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Economic Quarterly— Volume 104, Number 2— Second Quarter 2018— Pages 53–77 The Decline in Currency Use at a National Retail Chain Zhu Wang and Alexander L. Wolman T he composition of US retail payments is changing rapidly. According to the Federal Reserve’s triennial Payments Study (2013, 2016), from 2012 to 2015 the value of debit and credit card payments increased at annual rates of 7.1 percent and 7.4 percent, respectively. Over this same period, nominal GDP rose at less than a 4 percent annual rate, which suggests that the increase in card payments came at the expense of some other form(s) of payments, the obvious candidates being checks and cash. The value of check payments did fall over this period, but it is possible that the fall in check payments was o¤set by an increase in ACH rather than card payments; ACH tends to be used in business and …nancial transactions while cards are used in consumer payments. The Payments Study covers only noncash payments, but Wang and Wolman (2016a) provide direct evidence about cash use at a large discount retailer, …nding that the cash share of the number of payments fell by 2.46 percentage points per year from 2010 to 2013. In their study, an increase in card use was almost the mirror image of a decrease in cash use. At least four sets of factors could be contributing to the apparent shift from cash to card in retail payments. First, Wang and Wolman (2016a) documented a negative relationship between transaction size and the share of cash transactions; thus, some of the decline in observed cash shares could be due to an increase in average transaction size. Second, Wang and Wolman also documented systematic relationships between the cash share of payments in a location and the The views in this paper are those of the authors and do not necessarily represent the views of the Federal Reserve Bank of Richmond, the Federal Reserve Board of Governors, or the Federal Reserve System. The authors thank Erica Paulos for excellent research assistance, and Mohamed Abbas Roshanali, Arantxa Jarque, Bruno Sultanum, and John Weinberg for helpful comments on an earlier draft. DOI: https://doi.org/10.21144/eq1040201 54 Federal Reserve Bank of Richmond Economic Quarterly demographic and economic characteristics of the location; over time, changes in those characteristics may explain changes in the cash share. Third, changes in technology may be reducing the cost and increasing the availability and security of debit and credit cards. And fourth, consumers’ perceptions of cards may be improving slowly, generating a gradual expansion in card use. This paper brings new evidence to bear on the contributions of the …rst two factors to the decline in cash payments. Using an updated version of the data from Wang and Wolman (2016a,b), we study the association between changes in payment shares and changes in the size of transactions as well as changes in location-speci…c economic and demographic variables over the period from February 2011 to February 2015. While we cannot distinguish the third and fourth factors listed above, the portion of the decline in cash shares that is unexplained by our analysis represents the sum of these two sets of factors. There are important public policy questions for which it matters what explains the decrease in cash use. Cash remains an important means of payment in the United States, and in the wake of the long recent experience with interest rates at their e¤ective lower bound, some economists have advocated policies that would reduce or even eliminate the availability of paper currency (Rogo¤ 2016). Without paper currency, the argument goes, monetary policy would no longer be constrained by a lower bound on nominal interest rates.1 Against this, the bene…ts of cash must be considered, and the accounting we provide for the decline in cash use can contribute to the debate over the bene…ts of cash. To the extent that the decline in cash use is accounted for by changing demographics or changing transaction size, there may be greater scope for concern about the e¤ects of a (hypothetical) elimination of currency on particular segments of society. In Wang and Wolman (2016a), and in this paper, we analyze transactions data from a discount retailer with thousands of stores across the US. In the earlier paper, we combined the transactions data with …xed demographic data and other data across locations.2 With almost two million transactions every day, we were able to precisely characterize the daily and weekly patterns of payment use. And, with thousands of zip-code locations, we were also able to precisely estimate the relationships between cash shares and location-speci…c variables. However, the fact that our data covered only three years meant that we could not 1 Rogo¤ (2016) also sees bene…ts from eliminating cash related to the fact that cash is heavily used in the underground economy. 2 In Wang and Wolman (2016b), we conducted a similar analysis that concentrated on retail outlets in the Fifth Federal Reserve District. Wang & Wolman: Decline in Currency Use 55 incorporate time variation in the location-speci…c data: the Census Bureau’s American Community Survey (ACS) data were not available at the zip-code level for more than one year in our dataset. In the current paper, we do not attempt to capture the daily variation in payment shares but instead focus on the “medium-term”shift in the cash share of transactions from February 2011 to February 2015, using only data from those two months. While we sacri…ce on one dimension, we are able to incorporate time variation in the location-speci…c data using the …ve-year ACS estimates at the zip-code level for 2011 and 2015. On average, across the stores in our study, the share of cash transactions fell by 8.6 percentage points from February 2011 to February 2015. Our statistical model attributes approximately 1.3 percentage points of that decline to increasing transaction sizes. Changes in demographic and other location-speci…c variables contribute between 0.5 and 1.3 percentage points, so our analysis attributes approximately three-quarters of the decline in cash use to a pure time e¤ect, which stands in for the third and fourth factors listed above, and any other factors omitted from our analysis. 1. TRANSACTIONS DATA: THE DECLINE IN CURRENCY USE Our payments data come from a US retail chain selling a wide variety of goods, with a majority of its revenue accounted for by household consumables such as food and health-and-beauty aides. The chain has thousands of stores and is located in most states. Although there is not a speci…c geographic focus, the stores tend to be located in relatively low-income zip codes.3 While the raw data are at the level of individual transactions (time and location, size, means of payment), our analysis uses aggregated data: for each zip code, we compare the shares of transactions in each of the four main payment types (cash, debit card, credit card, and check) in February 2011 to the corresponding shares in February 2015. One month is a long enough time period to get a relatively large number of transactions: most zip codes had more than 7,000 transactions in each of the two months. The total number of zip-code locations is more than 5,000. We chose February 2011 and February 2015 to balance two considerations. A longer time span provides a better sense of the trend decrease in cash use, but we needed 3 See Wang and Wolman (2016a) for some additional information. Our use of the data is governed by a con…dentiality agreement that limits the degree of detail we may disclose. 56 Federal Reserve Bank of Richmond Economic Quarterly to choose years for which zip-code-level data are available from the ACS. Figure 1 is a scatterplot of the share of cash transactions in each zip code in 2015 and 2011, on the y- and x-axes respectively. The solid grey line is the locus of points for which the cash share is equal in the two years, and points below (above) the line indicate a decrease (increase) in the cash share. This …gure provides a nice overview of the data and the properties we want to study. First, there is signi…cant variation in the share of cash transactions in both years. Second, the share of cash transactions declined from 2011 to 2015 in almost every zip code, as indicated by the small number of observations that lie above the y=x line. And third, while the decrease in the cash share does not seem closely related to the level of the cash share, the decrease is also not constant across zip codes. The …rst and third properties— cross-zip-code variation in both the level and change in the cash share— provide motivation for using demographic and other zip-code-level variables in our statistical analysis. The second property— a signi…cant common component in the change in the cash share across zip codes— could partly re‡ect changes in demographics that are common across locations. However, the common component also re‡ects changes in payments technology and consumer perceptions that are not captured by our analysis. Table 1 displays summary statistics for the data in Figure 1, as well as the corresponding data for shares of debit, credit, and check transactions. From February 2011 to February 2015, the average cash share of transactions across zip codes declined from 78.2 percent to 69.5 percent, or 2.18 percentage points per year. Our focus is primarily on the decline in cash and the combined increase in credit and debit use; the total card share of transactions increased by an average of 2.3 percentage points per year, with the di¤erence, 0.12 percentage points per year, accounted for by a decrease in the share of transactions conducted with checks. Our data are not well-suited to distinguishing credit and debit transactions because the category we call “debit” includes only PIN debit transactions— signature debit and most prepaid cards are included in “credit.”4 PIN debit transactions increased by an average of 1.63 percentage points per year, approximately 70 percent of the overall increase in card use. Table 1 also shows that from 2011 to 2015 both the standard deviation of cash transaction shares and the interquartile range (di¤erence between the 75th and 25th percentiles) increased. This corresponds to 4 PIN debit is a debit card transaction that requires the consumer to enter a PIN number, whereas signature debit is a debit card transaction that requires the consumer to sign their name (like a credit card transaction). Wang & Wolman: Decline in Currency Use 57 Figure 1 Zip-Code-Level Cash Shares for 2015 and 2011 the third property noted in reference to Figure 1: the distribution of cash shares across zip codes did not shift down in a uniform manner. Figure 2 illustrates this explicitly, showing that the histogram of cash shares across zip codes was more spread out in 2015 than in 2011, in addition to shifting to the left. Dispersion across locations in the change in cash shares is illustrated in the third row of Table 1 and in Figure 3. Cash shares declined by an average of 8.6 percentage points, but there is signi…cant dispersion: in 25 percent of zip codes, the cash share decreased by at least 9.9 percentage points, and in 25 percent of zip codes the cash share decreased by less than 7.0 percentage points. As mentioned in the introduction, one factor that could help account for the changes in cash shares depicted in Figures 1 through 3 is a change in the distribution of transaction sizes. Our econometric analysis of the change in cash shares below will explicitly take into account transaction size, but for now we simply report on the distributions of median transaction size and change in median transaction size by location. Table 2 provides various statistics for the distributions: for example, the mean value of median transaction size rose from $7.26 to $7.96, and the mean change in median transaction size is $0.70. 