Figure 1: Price Change Dispersion Across Items Within Product Modules
The height of each line reflects the gap between the 75th and 25th percentile of national average log price changes across goods in that module. The figure excludes modules in
the general merchandise department and uses only prices recorded in the Retail Scanner data. Each barcode is importance weighted by its share of total expenditures so that each category's interquartile range captures 50% of spending rather than 50% of barcodes. Product modules with interquartile
ranges below the 10th or above the 90th percentile excluded to improve visibility.
The figure contains four lines, each conveying the degree of price change dispersion in a given year. It does this for a collection of about 800 product modules, which are ordered along the x-axis in increasing order of dispersion. The height of each line is given by the interquartile range of national average percent price changes across individual goods in that module. Lines all slope upward from zero with a distinct inflection point and increasing steepness for the last 100 product modules with the greatest price change dispersion. For the year 2019, conveyed by a green dotted line, the line slopes gently from zero to about seven percentage poitns before accelerating to an endpoint of about 13 percent. The lines for 2021 (blue long dashes) and 2023 (yellow dashes of alternating length) are similar to each other and are uniformly above the 2019 line. They increase gradually from zero to about 10 percentage points at the 700th module and rapidly thereafter to about 17 percentage points for modules with the most price change dispersion. A solid red line for 2022 is considerably above all other lines. It slopes upward from zero to about 14 percentage points by the 700th module and reaches a final value of about 23 percentage points. Fewer than 200 product modules have interquartile ranges below 5 percentage points and the line reaches 10 percentage points at the 500th ranked module.
Figure 2: Spread of Module Inflation Rates
Boxes summarize the interquartile range of household four-quarter inflation rates in the given product modules. Whiskers extend to the remaining variation, with dots representing observations beyond 1.5x
the interquartile range. Observations outside of 5th-95th percentiles are excluded to improve visibility. For households buying only one variety within the module in the base period (Q-4), inflation is computed using that good's price change between Q-4 and Q. For a household buying multiple
varieties, this is the Q-4 expenditure-weighted average price change across these goods.
The figure displays a box and whisker plot grouped into three regions, each conveying the spread of household inflation rates in a detailed product module: coffee, frozen chicken, or men's shaving cream. Within each region are four pairs of boxes/whiskers, one pair for each of the years 2019, 2021, 2022, and 2023. The pair for each year contains a solid colored or latticed box conveying the interquartile range (IQR) of Laspeyres or Paasche indices, respectively. Thin orange lines convey the average inflation rate across households. Whiskers extend to observations 1.5x the IQR, with other points conveyed by dots. In all cases, the Laspeyres and Paasche indices have similar IQRs and overall spreads (the largest deviation being a Paasche index whose upper tail is 4 percentage points lower than the accompanying Laspeyres index for frozen chicken in 2022.) Categories' average inflation rates are about zero for all goods in 2019 but rose in 2022 to between 10 and 15 percent. Over that same time, the spread of inflation rates increased. The ranges for 2019 (shown in green) are very small: boxes have heights of 5 percentage points (p.p.) or less. The entire range is contained within a spread of about 10 p.p. for coffee to about 18 p.p. for men's shaving cream (excepting a few outliers with a high Laspeyres index, who add 7 p.p. to the range). At the opposite extreme, the distributions for 2022 (in red) are very wide. In this year, indices have IQRs of about 4 to 22 percent for coffee and frozen chicken and about 5 to 27 percent for men's shaving cream. This occurs within an asymmetric range of 0 to 40 percent for coffee, 0 to 33 percent for frozen chicken, and zero to 36 percent for men's shaving cream. The distributions in 2021 (blue) and 2023 (yellow) are more dispersed than in 2019 but much less dispersed than in 2022. In 2021, the boxes (whiskers) have spreads ranging from about 6 (13) p.p. for coffee to 10 (16) p.p. for frozen chicken; the 20 p.p. range for men's shaving cream considerably exceeds an IQR of 8 p.p. In 2023, the IQRs run from about 5 p.p. for coffee to 10 p.p. for men's shaving cream, with overall spreads of about 18 p.p. in coffee and frozen chicken and 23 p.p. in men's shaving cream.
