Haipeng Daniel Levy
email: levyda@mail.biu.ac.il.homepage

Allan ) Chen

Sourav Ray

Mark Bergen

Gershon Alperovich

Ignazio Angeloni

Larry Ball

Bob Barsky

Susanto Basu

David Bell

Leif Danziger

Martin Eichenbaum

Ben Friedman

Xavier Gabaix

Vitor Gaspar

Wolter Hassink

Miles Kimball

Anil Kashyap

Saul Lach

John Leahy

Dongwon Lee

Andy Levin

Igal Milchtaich

Benoît Mojon

Monika Piazzesi

Akshay Rao

Ricardo Reis

Christina Romer

David Romer

Stephanie Rosenkranz

Avichai Snir

Bent Sorensen

Dani Tsiddon

Alex Wolman

Andy Young

Tao Zha

Chetan Agarwal

Manish Aggarwal

Ning Liu

Sandeep Mangaraj

Asymmetric price adjustment in the small

Keywords: E31, D11, D21, D80, L11, L16, M31 Asymmetric price adjustment, Price rigidity, Inflation, Rational inattention, Monetary policy

published or not. The documents may come

Introduction

A longstanding question in the price adjustment literature is whether or not prices adjust asymmetrically (Ball and Mankiw 1994, Carlton 1986, and Mankiw and Romer 1991). Although economists have devoted considerable attention to this issue (recent studies include Davis and Hamilton 2004, Rotemberg 2005, and Peltzman 2000), the link between asymmetry and the size of price changes has not received much attention. [START_REF][END_REF] This paper studies retail price data from a large US supermarket chain and offers evidence on a new and unusual type of asymmetric price adjustment. The dataset itself is quite large containing about 100 million weekly price observations for 18,037 products. The analysis of the data reveals a surprising regularity-small price increases are more frequent than small price decreases for price changes of up to about 10 cents. Furthermore, it turns out that inflation can explain some of the asymmetry. Inflation, however, fails to explain it fully. For example, the asymmetry is present even if one considers only a deflation-period sample, or if one focuses only on the products whose prices have not increased. The findings are robust across different measures of inflation (aggregate and disaggregate), and to lagged price adjustments.

The paper is organized as follows. Next section describes the data. Section 3 discusses the findings. Section 4 addresses robustness. Section 5 offers possible explanations. Section 6 concludes.

Data

The study uses scanner price data from Dominick's-a large supermarket chain in the Chicago metro area, operating 94 stores with a market share of 25 percent. In 1999 the US retail grocery sales reached $435 billion. Dominick's, thus, represents a major class of the retail trade. Moreover, the sales of large supermarket chains constitute about 14 percent of the total retail sales of about $2.25 trillion. Retail sales account for about 9.3 percent of the GDP, and thus our data represent as much as 1.3 percent of the GDP, which seems substantial.

The data set consists of up to 400 weekly observations of retail prices in 27 product categories representing 30 percent of the chain's revenue, from September 14, 1989 to May 8, 1997, although the length of individual series vary. [START_REF]I -Monthly Index 1[END_REF] The data contain the actual transaction prices paid at the cash register. 3 Table 1 displays the list of the product categories that are included in the dataset along with some general descriptive statistics.

Empirical Findings

Before presenting the findings, consider a sample series from the data. Figure 1 displays the weekly prices of Heritage House frozen concentrate orange juice, 12oz, from Dominick's Store No. 78. The series contain the following "small" price changes: (a) 1¢: 9 positive (weeks 13, 237, 243, 245, 292, 300, 307, 311, and 359) and 6 negative (weeks 86, 228, 242, 275, 386, and 387); (b) 2¢: 7 positive (weeks 248, 276, 281, 285, 315, 319, and 365) and 1 negative (week 287); (c) 3¢: 3 positive (weeks 254, 379, and 380) and 2 negative (weeks 203 and 353); (d) 4¢: 4 positive (weeks 23, 197, 318, and 354) and 1 negative (week 229); and (e) 5¢: 1 positive (week 280) and 1 negative (week 302). Thus, in this series there are more positive than negative price changes up to 4¢. Below the paper studies the pattern of price changes for the full sample as well as for the individual categories, to determine whether this pattern holds more generally.

Findings for the Full Sample

Figure 2 shows the cross-category average frequency of positive and negative price changes. A robust regularity is immediately apparent: there are more "small" price increases than decreases which we call asymmetry "in the small." The asymmetry lasts for price changes of up to about 10-15 cents, which is about 5 percent of the average retail supermarket price of about $2.50 (Levy, et al., 1997; Bergen, et al., 2008). Beyond that, the two lines crisscross each other and thus, the systematic asymmetry disappears.

Table 2 reports the category level asymmetry thresholds based on z-test results. Under the null, there should be equal number of price increases and decreases for each size of price change. We define an "asymmetry threshold" as the last point at which the asymmetry is supported statistically, that is, the last point at which the frequency of price increases exceeds the frequency of price decreases of the same absolute magnitude (z ≥ 1.96). 4 According to column 1 of Table 2, in four categories the asymmetry threshold falls below 5¢, and in two categories it exceeds 25¢. In most categories, however, the asymmetry threshold falls in the range of 5¢-25¢, averaging 11.3¢." 5

Findings for Low-Inflation and Deflation Periods

The most immediate explanation for these findings might be inflation. During the sample period, the US was experiencing a moderate inflation, with an annual rate of between 5 percent (the first year of the sample) and 2.5 percent (last year of the sample). 6 During inflation one expects to see more price increases than decreases (Ball and Mankiw, 1994). 7 Therefore, it will be useful to ask whether or not the asymmetry holds when inflationary periods are excluded from the data. Given our large sample, such an analysis is indeed feasible.

To answer this question, two specific analyses were conducted. The first analysis includes only those observations during which the monthly PPI inflation does not exceed 0.1 percent, which is defined here as a low-inflation period. The second analysis includes only those observations in which the monthly PPI inflation rate is non-positive, which is defined here as a deflation-period. 8 For the low-inflation sample (the middle column in Table 2), the asymmetry threshold is 8.2¢ on average. At the category level, the asymmetry holds in all but one category (bath soap), with some decrease in the thresholds, the majority falling between 2¢ and 20¢. In the deflation period sample (the last column in Table 2), the threshold is 6.2¢, on average. At the category level, asymmetry "in the small" is still found for all but two categories, bath soap and frozen entrees.

Thus, the asymmetry decreases from 11.3¢ in the full sample to 8.2¢ in the low inflation sample, and to 6.2¢ in the deflation sample, indicating that inflation accounts for about a half of the asymmetry. This suggests that inflation is indeed playing a role in the asymmetry. However, a sizeable fraction of the asymmetry still remains unexplained.

Asymmetry and Aggregate Inflation

In our data, deflation months are scattered throughout the sample period. To check further how asymmetry varies with inflation, therefore, the asymmetry threshold for each product category for each year was calculated (Table 3, columns A-G). This analysis revealed a negative relationship between asymmetry 4 Out statistical procedure allows for no asymmetry as well as for reverse asymmetry. The current analysis does not find any such case. Similarly, there are very few of them in later analyses (see Table 3 and the supplementary appendix). 5 Considering price changes of up to 50¢ is sufficient given our focus on small price changes. Indeed, price changes of all sizes were calculated, and it was found that most price changes are indeed smaller than 50¢. The full sample contains a total of 10,298,995 price increases and 9,438,350 price decreases, and thus in total, there are more price increases than decreases. Further, 1¢, 2¢, 3¢, 4¢, and 5¢ increases account for 3.60%, 3.50%, 3.39%, 3.30%, and 3.20% of all price increases, respectively. In other words, 17.09% of the price increases are of 5¢ or less. In contrast, 1¢, 2¢, 3¢, 4¢, and 5¢ decreases account for 2.49%, 2.88%, 2.75%, 2.99%, and 2.88% of all price increases, respectively. In other words, 14.00% of price decreases are of 5¢ or less. Thus, the asymmetry holds at the aggregate level as well. 6 These findings cannot be explained by promotions or sales, as promotions likely generate more price decreases than increases, which is opposite to what is observed in our data. In addition, a sale-related temporary price reduction is usually followed by a price increase (Rotemberg 2005). Price promotions, therefore, cannot produce the observed asymmetry. 7 A counter-argument to this idea is that if the reason for the asymmetry was inflation, then one would see the asymmetry not only "in the small" but also "in the large." The data, however, do not exhibit asymmetry "in the large." 8 The frequency plots for the low inflation and the deflation periods are included in the supplementary appendix available upon request. and inflation: over time, the asymmetry increased as inflation decreased (with PPI, t = 1.87, d.f. = 171, p = .03; with CPI, t = 3.15, d.f. =171, p < .01; with CPI-Chicago, t = 2.04, d.f. = 171, p < .05).