58 Federal Reserve Bank of Richmond Economic Quarterly Table 1 Payment Shares Across Zip Codes, February 2011 vs. February 2015 Mean Std. dev. Cash: 2011 2015 change 0.782 0.695 -0.086 0.056 0.063 0.025 Debit: 2011 2015 change 0.161 0.226 0.064 Credit: 2011 2015 change Check: 2011 2015 change 1% 25% 50% 75% 99% 0.636 0.532 -0.150 0.747 0.653 -0.099 0.787 0.699 -0.085 0.822 0.740 -0.070 0.891 0.824 -0.031 0.050 0.058 0.028 0.062 0.095 -0.016 0.127 0.187 0.049 0.156 0.222 0.065 0.192 0.261 0.081 0.292 0.380 0.128 0.047 0.074 0.027 0.034 0.049 0.029 0.008 0.015 -0.017 0.024 0.039 0.009 0.036 0.060 0.019 0.059 0.096 0.039 0.171 0.246 0.121 0.010 0.005 -0.006 0.011 0.006 0.006 0.000 0.000 -0.027 0.002 0.001 -0.008 0.006 0.003 -0.004 0.014 0.007 -0.001 0.051 0.026 0.001 Note: Rows titled “change” show distributions of changes in payment shares from 2011 to 2015. These may show di¤erent means than the change in the mean share for a particular payment type because the set of stores is not identical in the two years (e.g., for cash, change in mean is 0.087 and mean change is 0.086). Table 2 Median Size of Transactions Across Zip Codes, February 2011 vs. February 2015 2011 2015 Change Mean Std. dev. 1% 25% 50% 75% 99% 7.26 7.96 0.70 1.02 1.10 0.78 5.35 5.87 -1.28 6.56 7.20 0.27 7.15 7.88 0.67 7.81 8.66 1.12 10.12 10.90 2.52 Note: The third row is the distribution of change in median transaction size from 2011 to 2015. Figures 4 and 5 display histograms of the two distributions of median transaction size (Figure 4) and the distribution of changes in median transaction size (Figure 5). The distribution of transaction sizes shifted to the right from 2011 to 2015 and became slightly more spread out. The dispersion in changes in median transaction size (Figure 5) is indeed consistent with the behavior of transaction size accounting for some of the shift in the cash share distribution from 2011 to 2015. Wang & Wolman: Decline in Currency Use 59 Figure 2 Histograms of Zip-Code-Level Cash Share 2. LOCATION-SPECIFIC DATA Table 3 provides summary statistics for the location-speci…c data used in our analysis, comparing the 2011 and 2015 values. Wang and Wolman (2016a) provide a discussion of why one would expect these variables to be relevant for explaining payment choice, arguing that each consumer has a threshold transaction size below which they will use cash and above which they will use a noncash form of payment. The threshold may vary over the week, month, and year, and it will likely be related to the consumer’s …nancial situation, their demographic characteristics, and their surrounding environment (including banking options, population density, and crime rates). The overall cash share in a particular location at a particular time will thus depend on the characteristics of the consumers in that location, the characteristics of the location, and the size distribution of transactions. In Wang and Wolman (2016a), we used the same demographic variables to account for variation in cash shares across locations, but our data did not allow for the possibility of using changes in those variables to account for the change over time in cash shares; the location-speci…c variables were necessarily treated as …xed over the three-year sample of data due to limitations of the Census Bureau data. Here, the longer 60 Federal Reserve Bank of Richmond Economic Quarterly Figure 3 Histogram of Change in Zip-Code-Level Cash Share span of the transactions data means we can incorporate distinct demographic data for 2011 and 2015 for each zip code to decompose the changes in cash shares. Our earlier paper used forecasted nationwide changes in the location-speci…c variables to project future changes in cash shares and attributed up to 15 percent of the overall projected decline in cash shares to forecasted changes in location-speci…c variables. Below, we will compare that number to our decomposition of actual changes in cash shares. The demographic variables (sex, age, race, and education) and the housing variables in Table 3 are all from the ACS. We use ACS …ve-year estimates at the zip-code level for 2011 and 2015. Note that for age we report only the 2011 data. We …x the age data at 2011 levels because we think that cohort is more important than age for payment behavior.5 The banking variables— market concentration, as measured by the 5 In principle, we would like to use data on the distribution of cohorts in each year. However, because the age data in our regression are in relatively large bins (e.g., …fteen years), it will not provide an acccurate picture of how the cohort distribution changes across the four-year span of our data. In Section 4, we will use the estimated coe¢ cients together with more detailed age data to construct a rough measure of the cohort contribution to the change in cash shares. Wang & Wolman: Decline in Currency Use 61 Figure 4 Histogram of Zip-Code-Level Median Transaction Size Her…ndahl-Hirschman index (HHI), and the number of bank branches per capita— are from the FDIC’s Summary of Deposits. Banking HHI is calculated by squaring each bank’s share of deposits in a zip code and then summing these squared shares. We allow the HHI e¤ect to di¤er between rural and urban areas because of the possibility that high concentration in an urban area may re‡ect the presence of a small number of high-productivity banks. The robbery rate is from the FBI’s uniform crime report (note that the robbery rate is at the county level). In most cases, the changes from 2011 to 2015 appear to be small.6 However, the examples of median household income and education show that changes in location-speci…c variables have the potential to account for some of the decline in cash use. Across locations, Wang and Wolman (2016a) found that higher educational attainment and higher income were associated with lower cash use; Table 3 shows that both educa6 One exception is the HHI index. Note that in our earlier work the HHI was measured at the level of metropolitan statistical area (MSA) or rural county. Here it is measured at the zip-code level. In Wang and Wolman (2016b), we found that variation in HHI explained little of the variation in payment shares across zip codes. 62 Federal Reserve Bank of Richmond Economic Quarterly Figure 5 Histogram of Change in Zip-Code-Level Median Transaction Size tional attainment and income increased on average from 2011 to 2015, which would be consistent with a decrease in cash use assuming the relationship found by Wang and Wolman also holds across time. In the next section, we will report estimates of a statistical model similar to that in our 2016a paper using the variables in Table 3. Then in Section 4, we will quantify the contributions of changes in transaction size and in the demographic variables to the decline in cash use. 3. EMPIRICAL FRAMEWORK AND ESTIMATES In this section, we describe the statistical model used to analyze payment shares and provide a summary of the estimates. The statistical model is tailored to the properties of the variable we are seeking to explain: in a particular time period in a particular location, the shares of cash and other payment types are each between zero and one, and they must sum to one. These properties mean that linear regression is not appropriate. Wang & Wolman: Decline in Currency Use 63 Table 3 Summary Statistics of Zip-Code Variables Variable (unit) Banking HHI Banking HHI Metro Branches per capita (1/103 ) Robbery rate (1/105 ) Median household income ($) Population density (per mile2 ) Family households (%) Housing: Renter-occupied (%) Owner-occupied Vacant Female (%) Age: < 15 (%) 15-34 35-54 55-69 70 Race: white (%) black Hispanic Native Asian Pac-Islr other multiple Educ below high school (%) high school some college college Mean 2011 2015 0.43 0.28 0.38 13.17 43,221 1479 66.50 28.18 57.33 14.49 50.87 20.03 26.65 27.36 16.16 9.81 74.88 16.61 13.55 1.07 1.42 0.06 3.81 2.15 18.36 34.22 21.28 26.14 0.46 0.29 0.36 12.34 43,818 1484 65.52 30.14 55.28 14.58 50.74 75.62 15.85 15.26 1.06 1.58 0.06 3.31 2.51 16.89 33.62 21.76 27.72 Std. dev. 2011 2015 0.26 0.29 0.36 28.477 12,289 2614 8.65 11.21 12.86 8.59 2.87 4.08 5.88 3.28 3.77 3.81 22.80 21.65 19.39 4.20 2.34 0.28 6.31 1.76 8.70 7.33 4.34 10.18 0.26 0.30 0.32 26.02 12,621 2493 8.85 11.79 12.77 8.63 2.92 22.18 20.94 20.83 4.08 2.61 0.30 5.36 1.92 8.61 7.41 4.21 10.47 Note: The sum for race percentage is greater than 100 because Hispanic includes other categories. Description of model The purpose of the statistical model is to provide estimates of the relationship between the levels of payment shares and a set of explanatory variables comprising transaction size, the time- and location-speci…c variables, state-level …xed e¤ects, and year …xed e¤ects. We pool the data for the two years, restricting the relationship between payment and the explanatory variables to be the same across the two years. Changes in payment shares can be captured by changes in the explanatory variables and by the year …xed e¤ects. 64 Federal Reserve Bank of Richmond Economic Quarterly We assume that the relationship between payment shares and explanatory variables is captured by a fractional multinomial logit (FMLogit) model, which states the expected share of each payment type, conditional on the explanatory variables, is a multinomial logit function of the explanatory variables: exp(x0j;t E[sk;j;t j xj;t ] = 4 X k) exp(x0j;t ; (1) m) m=1 k = 1; 2; 3; 4: Before explaining each of the terms in this expression, it will be helpful to understand the subscripts: k and m denote the payment types, cash, debit, credit, and check; j denotes zip code; and t denotes year. The left-hand-side variable, E[sk;j;t j xj;t ]; is the expected value of the share of type k payments in zip code j in year t; conditional on the timeand location-speci…c variables xj;t (a vector), which can be thought of as including the state and the year as well as the median transaction size and the demographic and other variables summarized in Table 3. The right-hand side is a function of the explanatory variables as well as coe¢ cients; k is a vector of coe¢ cients that multiply the explanatory variables.7 By construction, the right-hand side is a number between zero and one as long as the data and coe¢ cients are real numbers. And, by X4 construction, the expected shares always sum to one: E[sk;j;t j k=1 xj;t ] = 1: Note, however, that from (1), for any k ; k = 1; 2; 3; 4; the expected shares are invariant to the transformation ~ k = k + c, where c is a vector the same length as k : In order to achieve identi…cation of k ; a normalization is needed. We use the standard normalization of setting 4 = 0; where k = 4 denotes cash. This implies 1 E[s4;j;t j xj;t ] = 1+ 3 X exp(x0j;t : (2) m) m=1 In the Appendix, we present this model in somewhat more detail and explain how the coe¢ cients can be estimated. 