Figure 3: Relationship of Household and Available Module Inflation Rates
Each dot represents a household's four-quarter average price change within a product module. Position on y-axis reflects household's Paasche-weighted average price change,
while position on x-axis shows price changes in same module in household's Nielsen DMA or at the retailer where the household purchases majority of this product.
The figure contains two scatter plots side by side, each relating households' realized Paasche inflation rates against an ``available" inflation rate computed by applying average prices within each product module in either their market area (left) or at the retailer where the household spends most on that product (right). The plots are similar in appearance: clouds of blue points exhibit much less variation over available inflation rates than realized inflation rates. A dashed black 45-degree line, steeper than an accompanying grey regression fit line, contextualizes the relationship conveyed by the scatter plots. The R-squared coefficient from each regression is displayed in the legend, and values of 0.10 (left) and 0.09 (right) emphasize the limited ability of differences across geography or shopping outlet to explain household inflation.
Figure 4: Distribution of Household Grocery Inflation Experiences
Household inflation rates computed using individual expenditure shares and applying national average price changes for each good. Tornqvist inflation rates apply average expenditure
shares in Q and Q-4. Measure includes spending in all NielsenIQ HomeScan categories except general merchandise. Distributions apply households' NielsenIQ projection factors, which weight sample to be nationally representative demographically.
The figure contains four lines, one for each of the years 2019 (dotted green), 2021 (dashed blue), 2022 (solid red), and 2023 (dot-dashed yellow). Each displays the density of households' four-quarter grocery inflation rates at the end of the given year. Densities are approximately normal in distribution and get wider as their midpoints increase. Midpoints rose from about 1 percent in 2019 to 5 percent in 2021 and 11 percent in 2022 before falling back to 2 percent in 2023. In 2019, the distribution is quite tight: 80\% of the distribution's mass falls between inflation rates of about -0.5 and 2.5 percent. The 2021 distribution is wider, with this same share of its mass running from 3 to 6.5 percent. By 2022 the distribution is very wide, with 80\% of the mass spanning inflation rates of 8 to 15 percent. This width reduced in 2023, running from about 0.5 to 4 percent.
Figure 5: Evolution of Grocery Inflation
Note: Mean and standard deviation of four-quarter Tornqvist grocery inflation computed across households in NielsenIQ HomeScan panel using national average price changes for all goods (excluding general
merchandise). Source: Bureau of Economic Analysis (PCE Food), author's calculations.
The figure contains three time series plots at a quarterly frquency running from 2012 through 2023. Two of its lines, for PCE Food inflation (in dotted black) and NielsenIQ average grocery inflation (solid grey), track each other closely. Both peak in 2022 at value of 11-12% and take their minimum values in 2016 at about -1%. They both achieve local maxima in 2012 (4%) and 2020 (5%) and local minima in 2021 (3%) and at the end of 2023 (2%). They deviate somewhat at other turning points, with PCE Food inflation exceeding NielsenIQ by about 1.5 percentage points in 2015 and 2018. A dashed yellow line plots the standard deviation of inflation rates across households over this same period. It is comparatively stable at between 1 and 2 percentage points, but rises noticeably in 2022 to 3 percentage points before declining to below 2 percentage points.
Figure 6: Distribution of Household Grocery Inflation by Demographic Group: 2022q4-2023q4
(a) Income
(b) age
(c) Employment
(d) Race
(e) Household Size
(f) ZIP Code Population
Density
The figure is a full-page composite of 6 individual figures, each conveying a set of inflation rate distributions in 2023 among households grouped by a given demographic: income, age, employment, race, household size, or ZIP code population density. All subfigures convey a common theme: the distributions of the various groups sit nearly on top of one another, with the bulk of their distributions running from about 0 to 4 percentage points and peaking near 2 percent, the average value of PCE Food inflation over that period (shown as a vertical dotted line).