Asymmetry and Disaggregate Inflation

Aggregate inflation during the sample period was not too variable. Therefore, a more disaggregated inflation measure was constructed by generating a weekly index (WI) of Dominick's category-level prices using the method of Chevalier, et al (2003). 9 From the WI two monthly (MI) and two annual (AI) indices were derived. The monthly indices MI1 and MI2 were formed by setting the monthly index equal to the weekly index value of the last week of the month, and to the average of the weekly indices over the month, respectively. Similarly, the two annual indices AI1 and AI2 were formed by setting the annual index equal to the weekly index of the last week of the year, and to the average of the weekly indices over the year, respectively.

Using the five category-level price indices, the deflationary periods were identified, and the asymmetry thresholds were calculated for each category. 10 The five new analyses generated a total of 135 (5x27) asymmetry thresholds. The findings, shown in columns H-L of Table 3, confirm the presence of asymmetry in the small: 92% (125/135) of the asymmetry thresholds are positive, while only 4% (5/135) are 0, and 4% (5/135) are -1. 11 The asymmetry thresholds range between 7.11¢ and 8.15¢, with an average of 7.72¢.

As an additional analysis, we run a linear cross-section regression of the category-level asymmetry thresholds on the category-level inflation using each of the five category-level inflation measures. The results suggest that there is no statistically significant relationship between asymmetry and inflation at the category-level. 12 4. Robustness

To check the robustness of this conclusion, five different tests of robustness were conducted. All confirm the conclusion that inflation at best offers a partial explanation for the asymmetry. These tests and the resulting findings are briefly discussed below. For more details, see the supplementary appendix.

Lagged Price Adjustment

The analysis so far assumed instantaneous price adjustment. To allow lagged adjustment, the analysis was repeated with 4-, 8-, 12-, and 16-week lags (Dutta, et al 2002; Bils and Klenow, 2004). The results suggest that the asymmetry holds for 25 of the 27 categories. In 99 of the 108 cases, i.e., in 92 percent of the cases, the thresholds are positive, averaging 6.6¢.

Alternative Measures of Inflation

The above analysis used the PPI. The analysis was repeated using CPI and CPI-Chicago. The latter is useful as it covers the area where most Dominick's stores operate. The findings of these analyses suggest that there is asymmetry in all but two categories, with the average threshold of 6.9¢.

Alternative Measures of Inflation with Lagged Price Adjustment

The analysis of 4.2 was repeated with 4-, 8-, 12-, and 16-week adjustment lags. The findings of these analyses indicate that in 185 of the 216 cases, i.e., in 86% of the cases, the asymmetry remains, with the average threshold of 4.5¢.

Products for Which Prices Have Not Increased

As another test, only the products for which prices have not increased during the sample period 9 See, Chevalier, et al., section for details. 10 The disaggregate price indices indicate greater variation in the inflation rates across categories in comparison to the aggregate inflation. For example, in our sample the average annual category-level inflation rate varies from -25.7 percent for analgesics to 21.9 percent for cookies. In contrast, the aggregate annual inflation rate during the sample period varied between 2 percent to 5.5 percent, on average. 11 The minus sign indicates a reverse asymmetry. The categories with 0 or reverse asymmetries are analgesics, bath soap, shampoo, and toothbrush. For the remaining 23 categories, the asymmetry thresholds are positive. The average asymmetry threshold across the 27 categories is positive in all five analyses (all t 26 's > 7.11, all p-values < .001).

12 For example, using MI2 to measure the category-level inflation, the estimates of the intercept and the slope are 11.3 and -137.3 with t-values 6.8 and -0.7, respectively. Thus the estimated slope is negative but statistically insignificant.

were considered. 13 The findings indicate that in 23 of the 27 categories, i.e., in over 85 percent of the cases, asymmetry is observed.

First Year vs. the Last Year of the Sample Period

The 1989-97period is characterized by a downward inflation trend. If inflation is causing the asymmetry, then the asymmetry should be stronger in the beginning of the sample period in comparison to the end of the sample period. Six product categories lack observations during the first year of the sample period. In 19 of the remaining 21 categories, i.e., in over 90 percent of the categories, a greater asymmetry is found in the last 12 months of the sample, averaging 9.0¢ in comparison to 0.6¢ in the first 12 months. A paired t-test comparing the asymmetry thresholds across the categories indicates statistical significance (t 20 = 4.799, p < .01).

Possible Explanations

The analyses in sections 3 and 4 suggest that inflation cannot fully account for the observed asymmetry. Next, the paper explores whether or not the existing theories of asymmetric price adjustment can explain it. Although these theories can explain asymmetric price adjustment in general, it appears that they are unable to explain the specific form of asymmetry the paper documents. For example, the theory of capacity constraints (Peltzman, 2000) emphasizes the asymmetry in the sellers' ability to adjust inventory to price fluctuations. The theory, however, predicts that asymmetry should be observed for large price changes because small price changes are less likely to make capacity constraints binding. This is the opposite of what is observed in our data. Similarly, theories of vertical channels and imperfect competition cannot explain asymmetry in the small because it is hard to see how market or the channel structure can vary between small and large price changes. Another possible explanation is menu cost under trend inflation. However, if the asymmetry were due to inflation and menu cost (Tsiddon, 1993; Ellingsen, et al., 2005), then one should not have seen asymmetry in periods of low-inflation, and even more so in periods of deflation. The asymmetry, therefore, is unlikely to be driven entirely by inflation. 14 The robustness of our findings and the possible challenges to explain their patterns make them particularly intriguing. As a possible explanation, we hypothesize that that time-constrained consumers may be inattentive to small price changes. 15,16 If, for example, the cost of processing information on a price change exceeds the benefit, then shoppers might choose to ignore-and not react to-small price changes. 17 The inattention creates along the demand curve around the current price a region where consumer sensitivity is low for both small price increases and decreases. This makes small price decreases less valuable to the retailer because the lower price does not trigger the consumer's response. A small price increase, however, is valuable to the retailer as the consumer will not reduce her purchases. Thus, the retailer has incentive to make more frequent small price increases than decreases. Large price changes, however, trigger consumer reaction, and therefore the retailer has no incentive to make asymmetric large price changes. 18,19 The idea that there exists a region of inattention around the current price along the demand curve is consistent with the findings of Fibich, et al. (2007) and Kalwani and Yim (1992), who show that promotional price changes must exceed a certain threshold to produce any effect. It is consistent also with the literature on "just noticeable difference" (Monroe, 2003) and "price indifference bands" (Kalyanaram and Little, 1994). For example, according to McKinsey, the price indifference band is 17 percent for health-and-beauty products and 10 percent for engineered industrial components. Consistent with this, the common managerial intuition is that price reductions of less than 15% do not attract enough customers to a sale (Della, et al 1981; Gupta and Cooper 1992). 20

Conclusion

The paper finds asymmetry for price changes of up to about 10¢. In other words, the paper finds downward price rigidity "in the small." This type of asymmetry has not been reported in the literature, often flying under the radar screen. For example, the data plots presented by Álvarez and Hernando (2004) and Baudry, et al. (2004) clearly indicate asymmetry "in the small" although the authors do not discuss it. These suggest that asymmetry in the small might be more prevalent than people think. 21 Our findings suggest that inflation can explain some of the asymmetry the paper documents, which is interesting because a long-standing question in the New-Keynesian macroeconomic theory is whether or not individual price setters respond to monetary policy or more generally to macro variables. The finding that some of the asymmetry in the small that the paper documents using product-and store-level individual transaction price data is explained by inflation, provides evidence that price-setters may be paying attention and reacting to monetary/macro developments.

There still remains a substantial portion of the asymmetry unexplained, even after accounting for inflation. While the existing theories of asymmetric pricing adjustment cannot explain the remaining asymmetry, it seems consistent with consumer inattention. To the extent that consumers' information processing costs depend on their opportunity costs, their ability to carry out the necessary calculations, their experience with doing this type of calculations and the amount of the calculations required, the asymmetry could vary with the level of customer attentiveness over shopping intensity (e.g., holiday vs. non-holiday periods) and across products and product categories. Therefore, studying settings in which the extent of inattention may vary will offer a more direct test of the empirical plausibility of the rational inattention explanation. Future research can incorporate models of reference point shift (e.g., Chen and Rao 2002; Eichenbaum, Jaimovich, and Rebelo 2008) to study the dynamics of information processing costs and their impact on firms' pricing behavior.

Our findings suggest that markets might respond differently to small and large changes, a notion consistent with the finding that prices react differently to small and large cost shocks (Ball and Mankiw 1994; Dutta, et al. 2002), and with recent field work that studies firms' conduct when they face decisions about small versus large price changes. 22 18 In a world inhabited by inattentive consumers, small price decreases are still possible. First, small price changes may be induced by competitive factors, such as price guarantees and price matches (Levy, et. al., 1997 and 1998), as well as by changes in supply conditions (Dutta, et. al., 1999 and 2002; Levy, et al., 2002) and demand conditions (Okun, 1981; Warner and Barsky, 1995; Chevalier, et. al., 2003). Second, many food items have expiration date, and they may go on sale as the expiration date approaches. And third, managers may be following simple pricing rules, such as "reduce all prices in a given category by 2%," which could lead to small price reductions. See also Lach and Tsiddon (1992, 1996, 2007), Danziger (1999, 2001) and Rotemberg (2008). 19 There is a limit on the surplus a retailer can extract from consumers. For example, if information-processing is costly, the customer may rely on the price for which she has last optimized. The retailer then can raise its price only to the upper bound of the region of inattention. Any additional increase beyond that will push the price far enough from the last optimization price to trigger a re-optimization. Thus, indefinite continuous small price increases are not feasible. 20 The possibility that consumers may be inattentive to small price changes is consistent with the observation that retailers alert the public about promotions by posting sale signs, to ensure that shoppers notice the price discounts. 21 Indeed, in his discussants' comments on this study, Cecchetti (2004) demonstrated that in Europe the phenomenon of asymmetric price adjustment in the small is widespread and is not limited to food store prices. See also Hoffmann and Kurz-Kim (2008). 22 See, for example, Zbaracki, et al. (2004, 2006). See also Cecchetti (1986), Rotemberg (1987), Basu (1995), Danziger (1999), Ball and Romer (2003), Konieczny and Skrzypacz (2005), Fisher and Konieczny (2006), and Barsky, et al. (2007).