7 As an alternative to the FMLogit model of payment shares, we could estimate a multinomial logit model at the individual transaction level. By aggregating transactions and modeling shares, we are able to use a larger number of transactions and smooth out the “noise” in individual transactions. Wang & Wolman: Decline in Currency Use 65 Basic results We follow the approach described in the Appendix to estimate the model in (1) and (2). In a linear regression model, the usual way to report results is in the form of the estimated coe¢ cients and Pvalues (or standard errors). With the nonlinear model used here, it is more informative to report marginal e¤ ects and their P-values; they are presented in Table 4.8 For continuous variables, the marginal e¤ect we report (on cash) is the derivative of the predicted share with respect to the variable. For the state and time …xed e¤ects (the former are not reported in the table), the marginal e¤ects we report are the di¤erence between the predicted cash share when the indicator variable is one and when it is zero. Many of the marginal e¤ects reported in Table 4 are highly signi…cant and have similar magnitudes to those reported in Wang and Wolman (2016a). For example, the median transaction e¤ect is -0.019, compared to -0.018 in the earlier paper. Some of the estimates do differ, however, and not all the marginal e¤ects reported in Table 4 are estimated precisely, in contrast to Wang and Wolman (2016a). The number of di¤erent zip codes is roughly comparable in the two papers, but here we use fewer days of data for each zip-code-level observation of the demographic variables. In our earlier paper there were more than 1,000 days of data for each observation of a demographic variable; here there is just one month of data— either February 2011 or February 2015, and this leads to the marginal e¤ects being estimated less precisely. With respect to age, as discussed above, we interpret the age distribution as the cohort distribution and therefore …x it at its 2011 value. Of course, this means we treat the cohort distribution as …xed so that it cannot explain any of the change in cash shares. In Section 4, we delve into the cohort e¤ect in more detail and present some calculations that represent a rough estimate of the contribution of changes in the cohort distribution to changes in the cash share. 8 The dependent variables are the fractions of each of the four general payment instruments used in transactions at stores in a zip code in February 2011 and February 2015. The independent variables take their values in 2011 and 2015. Metro is a dummy variable taking the value of one when the zip code is in an MSA, otherwise it is equal to zero. We rescale some of the variables relative to the levels reported in Table 3 in order to make the marginal e¤ects of common magnitude. Branches per capita is measured as the number of bank branches per 100 residents in a zip code. Robbery rate is de…ned as the number of robberies per 100 residents in a county. Median household income is measured in units of $100,000 per household in a zip code. Population density is measured in units of 100,000 residents per square mile in a zip code. All the demographic variables are expressed as fractions. 66 Federal Reserve Bank of Richmond Economic Quarterly Table 4 Marginal E ects on Cash Variable Med. transaction size (Year=2015) - (Year=2011) Banking HHI Banking HHI Metro Branches per capita Robbery rate Median household income ($) Population density (per mile2 ) Family households Housing: Owner-occupied Vacant Female Age: 15-34 35-54 55-69 70 Race: black Hispanic Native Asian Pac-Islr other multiple Educ: high school some college college Estimate at mean P-value -0.019 -0.068 -0.002 -0.022 -0.040 -0.062 -0.017 0.016 -0.089 -0.364e-04 .013 -0.027 -0.147 -0.114 0.016 6.80e-04 0.063 0.011 0.141 -0.062 -0.073 0.009 -0.001 -0.279 -0.463 -0.309 0.000 0.000 0.469 0.000 0.127 0.005 0.153 0.535 0.000 0.969 0.178 0.186 0.000 0.000 0.531 0.981 0.000 0.050 0.000 0.007 0.627 0.434 0.964 0.000 0.000 0.000 Turning to the model’s overall …t, Figures 6 and 7 show that it does a reasonable job of explaining the variation in cash shares across time and locations: Figure 6 compares the actual distribution of 2011 cash shares to the model’s predicted distribution, and Figure 7 does the same thing for 2015. The pseudo-R2 values are 0.55 for 2011 and 0.59 for 2015. 4. ANALYSIS OF DECLINE IN CASH SHARES Table 1 shows that the mean cash share of transactions declined by 8.7 percentage points from 2011 to 2015. Our model does a good job of capturing this decline: the predicted cash share evaluated at the means of the 2015 data is 8.8 percentage points lower than the predicted cash share evaluated at the means of the 2011 data. Alternatively, we can calculate the predicted cash share for every observation and compare the mean predicted shares for 2011 and 2015: the di¤erence is Wang & Wolman: Decline in Currency Use 67 Figure 6 2011 Actual (Green) and Predicted Cash Shares 8.7 percentage points. In a linear regression, these two objects would be identical, but because the FMLogit model is nonlinear, the mean predicted value may di¤er from the predicted value evaluated at the mean of the explanatory variables. We will report both numbers at various points below; they never di¤er by much. The empirical framework suggests three types of factors to account for the decline in cash shares from 2011 to 2015. First, given a relationship between transaction size and cash shares, an upward shift in the distribution of median transaction sizes (Figure 4) can account for some of the decline in cash shares. Second, given a relationship between demographic variables and cash shares (Table 4), changes in the demographic variables might account for some of the decline in cash shares. And …nally, a portion of the decline in cash shares is accounted for by the year dummy; this portion is e¤ectively unexplained and likely attributable to changes in the attributes of noncash payments (e.g., cost, availability, and security) and changing preferences on the part of consumers. 68 Federal Reserve Bank of Richmond Economic Quarterly Figure 7 2015 Actual (Green) and Predicted Cash Shares Increasing average transaction size The average value of median transaction size increased by $0.70 from 2011 to 2015. A simple measure of the contribution of changing transaction size to the decline in cash shares is the product of the $0.70 increase with the marginal e¤ect for transaction size, -0.019. According to this measure, increasing transaction size can account for a decrease of 1.35 percentage points in the cash share, roughly 15 percent of the total decline. This simple measure ignores nonlinearity in the empirical model. We can take into account the nonlinearity by comparing 2011 predicted cash shares to the shares the model would predict if transaction size changed to its 2015 level but all other variables were …xed at their 2011 values. This approach yields a decrease of 1.33 percentage points in the predicted cash share evaluated at the mean of the explanatory variables and a decrease of 1.44 percentage points in the mean predicted cash share across zip codes. Thus, the linear approximation (1.35 percentage points) turns out to be quite accurate. The smoothed density functions in Figure 8 are based on the same approach: the black line represents the density function of predicted cash shares for 2011, whereas the red line represents the density func- Wang & Wolman: Decline in Currency Use 69 Figure 8 The Transaction Size E ect tion of counterfactual predicted values, calculated with 2015 transaction size but 2011 values of all other variables. There is a notable leftward shift in the distribution explained by the increase in transaction size, but the shift is small relative to the overall change shown in Figure 2. Note …nally that our estimates of the contribution of increasing transaction size to the decrease in cash shares may be a¤ected by correlations between transaction size and some of the zip-code-level variables. This means that a portion of the e¤ect attributed to transaction size could instead be attributed to changes in the zip-code-level variables. In Wang and Wolman (2016b), we explore this idea in more detail by regressing transaction size on the zip-code-level variables and then including the residual portion of transaction size in the FMLogit regression in place of actual transaction size. We …nd that indeed the marginal e¤ects of other variables change when they include indirect e¤ects of transaction size. Changing demographic and other variables Table 4 shows that many location-speci…c variables have a systematic relationship with the cash share of transactions. Since these variables 70 Federal Reserve Bank of Richmond Economic Quarterly Figure 9 The Zip-Code-Level E ect take on di¤erent values in 2011 and 2015, they may be able to account for some of the decline in cash shares over that period. In contrast, Wang and Wolman (2016a) used only a three-year span of data with …xed values