Figure 7: Discrete Choice Preferences Within Product Category
Figure shows two agents' utilities for products $$a$$ and $$b$$ in module $$m$$ (here coffee). Navy lines represent indifference curves, whose slopes are given by
$$-\frac {e^{(\varphi _{ma}+\varepsilon _{hma})}}{e^{(\varphi _{mb}+\varepsilon _{hmb})}}$$. Household 1 strongly prefers Coffee B, which received a price increase (red dashed to purple dotted budget lines). Household 2
prefers Coffee A and was unaffected.
The figure contains two panels, conveying discrete choice utility preferences for two hypothetical households. Each panel presents tradeoffs between quantities of Coffee A (x-axis) and Coffee B (y-axis). Two downward-sloping budget lines, identical in each figure, represent in the amounts of each coffee that can be purchased under relative prices experienced in period t-1 (dashed red) and t (dotted purple). They both intersect the x-axis at a common point (the price of Coffee A in both periods). The dashed red line is steeper, intersecting the y-axis at a higher point reflective of the lower price of Coffee B in period t-1. Household preferences are shown in the form of linear indifference curves (solid blue lines). Panel 1 contains two indifference curves, each intersecting the y-axis at the price of Coffe B in the two periods. Household 2 has one, steeper indifference curve which intersects the x-axis at the price of Coffee A.
Figure 8: Cobb-Douglas Preferences Across Product Group
Figure shows two agents' identical Cobb-Douglas upper utility tiers. Agent 1 strongly prefers a coffee brand which received a price increase and continues to purchase it, while Agent 2
prefers a good which received no price change.
The figure contains two panels, conveying Cobb-Douglas utility preferences for two hypothetical households. Each panel presents tradeoffs between quantities of Coffee (x-axis) and Milk (y-axis). The households have identical preferences over these two goods, reflected in three convex solid blue lines. Their curvature decreases from an almost hyperbolic near the origin to almost linear at the opposite corner. A red dashed line cuts from the upper left to bottom right corner of both panels, representing the relative prices of coffee and milk in period t-1. The middle indifference curve is tangent to this line at a point just down and to the right of center. In the left panel only, a dotted purple line with a steeper slope extends from the upper left corner to a point about two-thirds down the x-axis. It shows the relative price of milk and coffee for household 1 in period t, which did not change for household 2. An additional indifference curve, in grey, is tangent to this line just below the midpoint of the figure.
Figure 9: Relative Variance of Cumulative Cost of Living Indices
Figure presents the variance ratio (see equation 34) over horizons ranging from 1 to 12 quarters. Formally, the variance ratio at a horizon of $$k$$ quarters expresses the cross-sectional variance of household inflation indices cumulating inflation to $$k$$ quarters relative to $$k$$ times the one-quarter variance of household inflation rates. A random walk process would exhibit a variance ratio of 1 at all horizons. A process with perfect mean reversion would exhibit a variance ratio of 0.5 for
$$k=2$$.
The figure plots the variance ratio (see equation 34) over horizons ranging from 1 to 12 quarters. It declines at a constant rate from a value of 1 (by definition) at the one-quarter horizon to a value of 0.75 at a four-quarter horizon. Its slope flattens significantly thereafter, reacing a value of 0.7 at an eight-quarter horizon, where it remains thereafter.
Figure 10: Distribution of Changes to Expected Lifetime Utility of Total Consumption
$$\mathcal {V}(\left \{\mathcal {C}_{h\tau }\right \}_{\tau =t}^T)$$
Each line translates the empirical distribution of household Q4/Q4 Tornqvist inflation experiences into changes in lifetime utility of
consumption (across both grocery and non-grocery categories). Estimates assume fixed lifetime resources, and an elasticity of intertemporal substitution
$$\theta = 0.25$$. Results assume households' costs of living take linear paths back to the average cost of living over three years.
The figure contains four lines, each conveying the distribution of changes to expected lifetime utility from realized four-quarter inflation draws in a given year. The years plotted are 2019 in dotted green, 2021 in dashed blue, 2022 in solid red, and 2023 in dot-dashed yellow. Distributions are roughly normal, with peaks at approximately 0 in 2019, -0.1 in 2021, -0.2 in 2022, -0.05 in 2023. The distributions encompass exhibit an interquartile range of about 0.1 percentage points in 2019, 2021, and 2023 and 0.2 in 2022.
