb −ra ) 2 k k 2 ((1+ra )(1+rb )e P0 +P1 P0 e (rb −ra )+P12 (1−ek ))(1+rb )2

+

Region III-VII Region IV-VII Region V-VI Region VI-VIII Region VI-IX Region VI-VII Region VII-IX Region VIII-IX

4(ek −1)(1+ra )(1+rb )+ek (rb −ra )2 +P0 (1+rb ) ek P0 ek −1 only if k = log PP01(1+r a )+P1 k P0 eke−1 1 1 e2k P 1 0 2 e 2 k −1 ³ ´ √ ek (2+ra +rb )+2 ek (1+ra )(1+rb ) (1 + rb ) P0 4(ek −1)(1+ra )(1+rb )+ek (rb −ra )2 P0 (1+rb ) 1+rb rb −ra only if k = log 1+ra k P0 eke−1 ek P ³ 0 ek −1 ´ √ −4ek (1+ra )(1+rb )+(1+ra )(1+rb )+(1+rb )2 −2 ek (ra +1)3 (1+rb ) 1

Region IX-X

1 2e 2 k −1 2 P0 e 12 k −1

4(1−ek )(1+ra )(1+rb )+(rb −ra )2

34

P0

Table: Continuity Purchase Indifference II

Credit purchase - no purchase indifference: Values of W if not otherwise noted

Intersection Region I-II Region II-III Region II-V Region II-VI Region III-IV Region III-VI Region III-VII Region IV-VII Region V-VI Region VI-VIII Region VI-IX Region VI-VII

P1 1 ³ ´ 2 (1+r ) e 12 k −1 b 1 ek (1+rbP)(e k −1) ³ ´ √ −2−(ra +rb )−2 ek (1+ra )(1+rb ) P1

(ra −rb )2 −4(ek −1)(1+ra )(1+rb ) 1+rb P1 rb −ra only if k = log 1+ra k Y = P1 eke−1 P1 (ek −1)(1+ra ) P (1+r )+P (1+r )(1+r ) P1 only if k = log P11(1+rab )+P00 (1+rbb )(1+raa ) (ek −1)(1+ra ) P1 (ek −1)(1+ra ) ³ ´ √ k e (2+ra +rb )+2 ek (1+ra )(1+rb ) P1

4(ek −1)(1+rb )(1+ra )+ek (rb −ra )2 1 1 e2k 2 P1 (e 12 k −1)(1+r ) a P0 (1+rb ) a )(1+rb ) only if k = 2 log 2P0 2P0 (1+r(1+r rb −ra a )(1+rb )−P1 (rb −ra ) k e (2+rb +ra )(P1 +P0 (1+rb ))−2(1+ra )(1+rb )P0 + k k 2 q 4(e −1)(1+ra )(1+rb )+e (ra −rb ) 2 k 2 −(1+ra )(1+rb )((e −1)P0 (1+ra )(1+rb )−ek P0 P1 (rb −ra )−P12 ek )

+

Region VII-IX

4(ek −1)(1+ra )(1+rb )+ek (ra −rb )2 1k 2 1 e ³ ´ 2 P1 e 12 k −1 (1+r ) a 1

Region VIII-IX Region IX-X

1

P0 (e 2 k −1)(1+rb )+P1 e 2 k

1 1 2(e 2 k −1)+(rb −ra )(e 2 k −1)

1 2

· P0 +

1

e2k

1

(e 2 k −1)(1+ra )

P1

¸

35

B

Exact solution: cash-credit indifference

In this Appendix we list algebraic formulae defining the combinations of current and future resources that make a consumer indifferent between purchasing the durable good with cash or with installment credit. Recall from the discussion in the text that cash purchase is never optimal in region I whereas credit purchase cannot be optimal in region X. We omit details of the derivation, which are in general quite similar to those discussed in the paper for region IV. The sign of the root in the solution is always uniquely determined by considering that the cash-credit indifference locus has to be upward sloping. This can be shown by an argument similar to the one used to prove the weakly negative slope of the purchase indifference locus. Suppose the cash-credit indifference locus were negatively sloped. Then it would separate the W − Y space in two regions one of which would be further away from the origin than the other. i) Imagine a pair (W0 , Y0 ) for which it is optimal to purchase the durable with cash. Then the pair

must lie above the negatively sloped frontier. Otherwise, a sufficiently large increase of present resources