Based on our findings, we speculate that asymmetry in the small will be present in settings where low-priced consumer goods are sold (Target, Wal-Mart, etc.). It is unclear, however, how generalizable our findings are to other setting. It is known that in some markets, such as in financial and in business-tobusiness markets, attention is critical because transactions often involve large quantities of the same asset. Similarly, in markets for big-ticket items people might be more attentive because of the large expenditures (Bell, et al., 1998). Even then, however, buyers might ignore some rightmost digits (Lee et. al., 2006). Thus, a car buyer may focus on "fourteen thousand eight hundred" dollars when the actual price is $14,889, creating some room for asymmetric price adjustment in the small. In future work, therefore, it will be valuable to study other data sets, products, and markets. 14, 1989-May 8, 1997 (Source: Dutta, et al. 2002, and Levy, et al. 2002) Notes:

1.

Week 1 = the week of September 14, 1989, and Week 399 = the week of May 8, 1997 2.

There are 6 missing observations in the series.

3.

The series contain many small price changes. Some of them are indicated by the circles. 4.

Section 3 of the text provides the exact list of all price changes of 5 cents or less. 1. PPI = Producer Price Index.

2. Low inflation sample includes the periods during which the monthly change in the PPI does not exceed 0.1 percent. 3. Deflation sample includes the periods during which the monthly change in the PPI does not exceed percent. 4. The figures reported in the table are the cutoff points of what might constitute a "small" price change for each category. The cutoff point is the last point at which the asymmetry is supported statistically (z ≥ 1.96). Thus, for example, in the Analgesics category, when the full sample is used, there is asymmetry (more frequent price increases than decreases) for price changes of up to 30 cents. 5. "0" means that there is no asymmetry.

Figure 1 .

 1 Figure 1. Price of frozen concentrate orange juice, Heritage House, 12oz (UPC = 3828190029, Store 78), September14, 1989-May 8, 1997 (Source: Dutta, et al. 2002, and Levy, et al. 2002)

  Figure R1. Average Frequency of Positive and Negative Price Changes All 29 Categories, Low/Zero Inflation Period

Table 1 .

 1 Descriptive statistics of Dominick's data The data are available at: http://gsbwww.uchicago.edu/kilts/research/db/dominicks/

		Number of	Proportion	Number of	Number of	Mean	Std.	Min.	Max.
	Category	Observations	of the Total	Products	Stores	Price	Dev.	Price	Price
	Analgesics	3,059,922	0.0310	638	93	$5.18 $2.36 $0.47 $23.69
	Bath Soap	418,097	0.0042	579	93	$3.16 $1.60 $0.47 $18.99
	Bathroom Tissue	1,156,481	0.0117	127	93	$2.10 $1.68 $0.25 $11.99
	Beer	1,970,266	0.0200	787	89	$5.69 $2.70 $0.99 $26.99
	Bottled Juice	4,324,595	0.0438	506	93	$2.24 $0.97 $0.32	$8.00
	Canned Soup	5,549,149	0.0562	445	93	$1.13 $0.49 $0.23	$5.00
	Canned Tuna	2,403,151	0.0244	278	93	$1.80 $1.07 $0.22 $12.89
	Cereals	4,747,889	0.0481	489	93	$3.12 $0.76 $0.25	$7.49
	Cheeses	7,571,355	0.0767	657	93	$2.42 $1.12 $0.10 $16.19
	Cigarettes	1,810,614	0.0183	793	93	$7.69 $7.90 $0.59 $25.65
	Cookies	7,634,434	0.0774	1,124	93	$2.10 $0.63 $0.25	$8.79
	Crackers	2,245,305	0.0228	330	93	$2.01 $0.57 $0.25	$6.85
	Dish Detergent	2,183,013	0.0221	287	93	$2.34 $0.90 $0.39	$7.00
	Fabric Softeners	2,295,534	0.0233	318	93	$2.82 $1.45 $0.10	$9.99
	Front-End-Candies	3,952,470	0.0400	503	93	$0.61 $0.24 $0.01	$6.99
	Frozen Dinners	1,654,051	0.0168	266	93	$2.37 $0.89 $0.25	$9.99
	Frozen Entrees	7,231,871	0.0733	898	93	$2.33 $1.06 $0.25 $15.99
	Frozen Juices	2,373,168	0.0240	175	93	$1.39 $0.45 $0.22	$6.57
	Grooming Products	4,065,691	0.0412	1,381	93	$2.94 $1.37 $0.49 $11.29
	Laundry Detergents	3,302,753	0.0335	581	93	$5.61 $3.22 $0.25 $24.49
	Oatmeal	981,106	0.0099	96	93	$2.65 $0.66 $0.49	$5.00
	Paper Towels	948,550	0.0096	163	93	$1.50 $1.41 $0.31 $13.99
	Refrigerated Juices	2,176,518	0.0221	225	93	$2.24 $0.91 $0.39	$7.05
	Shampoos	4,676,731	0.0474	2,930	93	$2.95 $1.79 $0.27 $29.99
	Snack Crackers	3,509,158	0.0356	420	93	$2.18 $0.57 $0.10	$8.00
	Soaps	1,834,040	0.0186	334	93	$2.51 $1.48 $0.10 $10.99
	Soft Drinks	10,547,266	0.1069	1,608	93	$2.34 $1.89 $0.10 $26.02
	Toothbrushes	1,852,487	0.0188	491	93	$2.18 $0.85 $0.39	$9.99
	Toothpastes	2,997,748	0.0304	608	93	$2.43 $0.89 $0.31 $10.99
	Total	98,691,750	1.0000	18,037	93
	Notes:
	1. The data are weekly.
	2. The figures in the table are based on all price data of Dominick's in its 93 stores for 400 weeks from September 14, 1989 to
	May 8, 1997.
	3.

Table 2 .

 2 Asymmetry thresholds in cents based on PPI-measure of price level

	Category	Full Sample	Low-Inflation Sample	Deflation Sample
	Analgesics	30	10
	Bath Soap	6	0
	Bathroom Tissues	6	4
	Bottled Juices	12	15
	Canned Soup	12	12
	Canned Tuna	1	2
	Cereals	29	24
	Cheeses	9	9
	Cookies	11	11
	Crackers	10	2
	Dish Detergent	5	4
	Fabric Softeners	5	11
	Front-end-candies	5	5
	Frozen Dinners	2	10
	Frozen Entrees	20	22
	Frozen Juices	9	9
	Grooming Products	20	12
	Laundry Detergents	16	13
	Oatmeal	25	2
	Paper Towels	2	2
	Refrigerated Juices	15	9
	Shampoos	0	10
	Snack Crackers	11	2
	Soaps	1	1
	Soft Drinks	5	3
	Tooth Brushes	20	3
	Tooth Pastes	18	14
	Average	11.3	8.2	6.2

Notes:

Table 3 .

 3 Relationship between asymmetry and inflation, asymmetry thresholds in cents Chicago 4 week lag; M-CPI-Chicago 8 week lag; N -CPI-Chicago 12 week lag; O -CPI-Chicago 16 week lag; 6. P -Products for which the first 4 week prices are greater than or equal to the last 4 week prices; 7. Q -First 12 months of the sample period; R -Last 12 months of the sample period. 8. The figures in parentheses indicate a reverse asymmetry. A zero-entry means that there is no asymmetry.