Figure A.1: Household Inflation 2018q4-2019q4 by Price Change Source
Notes for Figures A.1 - A.4: each line conveys a measure of inflation constructed using a different
source of price change information. The "matched goods, hh prices" line employs only price information reported by households, as in Kaplan & Schulhofer-Wohl (2017), and as a result is limited to the set of goods purchased in both the base and reference periods. The "goods with retailer prices
or better" line uses household price reports when available, but applies the average quarterly price change at the store or retailer of purchase for goods not purchased in both periods. The "all goods, best prices" measure continues to layer in the next best available price, either specific to the
household's market area or the national average. In contrast to each of these measures, the "all goods, best non-hh prices" construction similarly uses the best available price for a good, but skips the household's individual price reports. As a result, it excludes idiosyncratic price differences
across households (for instance arising from promotional sales and purchase timing) but retains heterogeneity driven by outlet or location. The "only using DMA prices" line applies average price changes for each item within its market area while the baseline measure assumes that all households face
the same average price change for a given barcode.
The figure shows six density plots of households' inflation experiences overlaid on one graph. Its x-axis, which conveys 2018Q4-2019Q4 Tornqvist inflation in percentage points, runs from -4 to 20. Its y-axis, showing the percent of households in each bin, spans 0 to 8 percent. A legend distinguishes lines by the sources used for price changes when computing each distribution, and it lists the percent of spending having non-missing price changes in that data source. Distributions are approximately normal with similar central tendencies, but they differ strongly in their widths. The widest distribution, in green, shows inflation rates computed only over matched goods (which the legend shows covers 36% of spending). The distribution is so wide that about 20% of its mass is truncated by the left edge of the graph. The 90th percentile of this distribution stretches to about 7.5 percent. This density peaks at a midpoint of 0% inflation, which captures the experience of 2 percent of households. A cluster of lines in yellow, magenta, and cyan are the next most dispersed, reaching peak densities of 3.4 percent of households. These apply DMA-average prices, retailer-average prices, or the best/most relevant available prices and capture 58%, 83%, and 93% of spending, respectively. A gold line summarizes the density when applying the best/most relevant price excluding the household's own. It captures 92% of spending and peaks just below 1 percent inflation at a density of 5.5 percent of households. A distribution using only national average price changes---the baseline estimate for this paper---is shown in blue, and is the tighest. It covers 93% of spending and peaks just above 1 percent inflation, which is experienced by about 7.7 percent of households. Figure caption contains the definition of best/most relvant prices. Table 3 summarizes the 10th, 25th, 50th, 75th, and 90th percentiles of the green, cyan, gold, and blue lines.
Figure A.2: Household Inflation 2020q4-2021q4 by Price Change Source
Notes for Figures A.1 - A.4: each line conveys a measure of inflation constructed using a different
source of price change information. The "matched goods, hh prices" line employs only price information reported by households, as in Kaplan & Schulhofer-Wohl (2017), and as a result is limited to the set of goods purchased in both the base and reference periods. The "goods with retailer prices
or better" line uses household price reports when available, but applies the average quarterly price change at the store or retailer of purchase for goods not purchased in both periods. The "all goods, best prices" measure continues to layer in the next best available price, either specific to the
household's market area or the national average. In contrast to each of these measures, the "all goods, best non-hh prices" construction similarly uses the best available price for a good, but skips the household's individual price reports. As a result, it excludes idiosyncratic price differences
across households (for instance arising from promotional sales and purchase timing) but retains heterogeneity driven by outlet or location. The "only using DMA prices" line applies average price changes for each item within its market area while the baseline measure assumes that all households face
the same average price change for a given barcode.