of the location-speci…c variables. As mentioned above, that paper included a rough forecasting exercise that took into account projected changes in the location-speci…c variables, but the projected changes were identical across locations. In order to quantify the e¤ect of the zip-code-level variables, here we use an analogous approach to that used for transaction size: we compare the predicted cash shares for 2011 with the predicted cash shares implied by holding …xed transaction size and the year dummy at their 2011 values but allowing all the locationspeci…c variables to take on their 2015 values. Comparing the predicted value of cash share conditional on 2011 means to that conditional on 2015 zip-code-level variable means, the 2011 year dummy, and 2011 mean transaction size yields a decline of 0.5 percentage points. This estimate does not change if we instead compare means of predicted values across zip codes. Figure 9 plots the smoothed density function for 2011 predicted cash shares and compares it to the density of predicted cash shares under the assumption that the zip-code-level variables take on their 2015 Wang & Wolman: Decline in Currency Use 71 Figure 10 Transaction Size and Zip-Code-Level E ects values but the year dummy and transaction size are …xed at their 2011 values. There is a small but discernible leftward shift in the distribution of predicted cash shares, consistent with the mean estimate. As discussed above, Figures 8 and 9 attribute any e¤ects of transaction size that work through zip-code-level variables to transaction size. In Figure 10, we combine both e¤ects, so that the precise decomposition is irrelevant: the black line is the density of 2011 predicted cash shares; the red line is the density of predicted cash shares holding …xed the year dummy at 2011 but allowing all other variables to change; and the blue line is the density of 2015 predicted cash shares. In Figure 10, the vertical lines represent the respective means. Consistent with our previous calculations, the combination of changes in transaction size and changes in zip-code-level variables accounts for a 1.8 percentage point decline in the mean predicted cash share across zip codes or 1.7 percentage points if we instead use the predicted change in the cash share at the means of the data. In Wang and Wolman (2016a), the forecasting exercise attributed a relatively large fraction of the projected decrease in the cash share to a cohort e¤ect: a shift in the population toward later-born cohorts who were accustomed to using cards would drive down the cash share of 72 Federal Reserve Bank of Richmond Economic Quarterly transactions. Thus far, the calculations here do not take into account that e¤ect because they hold …xed both the age and cohort distribution of the population and the coe¢ cients on age or cohort. Ideally, we would like to treat the cohort distribution just like the other zip-codelevel variables in our study: this would involve allowing the cohort distribution to change from 2011 to 2015, estimating a common cohort e¤ect, and then calculating the contribution of the changing cohort distribution to the change in the cash share. The di¢ culty with this approach is that our data are on age distribution, and in …fteen- and twenty-year bins. Age and cohort are interchangeable at a point in time; for example, the fraction of the population in 2011 that was between 15 and 34 years old (=age) is identical to the fraction of the population in 2011 that was born between 1977 and 1996 (=cohort). However, across time, cohort distributions and age distributions need to be tracked separately unless they are in one-year bins. For example, if we know the fraction of the population that was between 15 and 34 in 2011 and the fraction of the population that was between 15 and 34 in 2015, we have information about two di¤erent cohorts in the two years, not the same cohort. For 2011 we have the 1977 to 1996 cohort, and for 2015 we have the 1981 to 2000 cohort. If we knew the age distribution in one-year increments for 2011 and 2015, then it would be trivial to calculate the corresponding cohort distribution in one-year increments. Without precise data on how the cohort distribution evolved from 2011 to 2015, we nonetheless computed a rough estimate of the contribution of shifts in the cohort distribution to the decrease in cash shares from 2011 to 2015. The idea behind this estimate is to use aggregate census data on a …ner gradation of the age distribution to come up with an educated guess about how the cohort distribution changed from 2011 to 2015 across the large bins in our study. Then, we will combine that educated guess with our estimated marginal e¤ects for the di¤erent cohorts. Note …rst that, from Table 4, the cash marginal e¤ect for population aged 35-54 in 2011 is -0.114, compared to 0.016 for age 55-69. The 35-54 age group is the cohort born between 1957 and 1976, and the 55-69 age group is the cohort born between 1942 and 1956. For ages less than 34, the marginal e¤ect is even more negative, and for ages above 69, it is close to zero. According to nationwide census data, the 2011 population share of ages 50-54 was 7 percent. We thus pose the following question: How would the predicted cash share change if there were a 7 percentage point increase in the fraction of the population for whom the cash marginal e¤ect is -0.114, and a 7 percentage-point decrease in the fraction of the population for whom the cash marginal e¤ect is 0.016? The answer is that the predicted cash share would fall by 0.8 percentage points. Adding this to the 1.7 Wang & Wolman: Decline in Currency Use 73 percentage points accounted for by transaction size and other locationspeci…c variables would allow us to account for nearly 30 percent of the overall 8.7 percentage-point predicted decline in the cash share. The remainder of the predicted decrease in cash shares at the mean of the data— either 7 percentage points or 6.2 percentage points if we include the imputed age e¤ect— is attributed to the year dummy, although this decomposition is not exact: the marginal e¤ect for the year dummy is 6.8 percentage points, and if we compare predicted means for 2011 variables with the year dummy changing, the di¤erence is 6.6 percentage points. Regardless of how we measure it, between 70 and 80 percent of the decline in cash shares cannot be explained by either an increase in transaction size or changes in location-speci…c variables. We attribute that unexplained decline to a pure “time e¤ect,”which is standing in for all other factors that play a role in payment choice but are not included in the model. The leading candidates for these factors are wider availability, better security, and lower cost of cards, as well as evolving consumer perceptions of each of those factors. 5. CONCLUSION The cash share of transactions at a large national discount retailer declined by approximately 8.6 percentage points from February 2011 to February 2015. Following up on Wang and Wolman (2016a,b), we use a FMLogit model to study the cash share of transactions across time and locations. The geographic coverage is similar to our earlier paper: thousands of store locations, at the zip-code level. The time coverage is more sparse here: two months, four years apart, as opposed to three years of daily transaction shares in our earlier paper. By restricting the time dimension to low-frequency changes, in this paper we are able to introduce time variation in the zip-code-level variables. Previously, we measured the trend decrease in cash shares but were able to attribute it only to a pure time trend or an increase in transaction sizes. We used forecasts of demographic variables to produce a crude measure of the projected contribution of changes in those variables to changes in the cash share. The main contribution of this paper is to explicitly decompose the trend decrease in cash use into a component due to changes in demographic and location-speci…c variables, as well as a transaction-size component and a pure time e¤ect. We …nd that location-speci…c changes in demographic and other variables account for between 0.5 and 1.3 percentage points of the 8.6 percentage-point overall decline. Increasing transaction sizes account for 1.3 percentage points, which leaves between 70 and 80 percent of the decline in cash use unexplained. The unexplained portion is likely being driven 74 Federal Reserve Bank of Richmond Economic Quarterly by improved actual characteristics of payment cards as well as slowly evolving consumer perceptions of those characteristics. Referring back to the introduction, although we attribute a relatively small portion of the decline in cash use to location-speci…c factors, it would be premature to dismiss distributional arguments about the bene…ts of currency. First, evaluating those arguments requires quantifying the bene…ts of currency and payment cards to di¤erent groups; that is not part of our analysis and would require an economic model. Second, for the stores and time period in our study, the share of cash transactions declined from 78 percent to 70 percent. Whether our results would carry over to a much larger decline in cash use is an open question, to which time may help provide the answer. Finally, our focus has been on demographic and other location-speci…c factors across the store locations in our study. As discussed in Wang and Wolman (2016a), those stores are generally located in relatively low-income zip codes. It is possible that analysis of additional retailers in other locations would reveal that demographics account for a greater proportion of the change in cash shares; that is, part of the change in cash shares that we label unexplained may be accounted for by characteristics that are common to the stores and customers studied here but that are distinctive in the context of the entire US economy. Wang & Wolman: Decline in Currency Use APPENDIX: 75 THE FRACTIONAL MULTINOMIAL LOGIT MODEL The regression analysis in the paper uses the FMLogit model. The FMLogit model conforms to the multiple fractional nature of the dependent variables, namely that the fraction of payments for each instrument should remain between 0 and 1, and the fractions add up to 1. The FMLogit model is a multivariate generalization of the method proposed by Papke and Wooldridge (1996) for handling univariate fractional response data using quasi-maximum likelihood estimation. Mullahy (2010) provides more econometric details. Formally, consider a random sample of i = 1; :::; N zip-code-day observations, each with M outcomes of payment shares. In our context, M = 4, which corresponds to cash, debit, credit, and check. Letting sik represent the k th outcome for observation i, and xi , i = 1; :::; N , be a vector of exogenous covariates, the nature of our data requires that sik 2 [0; 1] Pr(sik = 0 j xi ) and k = 1; :::; M ; 0 and M X Pr(sik = 1 j xi ) 0; sim = 1 for all i: m=1 Given the properties of the data, the FMLogit model provides consistent estimates by enforcing conditions (3) and (4), E[sk jx] = Gk (x; ) 2 (0; 1); k = 1; :::; M ; M X m=1 E[sm j x] = 1; (3) (4) and also accommodating conditions (5) and (6), Pr(sk = 0 j x) Pr(sk = 1 j x) 0 0 k = 1; :::; M ; (5) k = 1; :::; M ; (6) where = [ 1 ; :::; M ]:9 Speci…cally, the FMLogit model assumes that the M conditional means have a multinomial logit functional form in 9 To simplify the notation, we suppress the “i” subscript in Eqs (3)-(9). 