			Asymmetry and Aggregate Inflation		Asymmetry and Disaggregate Inflation
		1990 1991 1992 1993 1994 1995 1996	WI	MI1	AI1	MI2	AI2
	Categories	A	B	C	D	E	F	G	H	I	J	K	L
	Analgesics	(1)	7	8	3	0	8	3	7	4	(1)	12	0
	Bath Soap	-	-	0	(1)	0	0	(1)	0	(1)	0	0	(1)
	Bathroom Tissues	3	1	1	4	6	9	5	12	6	6	12	12
	Bottled Juices	15	0	4	7	5	1	18	27	33	29	28	39
	Canned Soup	0	12	0	10	11	8	9	18	19	2	19	18
	Canned Tuna	1	1	2	2	1	0	2	11	7	7	6	10
	Cereals	4	24	0	25	19	1	12	4	4	2	2	10
	Cheeses	(1)	5	1	9	2	2	23	11	18	12	18	12
	Cookies	4	(1)	4	8	14	3	10	6	3	10	2	2
	Crackers	1	2	1	2	4	1	10	1	12	10	11	2
	Dish Detergent	(3)	2	2	10	4	2	11	2	6	2	2	2
	Fabric Softeners	0	5	11	5	1	1	1	20	4	10	20	20
	Front-end-candies	(1)	1	1	15	0	1	10	5	2	9	9	9
	Frozen Dinners	-	-	9	4	1	1	1	6	5	5	1	1
	Frozen Entrees	(1)	0	10	10	(1)	1	20	2	16	16	8	14
	Frozen Juices	0	(2)	2	3	9	9	9	10	12	3	12	1
	Grooming Prod.	-	-	12	20	5	1	16	2	3	3	2	3
	Laundry Detergent	(4)	3	2	9	1	1	2	2	1	2	2	1
	Oatmeal	-	5	12	4	1	2	9	9	13	6	19	16
	Paper Towels	1	0	1	1	2	9	1	3	3	13	4	5
	Refrigerated Juices	0	4	2	8	3	9	25	8	(1)	8	(1)	8
	Shampoos	-	-	6	20	2	(1)	(1)	11	2	2	2	2
	Snack Crackers	(2)	0	2	2	1	12	9	1	1	1	1	1
	Soaps	-	-	4	6	1	1	1	0	8	8	3	8
	Soft Drinks	1	(1)	(1)	5	3	4	13	1	1	1	1	1
	Tooth Brushes	(1)	8	8	(1)	3	7	1	20	18	20	16	18
	Tooth Pastes	1	7	0	6	2	12	(1)	9	6	6	6	6
	Average	0.8	3.8	3.9	7.3	3.7	3.9	8.1	7.70	7.59	7.11	8.04	8.15
	Notes:

F -CPI without lags; G -CPI 4 week lag; H -CPI 8 week lag; I -CPI 12 week lag; J -CPI 16 week lag; 5. K -CPI-Chicago without lags; L -CPI-

Asymmetric price adjustment has been studied for gasoline (e.g.,Davis and Hamilton,

2004), fruit and vegetables (e.g.,[START_REF] Ward | Asymmetry in retail, wholesale and shipping point pricing for fresh vegetables[END_REF], banking (e.g.,[START_REF] Hannan | The rigidity of prices: evidence from the banking industry[END_REF], processed food (e.g.,[START_REF] Ray | Asymmetric wholesale pricing: theory and evidence[END_REF]), manufacturing (e.g., Blinder, et al, 1998), and across a broad range of consumer product markets (e.g.,[START_REF] Peltzman | Prices rise faster than they fall[END_REF][START_REF] Müller | Asymmetric price adjustment: evidence from weekly product-level scanner price data[END_REF]).[START_REF]I -Monthly Index 1[END_REF] The findings for two categories, beer and cigarettes, are not discussed because the products included in these categories are highly regulated(Besley and Rosen, 1999, footnote 6). Their plots, however, are included in the supplementary appendix. See[START_REF] Barsky | What can the price gap between branded and private label products tell us about markups[END_REF] and[START_REF] Chevalier | Why don't prices rise during periods of peak demand? Evidence from scanner data[END_REF] for more details about the data.3 If the item was on sale or if the retailer's coupon was used, then the data reflect that. The prices are set on a chainwide basis but there is some variation across the stores. The analyses discussed in this paper, use the data available from all stores.

The average prices during the first and the last 4-weeks of the sample were compared. An 8-week window yielded similar results. In this comparison, the list prices are used in order to avoid any effect of sales on the results. In the asymmetry analysis, however, the actual prices are used to make the current results comparable with the previous results.

If one considers a broader notion of price adjustment costs including managerial costs, then price adjustment costs could lead to asymmetry: the cost of price increase could be higher than the cost of price decrease. The reason might be consumer anger or fairness[START_REF] Rotemberg | Monopolistic price adjustment and aggregate output[END_REF][START_REF] Rotemberg | Customer anger at price increases, changes in the frequency of price adjustment and monetary policy[END_REF][START_REF] Kahneman | Fairness as a constraint on profit seeking: entitlements in the market[END_REF], consumer goodwill loss[START_REF] Okun | Prices and Quantities: A Macroeconomic Analysis[END_REF][START_REF] Kackmeister | Yesterday's bad times are today's good old times: retail price changes are more frequent today than in the 1890s[END_REF][START_REF] Levy | The real thing: nominal price rigidity of the nickel Coke, 1886-1959[END_REF], or search triggered by a price increase. This, however, predicts more price decreases than increases.[START_REF] Rotemberg | Monopolistic price adjustment and aggregate output[END_REF] cost of price adjustment model implicitly assumes that consumers may prefer a series of small price changes to a single large price change. This idea could be consistent with our findings. However, that would depend on the size of menu costs: if the menu cost is too large, then a single large price change may be chosen over a series of several small price changes[START_REF] Mankiw | Small menu costs and large business cycles: a macroeconomic model of monopoly[END_REF][START_REF] Blinder | Asking about Prices: A New Approach to Understanding Price Stickiness[END_REF]).

See, for example, Ball, Mankiw and Reis (2005),[START_REF] Adam | Optimal monetary policy with imperfect common knowledge[END_REF],[START_REF] Mankiw | Sticky decisions vs. sticky prices: a proposal to replace the New Keynesian Phillips Curve[END_REF],[START_REF] Sims | Implication of rational inattention[END_REF],Reis (2006aReis ( , 2006b)),[START_REF] Woodford | Imperfect common knowledge and the effects of monetary policy[END_REF][START_REF] Shugan | The cost of thinking[END_REF].

Another explanation might be asymmetry in small shocks[START_REF] Ball | Asymmetric price adjustment and economic fluctuations[END_REF]. Prices may be reacting differently to shocks of different magnitudes, and in a world without inflation, asymmetric distribution of small shocks could lead to asymmetric price adjustment in the small. We thank the anonymous for pointing this out.

A recent news report offers anecdotal evidence: "The cost of General Mills cereals such as Wheaties, Cheerios, and Total is increasing an average of 2%. The price jump averages out to roughly 6 or 7¢ a box for cereals such as Chex, Total Raisin Bran ... which typically cost around $3 in the Minneapolis area, ... John French, 30, doubted he would even notice the higher prices for cereal on his next grocery trip. 'A few cents? Naw, that's no big deal,' said French, of Plymouth, MN" (our emphasis). Source: c."

Daniel Levy gratefully acknowledges the financial support from the Adar Foundation of the Economics Department at Bar-Ilan University.

Supplementary Appendix

1. Table R1 reports the results of five robustness tests, as discussed in the paper, in section 4.

2.

In Figure R1 we present the cross-category average frequency of positive and negative price changes in cents for the low/zero-inflation period sample.

3. In Figures R1.1a-R1.1c we present the frequency of positive and negative price changes in cents by categories for the low/zero-inflation period sample.

4. In Figure R2 we present the cross-category average frequency of positive and negative price changes in cents for the deflation period sample.

5. In Figures R2.1a-R2.1c we present the frequency of positive and negative price changes in cents by categories for the deflation period sample.


Matthew J Higgins 

Andrew T Young

Daniel Levy

Matthew J Higgins
email: matt.higgins@mgt.gatech.edu.

Federal, State, and Local Governments: Evaluating their Separate Roles in US Growth 

Keywords: O40, O11, O18, O51, R11, H50, and H70 Economic Growth, Federal Government, State

We use US county level data (3,058 observations) from 1970 to 1998 to explore the relationship between economic growth and the extent of government employment at three levels: federal, state and local. We find that increases in federal, state and local government employments are all negatively associated with economic growth. We find no evidence that government is more efficient at more decentralized levels. While we cannot separate out the productive and redistributive services of government, we document that the county-level income distribution became slightly wider from 1970 to 1998. For those who justify government activities in terms of equity concerns -perhaps even trading off economic growth for equity -the burden falls on them to show that the income distribution would have widened more in the absence of government activities. We conclude that a release of government-employed labor inputs to the private sector would be growth-enhancing.

Introduction

Does government contribute to or hinder economic growth? This question is of considerable interest to economists and is the subject of a large empirical literature.

However, government is a very broad concept. A more appropriate and tractable question is: what kinds of government contribute to or hinder economic growth? To address this question, we revisit and extend a number of results from Higgins et al.'s (2006) study of US growth determination. In that paper we use county-level data to assess the roles of federal, state, and local government, separately, in growth determination from 1970 to 1998. The analysis focuses on levels of centralization as the relevant kinds of government. Nearly all of the evidence that we present in [START_REF] Higgins | Growth and Convergence across the US: Evidence from County-Level Data[END_REF] suggests that government, at any level, hinders economic growth. Furthermore, there is no evidence that government's negative effect on growth diminishes at more decentralized levels.

In this paper we report how the basic [START_REF] Higgins | Growth and Convergence across the US: Evidence from County-Level Data[END_REF] findings are largely robust to examining (a) subsamples of metro and non-metro counties separately, (b) regional subsamples, and (c) individual US state subsamples. We also (d) employ a more careful selection of valid instruments for our regressions. The basic findings are again robust. We also include documentation that (e) the county-level income distribution widened slightly over the 1970 to 1998 period.