The figure shows six density plots of households' inflation experiences overlaid on one graph. Its x-axis, which conveys 2021Q4-2023Q4 Tornqvist inflation in percentage points, runs from -4 to 20. Its y-axis, showing the percent of households in each bin, spans 0 to 8 percent. A legend distinguishes lines by the sources used for price changes when computing each distribution, and it lists the percent of spending having non-missing price changes in that data source. Distributions are approximately normal with similar central tendencies, but they differ strongly in their widths. The widest distribution, in green, shows inflation rates computed only over matched goods (which the legend shows covers 33% of spending). The 10th and 90th percentiles of this distribution stretch to about -4 and 12 percent. This density peaks at a midpoint of about 3.5 percent inflation, which captures the experience of 1.8 percent of households. Magenta and cyan lines are the next most dispersed, reaching peak densities of 2.9 percent of households at inflation values of about 3.5 and 4.5 percent. These apply retailer-average prices or the best/most relevant available prices, capturing 79% and 91% of spending, respectively. A yellow line applying DMA average prices captures 73% of spending and reaches a peak of 4 percent of households with about 4.5 percent inflation. A gold line summarizes the density when applying the best/most relevant price excluding the household's own. It captures 90% of spending and peaks at a value of 4.5 percent of households at a 4.5 percent inflation rate. A distribution using only national average price changes---the baseline estimate for this paper---is shown in blue, and is the tighest. It covers 91% of spending and peaks just below 5 percent inflation, which is experienced by about 6 percent of households. Figure caption contains the definition of best/most relvant prices. Table 3 summarizes the 10th, 25th, 50th, 75th, and 90th percentiles of the green, cyan, gold, and blue lines.
Figure A.3: Household Inflation 2021q4-2022q4 by Price Change Source
Notes for Figures A.1 - A.4: each line conveys a measure of inflation constructed using a different
source of price change information. The "matched goods, hh prices" line employs only price information reported by households, as in Kaplan & Schulhofer-Wohl (2017), and as a result is limited to the set of goods purchased in both the base and reference periods. The "goods with retailer prices
or better" line uses household price reports when available, but applies the average quarterly price change at the store or retailer of purchase for goods not purchased in both periods. The "all goods, best prices" measure continues to layer in the next best available price, either specific to the
household's market area or the national average. In contrast to each of these measures, the "all goods, best non-hh prices" construction similarly uses the best available price for a good, but skips the household's individual price reports. As a result, it excludes idiosyncratic price differences
across households (for instance arising from promotional sales and purchase timing) but retains heterogeneity driven by outlet or location. The "only using DMA prices" line applies average price changes for each item within its market area while the baseline measure assumes that all households face
the same average price change for a given barcode.
The figure shows six density plots of households' inflation experiences overlaid on one graph. Its x-axis, which conveys 2021Q4-2022Q4 Tornqvist inflation in percentage points, runs from -4 to 20. Its y-axis, showing the percent of households in each bin, spans 0 to 8 percent. A legend distinguishes lines by the sources used for price changes when computing each distribution, and it lists the percent of spending having non-missing price changes in that data source. Distributions differ strongly in their widths. The widest distribution, in green, shows inflation rates computed only over matched goods (which the legend shows covers 32% of spending). The distribution has support over the entire x-axis and more than 10 percent of its mass is truncated by the right edge of the graph. No bin of this density contains more than 1 percent of households. A magenta line, computed using retailer average prices and covering 82% of spending, is similarly dispersed, barely exceeding 1 percent of households. It is shifted somewhat to the right of the green distribution and is centered at about 13 percent. All other distributions are considerably less dispersed. A cyan line, which applies the best/most relevant available prices, captures 93% of spending and peaks at an inflation rate of 12 percent with a density of 2 percent of households. Yellow, gold, and blue lines convey similar densities peaking at 3 percent of households around 11.5 percent inflation. These apply retailer-average prices, the best/most relevant available prices, or national average price changes and capture 79%, 93%, and 94% of spending, respectively. Figure caption contains the definition of best/most relvant prices. Table 3 summarizes the 10th, 25th, 50th, 75th, and 90th percentiles of the green, cyan, gold, and blue lines.