76 Federal Reserve Bank of Richmond Economic Quarterly linear indexes as exp(x E[sk j x] = Gk (x; ) = M X k) exp(x ; k = 1; :::; M: (7) m) m=1 As with the multinomial logit estimator, one needs to normalize for identi…cation purposes, and we choose the normalization M = 0. Therefore, Eq (7) can be rewritten as exp(x Gk (x; ) = 1+ M X1 k) exp(x ; k = 1; :::; M 1; (8) m) m=1 and 1 GM (x; ) = 1+ M X1 exp(x : (9) m) m=1 Finally, one can de…ne a multinomial logit quasilikelihood function L( ) that takes the functional forms (8) and (9) and uses the observed shares sik 2 [0; 1] in place of the binary indicator that would otherwise be used by a multinomial logit likelihood function, such that L( ) = N Y M Y Gm (xi ; )sim : (10) i=1 m=1 The consistency of the resulting parameter estimates ^ then follows from the proof in Gourieroux et al. (1984), which ensures a unique maximizer. In our regression analysis, we use Stata code developed by Buis (2008) for estimating the FMLogit model. Wang & Wolman: Decline in Currency Use 77 REFERENCES Buis, Maarten L. 2008. “FMLogit: Stata Module Fitting a Fractional Multinomial Logit Model by Quasi Maximum Likelihood.” Statistical Software Components, Department of Economics, Boston College (June). Federal Reserve System. 2013. “The 2013 Federal Reserve Payments Study.” https://www.federalreserve.gov/paymentsystems/frpayments-study.htm (December). Federal Reserve System. 2016. “The 2016 Federal Reserve Payments Study.” https://www.federalreserve.gov/paymentsystems/frpayments-study.htm (December). Gourieroux, Christian, Alain Monfort, and Alain Trognon. 1984. “Pseudo Maximum Likelihood Methods: Theory.” Econometrica 52 (May): 681–700 Mullahy, John. 2010. “Multivariate Fractional Regression Estimation of Econometric Share Models.” Working Paper 16354. Cambridge, Mass.: National Bureau of Economic Research. (September). Papke, Leslie E., and Je¤rey M. Wooldridge. 1996. “Econometric Methods for Fractional Response Variables with an Application to 401(K) Plan Participation Rates.” Journal of Applied Econometrics 11 (November/December): 619–32. Rogo¤, Kenneth S. 2016. The Curse of Cash: How Large-Denomination Bills Aid Crime and Tax Evasion and Constrain Monetary Policy. Princeton, N.J.: Princeton University Press. Wang, Zhu, and Alexander L. Wolman. 2016a. “Payment Choice and Currency Use: Insights from Two Billion Retail Transactions.” Journal of Monetary Economics 84 (December): 94–115. Wang, Zhu, and Alexander L. Wolman. 2016b. “Consumer Payment Choice in the Fifth District: Learning from a Retail Chain.” Federal Reserve Bank of Richmond Economic Quarterly 102 (First Quarter): 51–78 Economic Quarterly— Volume 104, Number 2— Second Quarter 2018— Pages 79–101 Idiosyncratic Sectoral Growth, Balanced Growth, and Sectoral Linkages Andrew Foerster, Eric LaRose, and Pierre-Daniel Sarte I n general, there is substantial heterogeneity in value added, gross output, and production patterns across sectors within the US economy. There is also considerable asymmetry in intermediate goods linkages; that is, some sectors are much larger suppliers of intermediate goods to di¤erent sectors, on average, than others. Such heterogeneity suggests that there may be signi…cant di¤erences in the extent to which shocks to individual sectors not only a¤ect aggregate output, but also transmit to other sectors.1 In this paper, in contrast to previous literature focusing on shorterrun variations in economic activity, we explore how longer-run growth in di¤erent sectors a¤ects other sectors and overall aggregate growth. We consider a neoclassical multisector growth model with sector-speci…c capital and linkages between sectors in intermediate goods. In particular, we investigate the properties of a balanced growth path where total factor productivity (TFP) growth is sector-speci…c. We derive a relatively simple formula that simultaneously captures all relationships between value-added growth and TFP growth across sectors. We then study the e¤ect of changes in TFP growth in one sector on value-added growth in every other sector. In addition, we can use the Divisia index for aggregate value-added growth to calculate the e¤ect of a change in TFP growth in a given sector on aggregate GDP growth. Finally, using The views expressed herein are those of the authors and do not necessarily re‡ect those of the Federal Reserve Bank of Richmond, the Federal Reserve Bank of San Francisco, or the Federal Reserve System. We thank Caroline Davis, Toan Phan, Santiago Pinto, and John Weinberg for helpful comments. 1 See, for instance, Acemoglu, Carvalho, Ozdaglar, and Tahbaz-Salehi (2012); Foerster, Sarte, and Watson (2011); Atalay (2017); and Miranda-Pinto (2018). DOI: https://doi.org/10.21144/eq1040202 80 Federal Reserve Bank of Richmond Economic Quarterly data on value-added growth for each sector over the period 1948-2014, we recover each sector’s model-implied mean TFP growth over this period and examine how sectoral changes in TFP growth in practice carry over to other sectors. In all three of the above exercises, we also consider a special case of our model without capital. This case collapses to the model considered by Hulten (1978), or Acemoglu et al. (2012). In that model, absent capital, the impact of a level change in sectoral TFP on GDP is entirely captured by that sector’s share in GDP.2 We show that a version of this result also holds in growth rates along the balanced growth path. In that special case, other microeconomic details of the environment become irrelevant as long as we can observe the distribution of valueadded shares across sectors. More generally, in the benchmark model, value-added growth and the e¤ects of changes in TFP growth in a given sector on GDP growth depend on that sector’s capital intensity, its share of value added in gross output, and the degree to which its goods are used as intermediates by other sectors. In this regard, in a multisector model with capital, it becomes important to have information pertaining to the underlying microeconomic structure of the economy beyond what is captured in shares. Fortunately, the model delivers a simple expression of relevant parameters that can easily be constructed from sectoral-level data provided by government agencies. Using such data, we can quantify the e¤ects of changes in sectoral TFP growth and compare these results to the special case of our model where a version of Hulten (1978) holds in growth rates. In the seven sectors we consider in this paper, sectors vary widely in their shares of capital in value added and value added in total output, and some sectors are considerably more important suppliers of intermediate goods than others. Overall, we …nd that adding capital to the model creates substantial spillovers across sectors resulting from TFP growth changes that, for every sector, substantially increase the responsiveness of GDP growth to such changes. These spillover e¤ects are larger for sectors more integral to sectoral linkages in intermediates, a …nding consistent with the literature we discuss below. 2 Pasten, Schoenle, and Weber (2018) and Baqaee and Farhi (2018) show that, even in a model without capital, this result may not hold due to factors such as heterogeneous price rigidity and nonlinearities in production. Foerster, LaRose, Sarte: Growth and Sectoral Linkages 1. 81 RELATED LITERATURE The modern literature on multisector growth models started with the real business cycle model presented in Long and Plosser (1983). In their model, a representative agent chooses labor inputs and commodity inputs to n sectors, with linkages between sectors in inputs and uncorrelated exogenous shocks to each sector. Taking the model to the data with six sectors, they found substantial comovement in output across sectors; furthermore, shocks to individual sectors generally led to large aggregate ‡uctuations, particularly for sectors that heavily served as inputs in production. For many years, there existed a sense that at more disaggregated levels than that of Long and Plosser (1983), idiosyncratic sectoral shocks should fail to a¤ect aggregate volatility. Lucas (1981), in particular, argued that in an economy with disaggregated sectors, many sector-speci…c shocks would occur within a given period and roughly cancel each other out in a way consistent with the Law of Large Numbers. Dupor (1999) helped formalize the conditions under which the intuition in Lucas (1981) would apply. He considered an n-sector economy with linkages between …rms in intermediates as well as full depreciation of capital. Assuming all sectors sold nonzero amounts to all other sectors, and that every row total in the matrix of linkages was the same (i.e., every sector is equally important as an input supplier to all other sectors), Dupor p found that aggregate volatility converged toward zero at a rate of n; the underlying structure of the input-output matrix was irrelevant as long as it satis…ed those conditions. Horvath (1998) countered that Dupor’s irrelevance theorem failed to hold because, in practice, sectors are not uniformly important as input suppliers to other sectors. He observed that at high levels of disaggregation in US data, the matrix of input-output linkages became quite sparse, with only a few sectors selling widely to others; consequently, sectoral shocks could explain a signi…cant sharepof aggregate volatility, which would decline at a rate much slower than n. (Horvath  showed that his earlier result still held in more general models including, among other things, linkages between sectors in investments.) Acemoglu et al. (2012) expand on Horvath’s idea by analyzing the network structure of linkages and conclude that it is the asymmetry, rather than the sparseness, of input-output linkages that determines the decay rate of aggregate volatility. In a multisector model with linkages between sectors in investment as well as intermediates, Foerster, Sarte, and Watson (2011) …nd evidence of a high level of asymmetry in the data, consistent with Acemoglu et al. (2012). They also show that, starting with the Great Moderation around 1983, roughly half the variation in aggregate output stems from sectoral shocks. 