Our 2006 paper primarily focuses on conditional convergence rates in the US.

Determinants of balanced growth paths are given second-order treatment in terms of detail and space. This paper allows us to highlight findings associated with the government employment variable subset of those growth determinants. We find that subset of particular interest and the elaboration of our original findings along the margins of (a) through (e) provides, we believe, a stronger and more thorough argument that government activities are growth-hindering in the US.

The paper is organized as follows. Section 2 elaborates on the questions of interest in terms of the existing literature and how we add to that literature. Section 3 then briefly discusses the econometric method and data that we employ. Section 4 discusses the empirical findings. Concluding remarks are made in Section 5.

2.

Growth and Government: The Existing Literature

When asking about kinds of US government there are two complementary (and not mutually exclusive) categorizations of interest. One categorization is kinds of government services and inputs to production and has received considerable attention in the literature. A seminal example is [START_REF] Aschauer | Is Public Expenditure Productive?[END_REF] who analyzes government military and non-military investment, finding that only non-military government investment is positively associated with state-level economic growth. Alternatively, [START_REF] Holtz-Eakin | Public-Sector Capital and the Productivity Puzzle[END_REF] finds that government investment has a negative effect or no effect on gross state product (GSP) growth. [START_REF] Evans | Are Government Activities Productive? Evidence from a Panel of US States[END_REF] confirm the Holtz-Eakin result even when considering investments in highways, water and sewer capital, and other infrastructure capital separately.1 However, Evans and Karras also find that increases in government educational services are positively related to growth.2 A more recent paper by [START_REF] Shioji | Public Capital and Economic Growth: A Convergence Approach[END_REF] revisits the effects of different types of government investment. He finds that the infrastructure component has a positive effect on growth. 3 The categorization we explore -namely kinds of government in terms of the level of decentralization -has not received as much attention in the literature. In a sample of 58 countries, an early paper by [START_REF] Oates | Fiscal Federalism[END_REF] finds that measures of fiscal decentralization (henceforth simply decentralization) positively correlate with real income per capita. 4 Davoodi and Zou (1998), on the other hand, find that decentralization is negatively associated with growth in developing countries, and not associated with growth in developed countries.

The literature is especially sparse in regards to the US economy. 5 Three notable exceptions are [START_REF] Xie | Fiscal Decentralization and Economic Growth in the United States[END_REF] who fail to find a relationship between decentralization and per capita income growth from 1948 to 1994 and [START_REF] Akai | Fiscal Decentralization Contributes to Economic Growth[END_REF] and [START_REF] Stansel | Local Decentralization and Local Economic Growth: A Cross-Sectional Examination of US Metropolitan Areas[END_REF] who find positive relationships between decentralization and growth. 6 Xie et al.

(1999) focus on the share of local and state government in aggregate government expenditures. [START_REF] Akai | Fiscal Decentralization Contributes to Economic Growth[END_REF] analyze expenditure and revenue shares of federal, state and local governments in state-level cross-section data. Finally, [START_REF] Stansel | Local Decentralization and Local Economic Growth: A Cross-Sectional Examination of US Metropolitan Areas[END_REF] 3 Different kinds of government services and inputs to production are also a common focus of crosscountry studies. Examples include [START_REF] Atkinson | The Welfare State and Economic Performance[END_REF], [START_REF] Slemrod | What Do Cross-Country Studies Teach Us about Government Involvement, Prosperity, and Economic Growth[END_REF], [START_REF] Agel | Growth and the Public Sector: A Critical Review Essay[END_REF]), Sala-i-Martin (1997aand 1997b[START_REF] Baldacci | Growth, Governance, and Fiscal Policy Transmission Channels in Low-Income Countries[END_REF]. 4 Despite [START_REF] Oates | Fiscal Federalism[END_REF] empirical finding, the relationship between economic growth and fiscal decentralization in general cannot be directly linked to the more specific "decentralization theorem" put forth by Oates in the same work. The decentralization theorem concerns the welfare advantages of localized public goods provision in the absence of "cost-savings from centralized provision" and "interjurisdictional externalities" (p. 72). The theorem only establishes "a presumption in favor of the decentralized provision of public goods with localized effects" [START_REF] Oates | An Essay on Fiscal Federalism[END_REF](Oates, , p. 1122)). Difficult questions involve, first, whether this implies a presumption that decentralization positively correlates with growth and, second, whether the assumptions underlying that presumption actually hold in a given economy. 5 Studies specific to countries other than the US include Carrion-i-Silvestre et al. (2008) for Spain and Zhang and Zhou (1998) for China. 6 For succinct overview of fiscal decentralization, see the introduction section of [START_REF] Brueckner | Fiscal Federalism and Economic Growth[END_REF]. See also, [START_REF] Hillman | Public Finance and Public Policy: Responsibilities and Limitations of Government[END_REF]. [START_REF] Martinez-Vazquez | Fiscal Decentralization and Economic Growth[END_REF] provides an insightful view of both the theoretical and empirical literature on fiscal decentralization.

examines the central city share of population, per capita municipalities, and per capita counties.

Our analysis complements the above studies. We focus on federal, state, and local government employment as a percent of a county's population. Our analysis of employment shares is novel in terms of both (i) the underlying concept of the public sector (or the dimensions of the public sector measured) and (ii) the economic unit of analysis (i.e., a county). Employment shares come from the county-level data that [START_REF] Higgins | Growth and Convergence across the US: Evidence from County-Level Data[END_REF][START_REF] Higgins | Robust Correlates of US County-Level Growth[END_REF] and [START_REF] Young | Sigma Convergence versus Beta Convergence[END_REF] exploit to study convergence in the US. In contrast to the 48-50 state-level observations, or the 314 metropolitan areas studied by [START_REF] Stansel | Local Decentralization and Local Economic Growth: A Cross-Sectional Examination of US Metropolitan Areas[END_REF], our data contain over 3,000 county-level observations.

Employment shares may provide a better measure of many government activities than expenditure shares do. They capture percentages of available labor input that are being directed by a specific level of the public sector (rather than by the decisions of private individuals and firms). To the extent that economically important government activities are labor-intensive, analyzing the effects of employment shares is of interest.

For example, regulatory environments are presumably inherently labor-intensive in that they are not effective without investigative monitoring and enforcement on the ground.7

Moreover, the large number of cross-sectional observations allows us to study the US as a whole as well as various sub-samples of interest. Metro and non-metro subsamples allow for the possibility that government's effect on economic growth varies with population density and provides a link to the metro area study of [START_REF] Stansel | Local Decentralization and Local Economic Growth: A Cross-Sectional Examination of US Metropolitan Areas[END_REF].

For instance, a higher population density may lead to negative externalities that the public sector is uniquely suited to deal with. Furthermore, regional and individual state samples (the later of which [START_REF] Higgins | Growth and Convergence across the US: Evidence from County-Level Data[END_REF]2008 Our analysis incorporates 38 variables (as well as state-dummies for the nationwide samples) on which to condition the growth rates of per capita income. These variables control for the effects of various levels of educational attainment, age, racial demographics, and industry composition.

Following [START_REF] Higgins | Growth and Convergence across the US: Evidence from County-Level Data[END_REF], we use a cross-sectional variant of the 3SLS-IV approach of [START_REF] Evans | Consistent Estimation of Growth Regressions[END_REF]. However, we improve upon [START_REF] Higgins | Growth and Convergence across the US: Evidence from County-Level Data[END_REF] in terms of the instruments used. [START_REF] Higgins | Growth and Convergence across the US: Evidence from County-Level Data[END_REF] employed lagged values of all conditioning variables as instruments. We employ an algorithm that chooses from those lagged values a set of instruments that (i) are valid and (ii) have the highest Shea partial R 2 .

Empirical specification and data

In this section we briefly describe the basic growth regression model and the estimation technique that we apply. We then briefly describe the US county-level data. Finally, since our estimation technique employs instrumental variables, we describe the algorithm that we use to optimally choose a set of instruments conditional on the available data.

The regression model and 3SLS-IV estimation procedure

Our growth regressions involve fitting cross-sectional data to the equation, ( 1)

n n n n x y g ν γ β α + ′ + + = 0 ,
where n g is the average growth rate of per capita income for economy n between years 0 and T; y n0 is initial (t = 0) per capita income; n x is a vector of other control variables including government employment shares; β and γ are coefficients; and ν n is a zero mean, finite variance error term. 8 Given our data, 0 ≡ 1970 and T ≡ 1998. 9

Caselli , et al. (1996) and [START_REF] Evans | Consistent Estimation of Growth Regressions[END_REF] show that applying OLS to (1) will yield consistent estimates only if the data satisfy highly implausible conditions. 10 Evans (1997) proposes a 3SLS-IV method that produces consistent estimates. In the first two stages, instrumental variables are used to estimate the regression equation,

(2)

n n n y g η β ω + ∆ + = ∆ 0 , where T y y T y y g n T n n T n n ) ( ) ( 1 , 1 , 0 , , - -- - - = ∆ , 1 , 0 0 - - = ∆ n n n y y y , n y , ω is a parameter,
and n η is the error. 11 ,12 8 For a derivation of this regression from a neoclassical growth model, see [START_REF] Barro | Convergence[END_REF].