Figure A.4: Household Inflation 2022q4-2023q4 by Price Change Source
Notes for Figures A.1 - A.4: each line conveys a measure of inflation constructed using a different
source of price change information. The "matched goods, hh prices" line employs only price information reported by households, as in Kaplan & Schulhofer-Wohl (2017), and as a result is limited to the set of goods purchased in both the base and reference periods. The "goods with retailer prices
or better" line uses household price reports when available, but applies the average quarterly price change at the store or retailer of purchase for goods not purchased in both periods. The "all goods, best prices" measure continues to layer in the next best available price, either specific to the
household's market area or the national average. In contrast to each of these measures, the "all goods, best non-hh prices" construction similarly uses the best available price for a good, but skips the household's individual price reports. As a result, it excludes idiosyncratic price differences
across households (for instance arising from promotional sales and purchase timing) but retains heterogeneity driven by outlet or location. The "only using DMA prices" line applies average price changes for each item within its market area while the baseline measure assumes that all households face
the same average price change for a given barcode.
The figure shows six density plots of households' inflation experiences overlaid on one graph. Its x-axis, which conveys 2018Q4-2019Q4 Tornqvist inflation in percentage points, runs from -4 to 20. Its y-axis, showing the percent of households in each bin, spans 0 to 8 percent. A legend distinguishes lines by the sources used for price changes when computing each distribution, and it lists the percent of spending having non-missing price changes in that data source. Distributions are approximately normal with similar central tendencies, but they differ strongly in their widths. The widest distribution, in green, shows inflation rates computed only over matched goods (which the legend shows covers 35% of spending). The distribution is so wide that well over 10% of its mass is truncated by the left edge of the graph. The 90th percentile of this distribution stretches to about 9 percent. This density peaks at a midpoint around 1% inflation, which captures the experience of 1.9 percent of households. A cluster of lines in yellow, magenta, and cyan are the next most dispersed, reaching peak densities of 3.3 percent of households. These apply DMA-average prices, retailer-average prices, or the best/most relevant available prices and capture 56%, 85%, and 96% of spending, respectively. A gold line summarizes the density when applying the best/most relevant price excluding the household's own. It captures 96% of spending and peaks just above 2 percent inflation at a density of 5.2 percent of households. A distribution using only national average price changes---the baseline estimate for this paper---is shown in blue, and is the tighest. It covers 96% of spending and peaks just at 2 percent inflation, which is experienced by about 6.3 percent of households. Figure caption contains the definition of best/most relvant prices. Table 3 summarizes the 10th, 25th, 50th, 75th, and 90th percentiles of the green, cyan, gold, and blue lines.
Figure A.5: Comparison of Baseline and Matched Goods Inflation
Figure plots the distribution of household Tornqivst inflation from 2020Q4-2021Q4 in a comparison of three measures. Its navy line, which assumes all households face the national
average price of each barcode, represents the paper's baseline measure. It computes inflation over all goods for which national average price changes can be observed. The green line applies only household-level price changes, necessarily restricting attention to matched goods that households
purchase in both 2020Q4 and 2021Q4. A magenta line decomposes the differences between these two measures by applying national average price changes (as in the navy line) to matched goods (as in the green).
The figure plots three densities of household inflation experiences. Its x-axis, which conveys 2020Q4-2021Q4 Tornqvist inflation in percentage points, runs from -4 to 20. Its y-axis, showing the percent of households in each bin, spans 0 to 8 percent. A legend distinguishes lines by the sources used for price changes and the goods used to compute each inflation distribution, listing the percent of grocery spending govered by those goods. A distribution using only national average price changes---the baseline estimate for this paper---is shown in blue, and is the tighest. It covers 91% of spending and peaks just below 5 percent inflation, which is experienced by about 6 percent of households. At the opposite extreme, a green distribution is most dispersed. It shows inflation rates computed using household-specific prices and is necessarily computed only over matched goods, which cover 33% of spending. It peaks at a midpoint of about 3.5 percent inflation, which captures the experience of 1.8 percent of households. An intermediate distribution, shown in magenta, more closely resembles the blue line but is more dispersed. It uses the same 33% of spending on matched goods as in the green line but applies national average price changes. Its distribution peaks around 4.5 percent inflation with a density of 3.4 percent of households. Table 3 summarizes the 10th, 25th, 50th, 75th, and 90th percentiles of the green and blue lines.