82 Federal Reserve Bank of Richmond Economic Quarterly As an additional perspective on the failure of sectoral shocks to average out, Gabaix (2011) also points out that the “averaging out” argument will not hold when the distribution of …rms (or sectors) is fat-tailed, meaning a few large …rms (or sectors) dominate the economy. In such a case, aggregate volatility decays at rate ln1n , and idiosyncratic movements can cause large variations in output growth. While it should be clear from this section that the literature on multisector growth models has mostly focused on the relationship between aggregate and sectoral volatility, this paper focuses instead on the relationship between aggregate and sectoral growth. The arguments of Horvath (1998), Acemoglu et al. (2012), and others regarding the nature of input-output linkages still hold relevance for sectoral growth. In that vein, the analysis herein builds more directly on the work of Ngai and Pissarides (2007). In that paper, the authors focus on the e¤ects of di¤erent TFP growth rates across sectors on sectoral employment shares. The model we present extends their work by explicitly capturing all pairwise linkages in intermediate goods in the economy while additionally allowing every sector to produce capital. 2. ECONOMIC ENVIRONMENT We consider an economy with n sectors. For simplicity, we assume that utility is linear in the …nal consumption good. Preferences are given by E0 1 X t Ct t=0 Ct = n Y j cj;t ; j j=1 n X j = 1; j=1 where Ct represents an aggregate consumption bundle taken to be the numeraire good. Gross output in a sector j results from combining value added and materials output according to yj;t = vj;t j j mj;t 1 j 1 j ; where yj;t , vj;t , and mj;t denote gross output, value added, and materials output, respectively, used by sector j at time t. Materials output in a given sector j results from combining di¤erent intermediate materials Foerster, LaRose, Sarte: Growth and Sectoral Linkages 83 from all other sectors, as described by the production function, mj;t = n Y mij;t n X ij ; ij i=1 = 1; ij i=1 where mij;t denotes the use of materials produced in sector i by sector j at time t. Value added in sector j is produced using capital and labor, j kj;t vj;t = zj;t 1 j 1 `j;t j ; j where zj;t denotes a technical shift parameter that scales production of value added, which we refer to as value-added TFP. Capital is sector-speci…c, so that output from only sector j can be used to produce capital for sector j, and it accumulates according to the law of motion, kj;t+1 = xj;t + (1 ) kj;t ; where xj;t represents investment in sector j at time t and depreciation rate of capital. Goods market clearing requires that cj;t + n X denotes the mji;t + xj;t = yj;t ; i=1 while labor market clearing requires that n X `j;t = 1: j=1 Here, we assume that aggregate labor supply is inelastic and set to one. We also assume that labor can move freely across sectors so that workers earn the same wage, wt , in all sectors. Finally, we assume that TFP growth in sector j, ln zj;t , follows an AR(1) process, ln zj;t = (1 where 3. < 1 and j;t ) gj + ln zj;t 1 + j;t ; D with mean zero for each j. PLANNER'S PROBLEM The economy we have just described presents no frictions, so that decentralized allocations in the competitive equilibrium are optimal. Thus, we derive these allocations by solving the following planner’s 84 Federal Reserve Bank of Richmond Economic Quarterly problem: 1 X max L = t t=0 n Y j cj;t (1) j j=1 such that 8 j and t, cj;t + n X vj;t mji;t + xj;t = j i=1 mj;t = n Y mij;t j kj;t ; (2) (3) 1 `j;t j ; (4) j kj;t+1 = xj;t + (1 n X ; ij 1 j and 8 t, j ij i=1 vj;t = zj;t 1 mj;t 1 j j ) kj;t ; `j;t = 1: (5) (6) j=1 x Let pyj;t , pvj;t , pm j;t , and pj;t denote the Lagrange multipliers associated with, respectively, the resource constraint (2), the production of value added (4), the production of materials (3), and the capital accumulation equation (5) in sector j at date t. The …rst-order conditions for optimality yield j Ct cj;t = pyj;t : This expression also de…nes an ideal price index, 1= n Y pyj;t j : (7) j=1 We additionally have that y j pj;t yj;t : pvj;t vj;t = Likewise, pm j;t mj;t = 1 j pyj;t yj;t : The above two expressions de…ne a price index for gross output, pyj;t = pvj;t j pm j;t 1 j : Foerster, LaRose, Sarte: Growth and Sectoral Linkages 85 In addition, we have that pyi;t mij;t = m ij pj;t mj;t ; which gives material prices in terms of gross output prices, n Y ij pm = ; pyi;t j;t i=1 and v wt `j;t = (1 j )pj;t vj;t ; where wt is the Lagrange multiplier associated with the labor market clearing condition (6). From the law of motion for capital accumulation, we have that pxj;t = pyj;t : Finally, the Euler equation associated with optimal investment dictates pvj;t+1 vj;t+1 pxj;t = Et j + pxj;t+1 (1 ) : kj;t+1 The …rst-order conditions give rise to natural expressions of the model parameters as shares that are readily available in the data. In particular, j represents the share of sector j in nominal consumption, and j represents the share of value added in total output in sector j, while ij represents materials purchased from sector i by sector j as a share of total materials purchased in sector j. Furthermore, 1 j equals the share of total wages in nominal value added in sector j, and consequently, j represents capital’s share in nominal value added. Nominal value added in sector j in this economy is then given P by pvj;t vj;t = i pyj;t yj;t , and it follows that GDPt = j pvj;t vj;t . In the remainder of this paper, we adopt the following notation: = diagf j g, d = diagf j g, = ( 1 ; :::; n ), and = f ij g. d Some Benchmark Results in Levels A special case of the economic environment presented above is one where j = 0 8j, which, absent any growth in sectoral TFP or shocks, reduces to the static economies of Hulten (1978) or Acemoglu et al. (2012). In this case, aggregate value added, or GDP, is given by the consumption bundle Ct and @ ln GDPt = svj 8t, @ ln zj;t where svj is sector j’s value-added share in GDP, and we summarize these shares in a vector, sv = (sv1 ; :::; svn ), given by sv = (I (I d) 0 ) 1 d: (8) 86 Federal Reserve Bank of Richmond Economic Quarterly As shown in Hulten (1978), in this special case, a sector’s value-added share entirely captures the e¤ect of a level change in TFP on GDP. Accordingly, Acemoglu et al. (2012) refer to the object (I (I 0 1 d) ) d as the in‡uence vector. A model with capital is dynamic but, in the long run, converges to a steady state in levels absent any sectoral TFP growth. With a discount factor close to 1, the e¤ect of a level change in sectoral log TFP on log GDP continues to be given primarily by sectoral shares, as in equation (8). In other words, Hulten’s (1978) result continues to hold in an economy with capital in that the variation in the e¤ects of sectoral TFP changes on GDP is determined by the variation in sectoral shares. In this case, however, sectoral shares need to be adjusted by a factor that is constant across sectors and approximately equal to the inverse of the mean employment share. With exogenous sectoral TFP growth, the economy no longer achieves a steady state in levels. Instead, with constant sectoral TFP growth, the steady state of the economy may be de…ned in terms of sectoral growth rates along a balanced growth path. Along this path, the effects of TFP growth changes on GDP growth involve additional considerations. In particular, sectoral linkages in intermediates mean that changes in sectoral TFP growth in one sector potentially a¤ect valueadded growth rates in every other sector and, therefore, can impact overall GDP growth beyond changes in shares. These sectoral linkages consequently create a multiplier e¤ect that, as we show below, can lead to a total impact of a TFP growth change in a given sector that is several times larger than that sector’s share in GDP. 4. SOLVING FOR BALANCED GROWTH We now allow for each sector to grow at a di¤erent rate along a balanced growth path. In particular, we derive and explore the relationships that link di¤erent sectoral growth rates to each other and study how TFP growth rates in one sector a¤ect all other sectors and the aggregate balanced growth path. Consider the case where zj;t is growing at a constant rate along a nonstochastic steady-state path, that is j;t = 0 and ln zj;t = gj 8j, t. Moreover, the resource constraint (2) in each sector requires that all variables in that equation grow at the same constant rate along a balanced growth path. Therefore, we normalize the model’s variables in each sector by a sector-speci…c factor j;t . In particular, we de…ne y~j;t = yj;t = j;t ; c~j;t = cj;t = j;t ; m ~ ji;t = mji;t = j;t ; and x ~j;t = xj;t = j;t . We show that detrending the economy yields a system of equations that is stationary in the normalized variables along the balanced growth path Foerster, LaRose, Sarte: Growth and Sectoral Linkages 87 and where the vector t = ( 1;t ; :::; n;t )0 can be expressed as a function of the underlying parameters of the model only. Detrending the Economy The capital accumulation equation in sector j can be written under this normalization as kj;t+1 = x ~j;t j;t + (1 ) kj;t ; so that k~j;t+1 = x ~j;t + (1 j;t 1 ) k~j;t ; j;t where k~j;t = kj;t = j;t 1 . Using this last equation, we can write value added in sector j as ! j 1 j k~j;t j;t 1 `j;t vj;t = zj;t : 1 j j P The aggregate labor constraint in each period, j `j;t = 1, implies that the labor shares, `j;t , are already normalized: `~j;t = `j;t . Then de…ning v~j;t = vj;t = zj;t j;t 1 j , the expression for value added becomes ! j 1 j k~j;t `j;t v~j;t = : 1 j j The equation for materials used in sector j can be written in normalized terms as n Y m ~ ij;t ij m ~ j;t = ; ij i=1 Q where m ~ j;t = mj;t = ni=1 i;tij . It follows that gross output in sector j becomes, in normalized terms, 11 j ! j0 Qn j ij v~j;t zj;t j;t m ~ @ j;t i=1 i;t A y~j;t j;t = ; 1 j j which may be rewritten as y~j;t = v~j;t j j m ~ j;t 1 j 1 j " zj;tj j j j;t 1 j;t n Y (1 i;t i=1 j) ij # : (9) Observe that for the detrended variables to be constant along a balanced growth path, it must be the case that the expression in square 88 Federal Reserve Bank of Richmond Economic