W induces a change from cash to credit purchase. This, however can never be optimal. Hence, the region where cash purchase is optimal must be the region further away from the region. ii) Imagine a pair (W0 , Y0 ) for which it is optimal to purchase the durable on credit. Then the pair must lie above the negatively sloped frontier. Otherwise, it would be possible that an increase of future resources

Y induces a change of credit to cash purchase. This, however, can never be optimal. Now, i) implies that in the region above the cash-credit indifference locus cash purchase is optimal, and ii) implies that in this region credit purchase is optimal. By contradiction, the cash-credit indifference locus cannot be downward sloping. The indifference locus is continuous and continuously differentiable at the boundaries of the regions. Its values at all boundary points are reported in a Table at the end of this Appendix.

B.1

Region II, III

In Region III the frontier is characterized by indifference between buying the durable cash and borrowing, versus buying the durable on credit and depleting assets, i.e.,

1 (Y + (W − P0 )(1 + rb ))2 ek = W (Y − P1 )ek 4(1 + rb ) which yields

B.2

Region IV

p Y = (W + P0 )(1 + rb ) − 2 W (1 + rb ) (P0 (1 + rb ) − P1 ).

In Region IV the frontier is characterized by indifference between buying the durable cash or on credit where the consumer holds zero assets in both cases, i.e.,

log(W − P0 ) + log (Y ) + k = log(W ) + log (Y − P1 ) + k so that Y =

P1 W . P0 36

B.3

Region V, VI, VIII

In Regions V, VI, and VIII the frontier implies indifference between buying the durable cash and borrowing, versus buying the durable credit and lending. This yields

Y =

B.4

(P1 − P0 (1 + ra )) (1 + rb ) +

p (1 + rb ) (1 + ra ) (W (rb − ra ) − P0 (1 + rb ) + P1 ) rb − ra

Region VII, IX

In Region VII and IX the frontier is characterized by indifference between buying the durable cash and borrowing, versus buying the durable credit and lending, which results in:

B.5

p Y = (1 + ra )(W − P0 ) + (P1 − (1 + ra )P0 ) + 2 (P1 − (1 + ra )P0 )(W − P0 )(1 + ra ) . Continuity

Table: Continuity Cash-Credit Indifference Intersection Cash-Credit Frontier: Values of W Region II-III Region III-IV Region V-II Region V-III Region V-VI Region VI-III Region VI-IV Region VI-VII Region VII-IV Region VIII-VI Region VIII-IX Region VIII-VII Region IX-VII

2 1 (1+rb )P0 4 (1+rb )P0 −P1 1+rb P02 P0 (1+r b )−P1 −(P1 −(1+rb )P0 ) √ 1+ra −2 (1+ra )(1+rb )+(1+rb ) p P1 P1 1 ((1 + rb ) + (1 + rb ) (1 + ra )) rb −ra only if P0 = 2√ (1+rb )[P0 (1+ra )−P1 ]+ ((1+rb )(1+ra ))[P0 (1+rb )−P1 ] ³√ ´ ((1+rb )(1+ra ))−(1+rb ) (rb −ra ) ³ ´ √ 1+ra + ((1+rb )(1+ra )) (P0 (1+rb )−P1 ) ³√ ´ ((1+rb )(1+ra ))−(1+ra ) (rb −ra ) p P1 +P0 (1+rb ) 1 only if P ((1 + rb ) (1 + ra )) rb −ra P0 =

√

(1+rb )(2P0 (1+ra )−P0 (1+rb )−P1 )+(P0 (1+rb )−P1 ) ((1+rb )(1+ra )) ³√ ´ ((1+rb )(1+ra ))−(1+rb ) (rb −ra ) P1 P1 −P0 (1+ra ) P0

√

(1+rb )[P0 (1+ra )−P1 ]+ ((1+rb )(1+ra ))[P0 (1+rb )−P1 ] ³√ ´ ((1+rb )(1+ra ))−(1+ra ) (rb −ra ) ³ ³ ´´ √ 2 (P1 −(1+ra )P0 ) (1+ra )+ (1+ra )(1+rb ) +P1 ((1+rb )−2(1+ra ))P0 + (rb −ra ) (rb −ra ) (1+ra )(1+rb ) P1 b √ only if P0 r1+r = 2 P0 b −ra 1+rb + ((1+rb )(1+ra )) P12 1 4 (1+ra )(P1 −P0 (1+ra ))

37

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