Figure A.6: Distribution of Laspeyres and Paasche Inflation
Distributions of Laspeyres and Paasche Q4/Q4 inflation rates among households in NielsenIQ HomeScan panel. Inflation computed using national average price changes for all goods except
those in general merchandise categories.
The figure displays four pairs of density functions: one pair for each of the years 2019 (lime and forest green), 2021 (dusty and Dodger blue), 2022 (coral and brown), and 2023 (yellow and goldenrod). All lines convey the distributions of household-level Q4/Q4 inflation rates in the respective years. The two lines in each year are distinguished by whether they compute Laspeyres or Paasche indices (shown in light and dark shades, respectively). In 2019, 2021, and 2023, the Laspeyres and Paasche indices form very similar densities in both width and central tendency, being separated by not more than 0.5 percentage points. By contrast, the 2022 distributions are slightly offset, with the coral Laspeyres index distribution having a midpoint more than 1 percentage point below the sienna Paasche index distribution. The 10th, 25th, 50th, 75th, and 90th percentiles of each distribution are well-approximated by the Tornqvist inflation rates summarized in Table 3.
Figure A.7: Distribution of Spending Shares in Modules With Laspeyres > Paasche
Figure constructed by evaluating, for each household and product module, whether the household's Laspeyres index in that module exceeds (Lasp. > Psch.) or is
within one percentage point (Lasp. - Psch. > 1) of its Paasche index. The statistic whose distribution these lines plot are households' shares of grocery expenditures meeting that condition.
This figure tracks the density of households who exhibit particular substitution patterns. Its x-axis tracks the share of spending (from 0 to 1) households allocate to product modules exhibiting the substitution pattern specified in each line. There are two pairs of lines (four in total) one pair in blue and another in red. Light-colored lines track how many households devote the specified share of spending to product modules whose Laspeyres index exceeds its Paasche index. Dark-colored lines track a similar statistic, but require that the Laspeyres index exceed the Paasche index by more than 1 percentage point. Blue lines summarize these distributions from 2012-2019 and red lines summarizes them from 2020-2023. All distributions have the bulk of their mass above 0.5, implying that it is typical for Laspeyres indices to exceed Paasche indices. The light-colored densities, only slightly left-skewed, reach a peak density of 14 percent of households at a spending share of about 0.68. The tails shrink to cover less than 2 percent of households at values of about 0.4 and 0.95. Light blue and red distributions are similar, conveying a similar across the 2012-2019 and 2020-2023 time periods. Both dark-colored distributions are more strongly skewed: the density falls below below 2 percent to the left of a spending share of 0.55 and 3 percent of households meet the criteria for all of their spending. The dark blue and dark red densities both peak at spending shares of about 0.85. The dark blue density for 2012-2019 peaks at a higher value of about 19 percent of households, while the dark red density peaks at a density of about 17.5 percent of households. Mass from the dark red density shifts to the left, showing that households had Laspeyres indices exceeding their Paasche indices in fewer product modules from 2020-2023.
Figure A.8: Distribution of Household Inflation by Demographic Group: 2020q4-2021q4
(a) Income
(b) Age
(c) Employment
(d) Race
(e) Household Size
(f) Population Density
The figure is a full-page composite of 6 individual figures, each conveying a set of inflation rate distributions in 2021 among households grouped by a given demographic: income, age, employment, race, household size, or ZIP code population density. All subfigures convey a common theme: the distributions of the various groups sit nearly on top of one another, with the bulk of their distributions running from about 3 to 7 percent and peaking near 5 percent, just below the average value of PCE Food inflation over that period (shown as a vertical dotted line). Noticeable deviations exhibit gaps smaller than 2 percent of households for a given value of inflation. These include the income < 25k group in the income plot and the 30-39 years old group in the age plot, both of whom feature distributions translated about 0.5 percentage points to the right in portions of the distribution. Non-white groups in the race plot feature a distribution shifted about 0.5 percentage points to the left in the regions of 3-4 percent and 6-7 percent inflation, though these groups exhibit noisier inflation values.