Quarterly brackets is also constant along that path. Thus, we can use equation (9) to solve for j;t as a function of the model parameters. In particular, we can rewrite the term in square brackets as zj;tj j j j j j;t 1 j;t j j j;t 1 n Y (1 j ) ij i;t ; i=1 where we aim for the growth rate of j;t to be constant. Thus, without loss of generality, we choose j;t such that zj;tj j j j;t n 1Y (1 j ) ij = 1; i;t i=1 which in logs gives j ln zj;t + j 1 ln uj;t + j n X 1 j ij ln i;t = 0: (10) i=1 In matrix form, with zt = (z1;t ; :::; zn;t )0 , equation (10) becomes d ln zt +( I) ln d d t + (I d) 0 ln t = 0: It follows that along a balanced growth path, ln t = I d d (I 0 d) 1 d gz ; (11) where gz = (g1 ; :::; gn )0 . Sectoral Value Added and GDP along a Balanced Growth Path Having derived expressions in terms of the normalizing factors for j;t , we now derive the normalizing factors for value added in each sector. By construction, these factors in turn will grow at the same rate as value added in each sector. As given above, the normalizing factor for value added in sector j, denoted as vj;t , is zj;t j;tj 1 . In vector form, this becomes ln v t = ln zt + (I d) so that along a balanced growth path, h ln vt = I + d I (I d d d) d I d d 1 0 ln zt d 0 1 d i gz : 1; (12) In other words, in this economy, TFP growth in each sector potentially a¤ects value-added growth in every other sector through a matrix that summarizes all linkages i in the economy, h 1 0 I + d (I (I ) d d d) d : Moreover, these e¤ects may be Foerster, LaRose, Sarte: Growth and Sectoral Linkages 89 summarized analytically by h i @ ln vt 1 0 = I+ d I (I ) (13) d d d d ; @gz where the element in row i and column j of this matrix represents the e¤ect of an increase in TFP growth in sector j on value-added growth rates in sector i: @ ln vi;t = 1 + i j ij if i = j; @gj where (I d d (I d) @ ln @gj 0) 1 v i;t = =f ij g, i j ij or if i 6= j: As mentioned above, growth rates in every sector depend on TFP growth rates in every sector because of the linkages between sectors 0) 1 in intermediate goods. The matrix (I (I d d d) d suggests that, all else equal, TFP growth changes in sectors that are more capital intensive (i.e., where j is higher) and have higher shares of value added in gross output (i.e., where j is higher) will tend to have larger e¤ects on other sectors. Additionally, more capital-intensive sectors will tend to have larger responses to TFP growth changes in other sectors. The expression for GDP gives us n X GDPt = pvj;t vj;t : j=1 Using a standard Divisia index, we can express aggregate GDP growth as a weighted average of sectoral growth rates in real value added, n X ln GDPt = svj;t ln vj;t ; (14) j=1 where svj;t is the share of sector j in nominal value added,3 pvj;t vj;t svj;t = Pn : v j=1 pj;t vj;t De…ne ln vt = ln vt along the balanced growth path. We may then substitute our expression for ln vt in terms of TFP to obtain the 3 These shares also hold in normalized form, so that svj;t = p ~v v ~ Pn j;t vj;t , ~j;t v ~j;t j=1 p and are constant along the balanced growth path. Here we take the shares as exogenous parameters given in the data, but they can alternatively be solved as part of the steady state in normalized variables. 90 Federal Reserve Bank of Richmond Economic Quarterly balanced growth rate of real aggregate GDP in terms of TFP growth: h i 1 0 ln GDPt = sv I + d I (I d d d) d gz : This last expression implies that, with constant shares, h @ ln GDPt 0 = sv I + d I (I d d d) @gz 1 d i ; (15) with the e¤ect of a change in TFP growth in sector j on GDP growth then given by the jth element, ! n X @ ln GDPt = svj + svi i j ij : @gj i=1 The above equation shows that TFP changes in sectors with higher shares of value added in gross output, and whose intermediates are more heavily used by other sectors, will have larger e¤ects on changes in GDP growth. Balanced Growth with No Capital Consider the special case of our model with no capital accumulation, j = 0 8j. Then the formula for value added in sector j becomes vj;t = zj;t `j;t : Since labor supply, `j;t , is already normalized as implied by the labor supply constraint, the normalizing factor for value added in sector j at time t, vj;t , is simply vj;t = zj;t , so that along a balanced growth path ln vt = gz . Then we have @ ln @gz v t = I; (16) so a change in TFP growth in sector j changes value-added growth in sector j by the same amount and has no impact on value-added growth in other sectors, even though sector j is linked to other sectors through intermediate goods. From equation (16), in the model without capital, we then have along a balanced growth path @ ln GDPt = sv ; @gz (17) which has jth element svj . Put another way, a change in TFP growth in sector j increases the growth rate of real aggregate GDP by that sector’s share of value added in GDP. To a …rst order, the intermediate goods matrix and other details are irrelevant as long as we know the value-added distribution of sectors. Foerster, LaRose, Sarte: Growth and Sectoral Linkages 91 In the rest of this paper, we match this model to the data with n = 7 sectors in order to quantify equations i(13) and (15), and we also invert h 0) 1 I + d (I (I d d d) d in equation (11) to obtain the implied TFP growth rates in each sector. We also use equations (16) and (17) to compare our quantitative benchmark results to those in the case without capital. 5. DATA As described above, the natural expressions of several model parameters as shares make it easy to match this model to available data. All of the model parameters, consisting of the matrix, the j ’s, and the j ’s, can be obtained through the Bureau of Economic Analysis (BEA), which provides data at various levels of industry aggregation going back to 1947. The highest level of aggregation reported by the BEA is the …fteenindustry level. We drop one industry corresponding to Government, and then we consolidate the fourteen remaining industries into seven broader sectors: Agriculture, Forestry, Fishing, and Hunting; Mining and Utilities; Construction; Manufacturing; Wholesale and Retail Trade; Transportation and Warehousing; and Services. The sevensector level is a high enough level of aggregation to give us a broad overview of the economy, and these constructed sectors closely match the six sectors examined by Long and Plosser (1983). To assemble the matrix for our benchmark year, 2014, we rely on data from the BEA’s Make-Use Tables, which at the …fteen-industry level provide a …fteen-by-…fteen matrix showing all pairwise combinations of intermediate goods purchases by one industry from another. From here, we sum intermediate goods purchases across all industries in a sector and then calculate shares of nominal intermediates from sector i in sector j’s total nominal intermediates accordingly (dropping intermediate purchases from the Government sector from the total). In addition to calculating the matrix for 2014, we also calculate it for 1948, the earliest year for which data on value-added growth are available. Later on, we will be interested in comparing our results when using the matrix for 1948 to those using the matrix for 2014 to see how changes in intermediate purchases patterns across sectors have a¤ected growth and TFP throughout the economy. The BEA provides the pairwise intermediates purchases at a higher level of disaggregation in 1948, with forty-six industries. Since every industry at the …fteenindustry level is a grouping of industries at the forty-six-industry level, we can sum intermediate goods purchases across industries in a sector as before. 92 Federal Reserve Bank of Richmond Economic Quarterly We also use the BEA’s Make-Use Tables to calculate each sector’s share of nominal value added in nominal gross output, j , for 2014 by summing total value added and total gross output across industries in a sector and dividing accordingly. To calculate shares of capital in nominal value added, j , we use the BEA’s data on GDP by industry, which breaks down value added within an industry into the sum of wages paid to employees, a gross operating surplus, and taxes minus subsidies. We sum the …rst two components across industries in a sector, ignoring taxes and subsidies, and calculate j as sector j’s gross operating surplus divided by the sum of its gross operating surplus and wages. Finally, the BEA’s GDP data include the total nominal value added for each industry at the …fteen-industry level for each year going back to 1947. We use the BEA’s chain-type price indexes for value added in each industry to calculate these numbers in real terms, then sum across industries in a sector to obtain real value added for each sector. From here, we can easily calculate the real value-added growth rates for each sector for each year from 1948 through 2014 and take an average for each sector over this period to get mean value-added growth rates. Additionally, we can calculate a sector’s share in nominal value added for each year (excluding value added from the Government sector in total value added) and average across years to obtain each sector’s mean share in nominal value added. Table 1 displays the share of nominal value added in nominal gross output, j , and the share of capital in nominal value added, j , for each sector. Some of these results are fairly intuitive; for instance, Construction and Wholesale and Retail Trade have the lowest (highest) shares of capital (labor) in value added, while Agriculture, Forestry, Fishing, and Hunting, and Mining and Utilities are the most capitalintensive. There is somewhat less variation in the shares of nominal value added in nominal gross output, with Manufacturing having the lowest share and Mining and Utilities having the highest. Table 2 displays the matrix summarizing intermediate goods linkages, , calculated for 2014, where the element in row i and column j represents the percentage of all intermediate goods purchased by sector j that come from sector i. First, it is not surprising that most sectors purchase a large share of intermediate goods from within their own sector: …ve of seven sectors have jj values above 20 percent, with the Services sector purchasing over 75 percent of its intermediates from itself. It is also important to note that, in general, the matrix displays substantial asymmetry. The average sector buys approximately 35 percent and 29 percent of its intermediates from Services and Manufacturing, respectively. If we exclude the diagonal entries of , these Foerster, LaRose, Sarte: Growth and Sectoral Linkages 93 Table 1 Parameter Values for Each Sector Sector Sector Number Agriculture, Forestry, Fishing, and Hunting Mining and Utilities Construction Manufacturing Wholesale and Retail Trade Transportation and Warehousing Services (1) (2) (3) (4) (5) (6) (7) Table 2 j j 0.4139 0.6845 0.5419 0.3462 0.6558 0.4795 0.6123 0.7493 0.7337 0.3659 0.5205 0.3680 0.3865 0.4556 in 2014, with All Numbers Expressed as Percentages Sector Number (1) (2) (3) (4) (5) (6) (7) (1) (2) (3) (4) (5) (6) (7) 39.72 2.88 0.96 29.16 10.30 5.58 11.39 0.04 32.76 3.86 21.40 4.10 9.27 28.57 0.27 2.47 0.03 52.72 24.00 3.85 16.65 7.20 15.70 0.36 50.37 8.03 4.11 14.24 0.31 1.66 0.41 9.12 7.26 12.53 68.70 0.02 1.84 1.01 31.90 9.23 23.85 32.15 0.19 2.65 2.64 12.98 3.31 2.73 75.51 numbers are still 29 percent and 26 percent. On the other hand, Agriculture, Forestry, Fishing, and Hunting, and Construction stand out as relatively unimportant suppliers of intermediate goods to other sectors. 