For exposition of this model in greater detail see [START_REF] Higgins | Growth and Convergence across the US: Evidence from County-Level Data[END_REF]. 9 Though the period of growth under consideration is 28 years, the analysis is essentially cross-sectional because average growth over those 28 years is the single dependent variable. 10 For a list of these conditions see [START_REF] Higgins | Growth and Convergence across the US: Evidence from County-Level Data[END_REF] and [START_REF] Evans | Consistent Estimation of Growth Regressions[END_REF].

11 An immediate concern with the use of ( 2) is the reliance on the information from the single difference in the level of income (1969 to 1970) to explain the difference in average growth rates over overlapping time periods (1969 to 1997 and 1970 to 1998). Given that the growth determination is a stochastic process, the potential problem is basically one of a high noise to signal ratio. We are relying on large degrees of Next, we use * β , the estimate from (2), to construct the variable,

(3)

0 * n n n y g β π - = .
In the third stage, we perform an OLS regression of n π on n

x :

(4)

n n n x ε γ τ π + + = ,
where τ and γ are parameters and ε n is the error term. This regression yields a consistent estimator, γ * .

US county-level data

The data we use are drawn from several sources but the majority comes from the Bureau of Economic Analysis Regional Economic Information System (BEA-REIS) and US Census data sets. The BEA-REIS data are largely based on the 1970, 1980and 1990 decennial Census files; decennial Census files;[START_REF] Oates | Fiscal Federalism[END_REF][START_REF] Buchanan | Democracy in Deficit: The Political Legacy of Lord Keynes[END_REF], 1982and 1987 Census of Governments; and the Census Bureau's City and County Book from various years. We exclude military personnel from all data.

Our data contain 3,058 county-level observations. The large number of observations allows us to explore possible heterogeneity in growth across the US by splitting the data into three sets of subsamples. The first set includes a metro subsample freedom to alleviate this problem and identify coefficients. Indeed, we obtain statistically significant β estimates for the full sample as well as for 32 individual states. 12 As [START_REF] Evans | Consistent Estimation of Growth Regressions[END_REF] shows, the derivation of this equation depends on the assumption that the x n variables are constant during the time frame considered, allowing them to be differenced out. Since the difference is only over a single year and these are variables representing broad demographic trends, this does not seem to be unwarranted. To make sure that this did not introduce significant omitted variable bias into our estimations, we ran the IV regression for the full US sample with differenced values of all conditioning variables included as regressors. The point estimate of β from the modified IV fell within the 95 percent confidence interval of the Evans method IV estimate. If β estimates are not significantly affected then neither are the third stage results (see below).

(867 counties) and a non-metro subsample (2,191 counties). 13 The second set includes 5 regional subsamples: Great Lakes, Northeast, Southern, Plains, and Western. Finally, the third set includes 50 US state subsamples.

We government employments as a percents of a county's population.

Choice of Instruments

Higgins et al. ( 2006) used lagged values of conditioning variables as instruments for ∆y n,0 in equation (2). A basic criticism of that paper is that no evidence was presented of those instruments' validity. It is well-known that a valid set of instruments includes variables that are correlated with the endogenous regressors (instrument relevance) but uncorrelated with the error term of the regression (instrument orthogonality). For a given set of instruments, an F-statistic test can be used to assess the former while overidentification tests can be used to assess the latter.

In the case of a single endogenous regressor, such as the case with ∆y n,0 in equation ( 2), [START_REF] Staiger | Instrumental Variables Regression with Weak Instruments[END_REF] have suggested declaring instruments to be weak (not relevant) if the first-stage F-statistic is less than ten [START_REF] Stock | Testing For Weak Instruments in Linear Regression[END_REF]. This "rule of thumb" for testing instrument relevance is supported by [START_REF] Baum | Instrumental Variables GMM: Estimation and Testing[END_REF]. In regard to instrument orthogonality, [START_REF] Deaton | The Analysis of Household Surveys: Microeconomic Approach to Development Policy[END_REF] and [START_REF] Stock | Testing For Weak Instruments in Linear Regression[END_REF] point out that when several instruments are used, overidentification tests can be applied. In the present case we use [START_REF] Sargan | Wages and Prices in the United Kingdom: a Study in Econometric Methodology[END_REF] overidentification test to assess whether the instruments are correlated with the error process. 15Given the size of our sample and number of independent variables we have a large number of potential instruments. In order to generate a set of valid instruments (i.e, those that address instrument relevance and orthogonality) we apply an algorithm that (i) selects various combinations of available (lagged) conditioning variables such that the system is overidentified, (ii) tests their validity, and (iii) if found valid, uses them in the regression analysis. Our algorithm does not maximize any particular function but rather aids in sifting through all various combinations of valid instruments (i.e., those where the first-stage F-statistic is greater than 10 and that satisfy the Sargan test). From this subset of valid instruments we selected the set that produced the largest first-stage [START_REF] Shea | Instrument Relevance in Multivariate Linear Models: A Simple Measure[END_REF] partial R 2 , 0.3510. Our final set of instruments consists of the following 13 lagged variables: "Age: 18-64 years", "Education: Bachelor +", "Federal government employment", "State government employment", "Local government employment", "Selfemployment", "Agriculture", "Construction", "Manufacturing: durables", "Manufacturing: non-durables", "Health services", "Personal services", and "Entertainment & Recreational Services".16

Empirical findings

We begin by presenting our empirical estimates of the effect of federal, state, and local government employment (as a percent of a county's population), separately, on US economic growth at the county-level. Estimates are presented for (a) country-wide samples, including metro and non-metro subsamples, (b) regional subsamples, and (c) individual state subsamples. We also informally consider the redistributive role of government which may account for the lack of positive growth effects of government employment.

Full US sample results

Table 2 presents the estimation results for the full US sample, as well as metro and non-metro samples. We find a negative and statistically significant partial correlation between the percent of the population employed by government and the rate of economic growth, regardless of whether one considers federal, state or local government. The coefficients for federal, state and local government employment percent of the population variables are -0.0226, -0.0153, and -0.0219 respectively, all significant at the 1-percent level. There is no clear pattern in the point estimates as the level of decentralization increases, and the 95 percent confidence intervals of the point estimates are overlapping.17

The negative effects are also present at all levels of government employment whether one considers only metro counties or only non-metro counties as indicated by the figures in Table 2. This suggests that government employment is not positively related to economic growth at higher population densities. Indeed the negative partial correlations are larger for the metro sample than for the non-metro sample at all levels.

Given the apparent robustness of a negative association of government employment with growth, one interpretation is that federal, state and local government activities misallocate resources into relatively (compared to the private sector) inferior production processes. Government lowers total factor productivity, thus lowering balanced growth paths. However, another interpretation may be entertained. Perhaps nongovernment wage growth outpaces government wage growth, and this drives the results. 18 These interpretations are not incompatible in a world where both government and non-government wages approximated marginal products. However, government employment, not being subject to the same market forces, may be different along certain intangible margins (e.g., greater job security; lower expectations of effort; higher psychic benefits associated with providing a public service).

In order to explore the above we have assembled government and nongovernment wage growth data for the sample period (Table 3). At the state and federal level, non-government wages outpace government wages in approximately 55 percent (a small majority) of all counties. At the local level, non-government wages grew faster in only about 30 percent of counties. Relative sluggishness of government wages at the state and federal levels is dominated by wage growth rates in the metro counties. For the nonpopulation; the total effect is negative until it employs over 60 percent! (In the data, however, only 9 out of 3,058 counties have a federal government employment share in excess of 30 percent.) 18 We thank Paul Rubin for bringing this possibility to our attention.

metro counties, which constitute a vast majority of 2,196 counties panel, non-government wages outpaced government wages in just over 50 percent of cases. 19 If a relatively sluggish growth of government wages story were important, then we would expect to find smaller estimated coefficients for metro counties than for nonmetro counties. This we do see. The coefficient estimates for the regression including only metro counties are -0.0300, -0.0264 and -0.0214 for federal, state and local governments respectively. For non-metro counties the corresponding estimates are -0.0179, -0.0081 and -0.0128. Note, however, that this pattern holds for the local government coefficients as well, despite local wages outpacing non-government wages in a majority of counties. At least at the most decentralized level of government, a relatively sluggish government wage growth story is unable to account for the negative partial correlation. Indeed, we find a negative relationship despite the relatively fast growth of government wages.

Regional Results and Individual State Results

Table 4 reports the results for our five regional sub-samples: Great Lakes, Northeastern, Southern, Plains and Western. Statistically significant results are harder to come by using these smaller samples, but those significant coefficient estimates are broadly consistent with our full sample findings. The significant coefficient estimates are all negative except for those associated with the non-metro Plains region. For the nonmetro Plains region, notably, positive effects are estimated for all levels of government.