6. QUANTIFYING BALANCED GROWTH RELATIONSHIPS @ ln v t As derived in equation (13), = @gz h i 1 0 I + d (I (I ) d d d) d in the benchmark model. Table 3 shows this matrix for our seven sectors. The element in row i and column j shows the percentage-point increase in value-added growth in sector i resulting from a 1 percentage point increase in TFP growth in sector j. Unsurprisingly, increases in TFP growth in sector j have by far the largest impact on value-added growth rates in that same sector; all the entries on the diagonal have magnitude greater than 1, with Mining and Utilities having the largest diagonal value and Construction having the smallest. However, the o¤-diagonal entries still indicate substantial e¤ects of TFP growth changes in one sector on value-added growth in another. For instance, a 1 percentage point increase in TFP 94 Federal Reserve Bank of Richmond Economic Quarterly Table 3 E ect of 1 Percentage Point Change in TFP Growth on Value-Added Growth in Percentage Points Sector Number (1) (2) (3) (4) (5) (6) (7) (1) (2) (3) (4) (5) (6) (7) 1.7131 0.0187 0.0135 0.0536 0.0048 0.0118 0.0075 0.2099 2.3645 0.0692 0.2371 0.0295 0.0653 0.0500 0.0160 0.0221 1.2507 0.0090 0.0035 0.0059 0.0090 0.2751 0.1456 0.1032 1.4316 0.0332 0.0925 0.0538 0.1512 0.0615 0.0669 0.0801 1.3409 0.0454 0.0252 0.0726 0.0572 0.0189 0.0405 0.0211 1.2808 0.0146 0.4271 0.3818 0.1502 0.2862 0.2065 0.2153 1.7053 growth in the Services sector increases value-added growth in Agriculture, Forestry, Fishing, and Hunting by about 0:43 percentage points. Overall, increases in TFP growth rates in the Services sector have particularly strong e¤ects on value-added growth rates in other sectors, re‡ecting the generally high usage of intermediate goods from Services by other sectors. On the other hand, changes in TFP growth in other sectors have small e¤ects on value-added growth in Services, in part because Services purchases a small fraction of its intermediates from other sectors. (These observations apply, to a somewhat lesser extent, to the Manufacturing sector as well.) Increases in TFP growth rates in sectors such as Construction and Agriculture, Forestry, Fishing, and Hunting, whose intermediates are not heavily used by other sectors, have tiny e¤ects on value-added growth in other sectors. Finally, it is worth noting that Mining and Utilities and Agriculture, Forestry, Fishing, and Hunting, whose j values are substantially higher than those of other sectors, are, on average, the most responsive to sectoral TFP growth changes. In the case with no capital, a TFP growth change in sector j changes value-added growth in sector j by the same amount and has no impact on value-added h growth in other sectors. Since all the i diagonal entries 1 0 of the matrix I + d (I (I ) d d d) d have values above 1, linkages increase the own-sector e¤ect of TFP growth rate increases on value-added growth rates in every sector. Given data on shares of each sector in nominal value added, we can then calculate the e¤ect of changes in TFP growth in each sector on changes in aggregate GDP in the benchmark model according to equation (15). As described above, we compile data on sectoral shares in nominal value added for each year in the period 1948–2014, and then we take the mean shares in nominal value added for each sector over t calculated from these mean shares this period. Table 4 shows @ ln@gGDP z Foerster, LaRose, Sarte: Growth and Sectoral Linkages 95 Table 4 E ect of 1 Percentage Point Change in TFP Growth on GDP Growth in Percentage Points Sector No Capital Benchmark Di¤erence Agriculture, Forestry, Fishing, Hunting Mining and Utilities Construction Manufacturing Wholesale and Retail Trade Transportation and Warehousing Services 0.0297 0.0457 0.0502 0.2332 0.1552 0.0425 0.4435 0.0695 0.2026 0.0712 0.3868 0.2505 0.0794 0.9020 0.0398 0.1569 0.0210 0.1536 0.0953 0.0369 0.4585 for both cases. The …rst column shows the case with no capital, where each entry just equals that sector’s mean share in total nominal value added. Two of the seven sectors, Services and Manufacturing, account for over two-thirds of total nominal GDP, on average. The second column shows the benchmark case, and the di¤erence between the two cases in the third column can be interpreted as the total multiplier e¤ect of a change in TFP growth in one sector on other sectors (including itself). t Figure 1 plots the mean value-added shares against @ ln@gGDP comz puted in the benchmark. The size of the deviation from the forty-…vedegree line indicates the size of the multiplier e¤ects on other sectors. In absolute terms, this multiplier e¤ect is by far the largest for the Services sector, in part re‡ecting the fact that the o¤-diagonal entries of 0) 1 the matrix (I (I d d d) d are, on average, the highest for the column corresponding to Services. There are also large increases for Manufacturing, another sector important in the production of intermediate goods, and Mining and Utilities, which has a multiplier e¤ect over three times as large as its share in GDP. This can be largely explained by the sector’s high share of capital in value added and its importance as an intermediate goods supplier to itself and to the second-largest sector, Manufacturing. To see the extent to which changes in the usage of intermediate goods across sectors, summarized in , have impacted the e¤ect of TFP growth changes in a sector on changes in the growth rate of GDP, t using the matrix in 1948. Figure 2 plots we also recompute @ ln@gGDP z @ ln GDPt calculated in the benchmark using from 2014 against the @gz values calculated from 1948. Because we hold the other parameters constant for each sector, any changes should result from changes in the relative importance of sectors as intermediate goods suppliers to other sectors. As noted by Choi and Foerster (2017), there have been 96 Federal Reserve Bank of Richmond Economic Quarterly Figure 1 Derivative of GDP Growth with Respect to Sector TFP Growth signi…cant changes in the US economy’s input-output network structure over this period. In particular, the Services sector is a markedly more important supplier of intermediate goods in 2014 than it was in 1948, driven by the increasing centrality of …nancial services, real estate, and other industries within this sector. On the other hand, sectors such as Manufacturing; Agriculture, Forestry, Fishing, and Hunting; and Mining and Utilities declined in importance over this period. Consistent with these observations, Services saw the largest abt over this period, while Manufacturing saw solute increase in @ ln@gGDP z the largest absolute decrease, and Agriculture, Forestry, Fishing, and Hunting saw the largest percentage decrease. On the other hand, bet cause @ ln@gGDP also depends on the shares of each sector in total z nominal value added, a sector may decline in overall importance, as measured by its row total in , over this period while still having t an increasing value of @ ln@gGDP . For example, Mining and Utilities z declines in overall importance between 1948 and 2014 but it is a much more important supplier of intermediates for the Manufacturing sector Foerster, LaRose, Sarte: Growth and Sectoral Linkages Figure 2 E ect of TFP Growth on GDP Growth, 1948 2014 97 vs. in 2014 than in 1948, largely explaining why Mining and Utilities sees t a slight overall increase in @ ln@gGDP . z As a …nal exercise, given data on value-added growth, we can invert 0) 1 the matrix I + d (I (I d d d) d to obtain the implied TFP growth rates in the benchmark: i 1 h 1 0 gz = I + d I (I ) ln vt : (18) d d d d With no capital, this expression simply becomes gz = ln v t: (19) For each of our seven sectors, we take an average of their real valueadded growth rates over the period 1948-2014 and then calculate the implied mean TFP growth rates over this period. Figure 3 plots observed mean value-added growth against the model-implied mean TFP growth in the benchmark case and the case with no capital, where in the latter case all points lie on the forty-…ve-degree line. In the benchmark, all points lie well to the left this line. The decrease is largest in absolute terms for Agriculture, Forestry, Fishing, and Hunting and, consistent with intuition, is generally larger for sectors with larger val- 98 Federal Reserve Bank of Richmond Economic Quarterly Figure 3 Implied Mean TFP Growth, 1948-2014 ues of j . The implied mean TFP growth for Mining and Utilities is just 0:08 percent. Additionally, for the benchmark case, we calculate implied mean TFP growth rates using the matrix for 1948 and compare the results to those using the matrix for 2014. As shown in Figure 4, changes in patterns of intermediate goods usage between 1948 and 2014 have very little impact on implied mean TFP growth rates. 7. CONCLUSION Our analysis suggests that linkages between sectors in intermediate goods, and capital intensities of di¤erent sectors, lead to substantial e¤ects of sector-speci…c TFP growth changes on value-added growth. TFP growth changes in sectors such as Manufacturing and Services, which account for a large share of the intermediate goods shares of other sectors, have especially large impacts on value-added growth in other sectors. On the other hand, changes in the input-output structure of the US economy from 1948 to 2014 have had a modest impact on Foerster, LaRose, Sarte: Growth and Sectoral Linkages Figure 4 Implied Mean TFP Growth, 1948 99 vs. 2014 TFP growth in each sector and on the e¤ect of TFP growth changes on GDP growth. It is worth noting that our analysis here relies on a very high level of aggregation, with only seven sectors, and every sector uses some positive amount of intermediate goods from every other sector. 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