The statistically significant coefficient estimates that are negative occur, variously, for 19 These results are based on data from 3,066 counties. Our regression analysis utilizes data for 3,058 counties. The difference is due to a lack of some data in eight counties.

federal, state, and local government. 20 (Of course, this could simply imply that government activity productivity growth was relatively high.)

Redistributive Activities

It is possible that the negative correlation between government employment and 20 The non-metro Plains region effects may be associated with relatively fast government wage growth. Federal, state, and local government wage growth, respectively, outpaced non-government wage growth in approximately 62 percent, 68 percent, and 80.32 percent of counties. 21 Given the extensive number of independent variables, our model was only identified in 32 of 50 states. By this we refer to the coefficient on initial income being statistically significant. In the absence of this the interpretation of conditioning variable coefficients is unclear.

economic growth is due to the redistributive activities of government. Redistributive activities may be detrimental towards average (per capita) income growth though effective in their specific goals. Furthermore, if greater income equality is valued in and of itself, then trading off growth for equity may be viewed as desirable.

Our analysis cannot separate out the productive and redistributive activities of federal, state, or local governments. However, under the hypothesis that the negative correlation between government employment and economic growth is due to redistributive activities, we can ask whether or not the economic growth has indeed been traded off for a more equitable distribution of income. In other words, we can evaluate a possible redistributive role for the various levels of government given the caveat that we can not control for how the distribution of income might have evolved in the absence of government. (Clearly this caveat is an important one.) [START_REF] Young | Sigma Convergence versus Beta Convergence[END_REF] provide evidence on this issue; here we briefly summarize their results. Figure 1 displays the 1970 and 1998 distributions of per capita income across US states. The distribution became wider. Also, the Gini coefficients associated with US counties ' 1970 and 1998 (log) incomes are 0.0167 and 0.0165 respectively -a decrease of about 1.2 percent. 22 At the county-level, although the dispersion of US per capita income widened from 1970 to 1998, it became a bit more equal. 23 However, changes in both the standard deviation and the Gini coefficient are small enough to suggest that changes in both dispersion and equality are negligible.

To try to understand further the evolution of the US county-level income distribution, Table 6 summarizes statistics computed from the 1970 and 1998 income distributions. From 1970 to 1998, the skewness of the distribution increased from -0.2244 (to the left) to 1.7240 (to the right). At the same time, kurtosis increased from 3.4334 to 10.3237, implying that the distribution has become more peaked. Cumulatively, this suggests that these two effects have been offsetting to a great extent.

In conclusion, the evidence presented in [START_REF] Young | Sigma Convergence versus Beta Convergence[END_REF] does not suggest that government activities have achieved (absolutely) a more equal distribution of income at the expense of lower rates of economic growth. Put differently, it suggests government failure at federal, state, and local levels in lieu of evidence that the distribution of countylevel per capita income would have widened significantly in the absence of government activities.

Concluding Remarks

We use US county data containing over 3,000 cross-sectional observations during the period 1970 to 1998 to explore the relationships between economic growth and the size and scope of government at three levels: federal, state and local. In contrast to the extant literature that has used taxes, government expenditures and government capital stocks to proxy for the extent of government, here we focus on government employment shares.

Following [START_REF] Higgins | Growth and Convergence across the US: Evidence from County-Level Data[END_REF], we use a 3SLS-IV estimation technique and report on the full sample of US counties, as well as metro and non-metro subsamples, and 5 regional and 32 individual state subsamples. Federal, state and local government employment shares almost always negatively correlate with growth (when statistical significance holds) across samples.

We find no evidence that government is more effective at more or less decentralized levels. Furthermore, while we cannot separate out the productive and redistributive services of government, we document that income inequality in the US has widened slightly from 1970 to1998. However, the change in both the standard deviation of per capita income and the Gini coefficient are small enough to suggest that both dispersion and equality remained essentially the same.

Given the ostensible goals of higher growth and/or a more equitable distribution of income, our findings are not supportive of expanding the roles of government, in terms of employment, at any level. While government is often invoked as a solution to market failures, the evidence points to government failure as a more important, and negative, influence in the economy. Our findings suggest a rolling back of government activities;

specifically for a release of government-employed labor inputs to the private sector.

The most important and obvious caveat to our findings is that we cannot disaggregate government employment into specific government activities, e.g. infrastructure development. Despite examining federal, state, and local government separately, we must view each level of government as a given package of activities and services. Still, our results are consistent with the conclusion that, as currently composed, a shrinking of government at any and all levels of decentralization would be a desireable policy. Significance levels are indicated by "a" for 1 percent, "b" for 5 percent, and "c" for 10 percent.

TABLES

TABLES (Cont.)

Fig. 1

 1 Fig. 1 Distribution of US county (log) per-capita incomes, 1970 and 1998

  use the BEA's measure of personal income net of government transfers expressed in 1992 dollars. Natural logs of real per capita income are used. The 38 conditioning variables are the same as in[START_REF] Higgins | Growth and Convergence across the US: Evidence from County-Level Data[END_REF] see Table 1) and account for various age and racial demographics, levels of educational attainment, and industry employment composition.

14

For the variables of interest to the present analysis, Table

1

reports some summary statistics for the various samples of US counties. The variables are average per capita income growth from 1970 to 1998 and federal, state, and local

Table 5

 5 reports the estimates for 32 of 50 individual US state sub-samples.21 Again, these results are broadly consistent with the conclusion that government employment, at all levels, is negatively correlated with economic growth. Almost all statistically significant effects are negative. The notable exception is North Dakota. North

	Dakota has large, statistically significant positive coefficients on federal and local
	government employment.
	It is not immediately apparent what might be peculiar about North Dakota. For
	example, in 1970, North Dakota ranked 21 st highest in terms of federal government
	employment (1.3 percent); 7 th in terms of state government employment (2.3 percent);
	and 4

th in terms of local government employment (4.7 percent). However, similar to the Plains region non-metro case, federal, state, and local government wage growth was higher than non-government wage growth in, respectively, about 57 percent, 75 percent, and 85 percent of counties. This exceptional government relative wage growth may account for the estimated positive effect of government employment on income growth.

Table 1

 1 Summary Statistics

	Samples	Counties Growth		Federal		State		Local
			Mean	σ	Mean	σ	Mean	σ	Mean	σ
	U.S. -Full	3058	0.0154	0.0067	0.0396	0.0402	0.0485	0.0470	0.0835	0.0289
	U.S. -Metro	867	0.0167	0.0062	0.0430	0.0429	0.0495	0.0508	0.0785	0.0233
	U.S. -Non-Metro	2192	0.0149	0.0068	0.0382	0.0389	0.0482	0.0455	0.0855	0.0306
	Great Lakes -Full	435	0.0148	0.0045	0.0265	0.0238	0.0404	0.0466	0.0782	0.0220
	Great Lakes -Metro	140	0.0160	0.0045	0.0235	0.0187	0.0349	0.0390	0.0697	0.0109
	Great Lakes -Non-Metro	295	0.0142	0.0044	0.0279	0.0259	0.0430	0.0496	0.0822	0.0247
	New England -Full	244	0.0159	0.0050	0.0365	0.0382	0.0513	0.0361	0.0759	0.0209
	New England -Metro	90	0.0166	0.0056	0.0302	0.0206	0.0381	0.0278	0.0793	0.0200
	New England -Non-Metro	154	0.0154	0.0046	0.0401	0.0451	0.0591	0.0381	0.0740	0.0212
	Plains -Full	832	0.0139	0.0075	0.0382	0.0388	0.0392	0.0457	0.0979	0.0318
	Plains -Metro	143	0.0171	0.0052	0.0469	0.0548	0.0447	0.0438	0.0857	0.0248
	Plains -Non-Metro	689	0.0133	0.0077	0.0363	0.0343	0.0380	0.0460	0.1004	0.0325
	Southern -Full	1009	0.0184	0.0053	0.0414	0.0398	0.0527	0.0435	0.0726	0.0220
	Southern -Metro	252	0.0200	0.0055	0.0486	0.0495	0.0515	0.0503	0.0709	0.0185
	Southern -Non-Metro	757	0.0179	0.0051	0.0390	0.0357	0.0530	0.0410	0.0732	0.0230
	Western -Full	538	0.0123	0.0075	0.0503	0.0499	0.0606	0.0558	0.0895	0.0325
	Western -Metro	242	0.0134	0.0065	0.0509	0.0396	0.0627	0.0631	0.0868	0.0284
	Western -Non-Metro	296	0.0114	0.0082	0.0498	0.0571	0.0588	0.0490	0.0916	0.0354
	"Growth" is per capita income growth from 1970 to 1998. Government variables are employment shares of total population. σ is the standard deviation

Table 2

 2 Estimated effects of government on growth: US-wide samples

	TABLES (Cont.)
	Sample	Variable	Coefficient
			(Standard error)
	Entire US	Federal government employment	-0.0252 a
			(0.0051)
		State government employment	-0.0153 a
			(0.0040)
		Local government employment	-0.0219 a
			(0.0052)
	Metro counties	Federal government employment	-0.0305 a
			(0.0108)
		State government employment	-0.0249 a
			(0.0076)
		Local government employment	-0.0240 a
			(0.0052)
	Non-metro counties Federal government employment	-0.0206 a
			(0.0060)
		State government employment	-0.0056
			(0.0048)
		Local government employment	-0.0145 a
			(0.0059)

Table 3

 3 Estimated effects of government on growth: US-wide samples

	Sample	Level of government	Percent of counties where
			government wages grew faster
			than non-government wages
	Entire US	Federal government employment	44.36
		State government employment	45.37
		Local government employment	70.35
	Metro counties	Federal government employment	34.14
		State government employment	38.85
		Local government employment	58.16
	Non-metro counties Federal government employment	48.41
		State government employment	47.95
		Local government employment	75.18

Non-government wages are the sum of private and farm wages.

Table 4

 4 Estimated effects of government on growth: US regional samples

	TABLES (Cont.)
	Sample	Variable	Coefficient
			(Stand. error)
			Great Lakes Northeast Plains	Southern Western
	Number of		435	244	832	1,009	538
	Counties
	Entire US	Federal	-0.0086	-0.0263	-0.0223 b	-0.0241 a	-0.0149
			(0.0159)	(0.0212)	(0.0101)	(0.0091)	(0.0109)
		State	-0.0211 b	-0.0118	0.0100	-0.0048	-0.0104
			(0.0100)	(0.0139)	(0.0075)	(0.0076)	(0.0097)
		Local	-0.0377 a	-0.0372	0.0031	0.0036	-0.0309 a
			(0.0135)	(0.0271)	(0.0084)	(0.0113)	(0.0119)
	Number of		140	90	143	252	242
	Counties
	Metro	Federal	0.0109	0.0195	-0.0121	-0.0392 b	-0.0117
			(0.0344)	(0.0345)	(0.0286)	(0.0211)	(0.0219)
		State	-0.0203	-0.0162	-0.0358 b	-0.0166	-0.0163
			(0.0214)	(0.0288)	(0.0154)	(0.0180)	(0.0158)
		Local	-0.0325	-0.1024 c	-0.0086	0.0067	-0.0391 c
			(0.0398)	(0.0571)	(0.0225)	(0.0353)	(0.0221)
	Number of		295	154	689	757	296
	Counties
	Non-metro	Federal	0.0055	-0.0286	0.0187 c	-0.0158	-0.0104
			(0.0201)	(0.0260)	(0.0102)	(0.0104)	(0.0147)
		State	-0.0238 b	0.0024	0.0161 c	-0.0035	-0.0066
			(0.0120)	(0.0158)	(0.0092)	(0.0086)	(0.0151)
		Local	-0.0356 b	-0.0453	0.0187 c	0.0096	-0.0335 c
			(0.0151)	(0.0345)	(0.0102)	(0.0118)	(0.0171)

Table 5

 5 Estimated effects of government on growth: individual US state samples

	State	Coefficient (std. err.)
		Federal	State	Local
	Alabama	0.0383 (0.0543)	-0.0130 (0.0482)	0.0377 (0.0536)
	Arkansas	-0.0153 (0.0499)	-0.0435 (0.0364)	-0.0385 (0.0443)
	California	0.0337 c (0.0292)	0.0275 (0.0399)	0.0334 (0.0651)
	Colorado	-0.0631 c (0.0358)	-0.0627 (0.0420)	-0.0224 (0.0361)
	Florida	0.0254 (0.0740)	0.0039 (0.0560)	-0.0217 (0.0868)
	Georgia	-0.0094 (0.0353)	0.0038 (0.0235)	0.0149 (0.0302)
	Idaho	-0.1187 c (0.0569)	0.0439 (0.0331)	0.0453 (0.0685)
	Illinois	0.0403 (0.0328)	-0.0101 (0.0189)	-0.0011 (0.0239)
	Indiana	-0.0695 (0.0498)	0.0025 (0.0231)	-0.0357 (0.0351)
	Iowa	0.0633 (0.0398)	-0.0479 b (0.0207)	0.0269 (0.0283)
	Kansas	0.0272 (0.0413)	0.0157 (0.0169)	0.0119 (0.0175)
	Kentucky	-0.0189 (0.0226)	-0.0037 (0.0200)	-0.0075 (0.0261)
	Louisiana	-0.0182 (0.0376)	-0.0207 (0.0353)	-0.1758 a (0.0633)
	Michigan	0.0714 (0.0492)	-0.0629 b (0.0270)	-0.0331 (0.0341)
	Minnesota	-0.0384 (0.0604)	-0.0102 (0.0232)	-0.0247 (0.0237)
	Mississippi	0.0060 (0.0630)	-0.0525 (0.0488)	-0.1657 a (0.0614)
	Missouri	-0.1032 a (0.0348)	-0.0779 a (0.0264)	-0.0194 (0.0324)
	Montana	-0.1149 (0.0718)	-0.0422 (0.0474)	-0.0161 (0.0478)
	New York	0.0942 (0.1194)	0.0518 (0.0439)	0.0077 (0.0669)
	North Carolina	-0.2145 a (0.0694)	-0.0621 b (0.0309)	-0.0165 (0.0682)
	North Dakota	0.2524 c (0.1416)	0.1183 (0.1023)	0.2537 b (0.1173)
	Ohio	-0.0233 (0.0569)	-0.0316 (0.0235)	-0.0205 (0.0470)
	Oklahoma	-0.1013 c (0.0529)	-0.0654 c (0.0352)	-0.0903 b (0.0370)
	Pennsylvania	0.0170 (0.0613)	0.0252 (0.0258)	-0.1347 b (0.0605)
	South Carolina	-0.0169 (0.0795)	-0.0692 (0.0957)	-0.0189 (0.1069)
	South Dakota	-0.0297 (0.0458)	0.0093 (0.0342)	-0.0410 (0.0439)
	Tennessee	-0.0039 (0.0360)	-0.0731 b (0.0361)	0.0072 (0.0390)
	Texas	-0.0442 (0.0301)	0.0035 (0.0174)	-0.0285 (0.0214)
	Virginia	-0.0161 (0.0411)	-0.0139 (0.0427)	0.0218 (0.0539)
	Washington	-0.0295 (0.0651)	-0.0662 (0.1226)	-0.1478 (0.2310)
	West Virginia	0.0858 (0.0717)	0.0146 (0.0354)	-0.0255 (0.0811)
	Wisconsin	0.0156 (0.0332)	-0.0199 (0.0311)	0.0466 (0.0348)

Table 6

 6 Summary statistics for the distribution of US county (log) per-capita income

	Statistic	1970	1998
	Standard deviation	0.2728	0.2887
	Gini Coefficient	0.1666	0.1654
	Skewness	-0.2244	1.7240
	Kurtosis	3.4334	10.3237
	FIGURES

[START_REF] Garcia-Milà | The Effect on Public Capital in State-Level Production Functions Reconsidered[END_REF] confirm these results.[START_REF] Richmond | Public Capital and Total Factor Productivity[END_REF], to the contrary, use a translog production function specification and find that government investment contributes to output growth while the government capital stock contributes to productivity growth.

On the other hand, changes in highway services, health and hospital services, police and fire services, and sewer and sanitation services do not have statistically significant effects.

There are, of course, also advantages to expenditure shares relative to employment shares. For government activities that are not labor intensive (e.g., industry subsidies) or for which associated labor is contracted for from the private sector (e.g., highway construction), expenditure shares may be a more appropriate measure.

Metro counties are defined as those that contain cities with populations of 100,000 or more, or border such counties[START_REF] Higgins | Growth and Convergence across the US: Evidence from County-Level Data[END_REF]

An appendix at the end of[START_REF] Higgins | Growth and Convergence across the US: Evidence from County-Level Data[END_REF] describes the data in more detail.

For Sargan's test see also[START_REF] Wooldridge | Econometric Analysis of Cross Section and Panel Data[END_REF].

All of these variables are in terms of percents of a county's population. SeeHiggins et al. (2006, Table 1) for full definitions of these variables and data sources.

The relationship may, however, be nonlinear. Government expansion may promote growth but only up to a point when further expansion is detrimental to growth at the margin; or it may be the case that government expansion aims to enhancing growth up to a point, and then expansion beyond that point aims at income redistribution[START_REF] Buchanan | Democracy in Deficit: The Political Legacy of Lord Keynes[END_REF]. To check this, we run the full US sample regression including both linear and quadratic government employment share terms. Only for the federal employment share is the quadratic term positive and significant, but the parameter estimates imply that the marginal effect is negative up until the federal government employs over 30 percent of a county's

The Gini coefficient is a number between 0 (perfect equality) and 1 (perfect inequality).

This statement is not to be confused with one concerning the distribution of US individuals' incomes.

Significance levels are indicated by "a" for 1 percent, "b" for 5 percent, and "c" for 10 percent.

Government, Local Government, and County-Level thank Jordan Rappaport for kindly sharing with us his data and computer codes from Rappaport (1999), and Arye Hillman and Avichai Snir for discussions. Daniel Levy gratefully acknowledges financial support from Adar Foundation of the Economics Department at Bar-Ilan University. Higgins gratefully acknowledges financial support from The Imlay Professorship.