Figure A.9: Distribution of Household Inflation by Demographic Group: 2021q4-2022q4
(a) Income
(b) Age
(c) Employment
(d) Race
(e) Household Size
(f) Population Density
The figure is a full-page composite of 6 individual figures, each conveying a set of inflation rate distributions in 2021 among households grouped by a given demographic: income, age, employment, race, household size, or ZIP code population density. All subfigures convey a common theme: the distributions of the various groups sit nearly on top of one another, with the bulk of their distributions running from about 8 to 15 percent and peaking near 11 percent, about the average value of PCE Food inflation over that period (shown as a vertical dotted line). Noticeable deviations are asystematic and limited to irregularities in the race and population density plots. Near the center of these distributions, the Asian and 5k-10k people per sq. mile groups have densities exceeding the other distributions by less than 2 percent of households.
Figure A.10: Total Utility of Consumption
The left panel shows the concavity of constant relative risk aversion (CRRA) preferences by plotting consumption on the x-axis against the flow utility of that consumption on the y-axis. It does this under three different values of the coefficient of relative risk tolerance. Blue, orange, and green lines show the curvatures for parameter values of 0.50, 0.25, and 0.17, respectively. All three lines share the common point (1, 0). The blue line features the least curvature: utility runs from -0.33 to 0.2 as as consumption varies from 0.75 to 1.25. The green line features the most curvature, rising steeply from about (0.75, -0.63) through (1.0, 0) before tapering off to a value about (1.25, 0.14). The orange line is an intermediate case, its endpoints being near the midpoints of the other lines' endpoints. The right panel shows the concavity of the intertemporal choice problem generated by constant relative risk aversion (CRRA) preferences, plotting indifference curves between next-period consumption on the x-axis against current consumption on the y-axis. It does this under three different values of the coefficient of relative risk tolerance. Blue, orange, and green lines show the curvatures for parameter values of 0.50, 0.25, and 0.17, respectively. All three lines share the common point (1, 1), which lay in the center of the plot along a downward-sloping dashed red diagonal line representing equal spending in each period. They all bend in a concave upward pattern open to the top-right quadrant of the graph. The blue line features the least curvature. Its endpoints pass through about (0.85, 1.25) and (1.25, 0.83). The green line features the most curvature, passing through about (0.9, 1.25) and (1.25, 0.9). The orange line is between these, its endpoints passing through about (0.87, 1.25) and (1.25, 0.86).
Figure A.11: Distribution of Changes to Flow Utility of Total Consumption
$$\nu (\mathcal {C}_{ht})$$
Each line translates the empirical distribution of household Q4/Q4 Tornqvist inflation experiences--as in Figure 4--into implied changes in flow
utility of total consumption across both grocery and non-grocery spending categories
$$\nu (\mathcal {C}_{ht})$$. Estimates assume a fixed budget, with expenditures not increasing to offset the price shocks, and a coefficient of relative risk tolerance
$$\theta = 0.25$$.
The figure shows four densities summarizing the distribution of implied changes to flow utility of total consumption based on households' realized grocery inflation rates and the assumptions that non-grocery inflation is zero and that expenditures remain constant. These densities reflect grocery inflation experiences in different years: 2019 (dotted green), 2021 (dashed blue), 2022 (solid red) and 2023 (dot-dashed yellow). Utility losses on the x-axis range from -2.4 percent to 0.4 percent. By far the widest distribution is in the red line: in 2023 households saw losses ranging from -2.2 to -0.6 percent. The distribution peaks around -1.35 percent with a density of 6 percent of households. For 2021 and 2023 the blue and yellow densities are narrower, each peaking around 12 percent of households at midpoints of -0.6 and -0.2, respectively. The density in 2019 is narrower still, peaking at 15 percent of households with a midpoint of -0.15. Almost the entire density falls between -0.5 and 0.2.
Figure A.12: Distribution of Lifetime Utility Changes for Shocks of Varying Persistence Absent Expenditure Increases
Each line translates the empirical distribution of household Q4/Q4 Tornqvist inflation experiences--as in Figure 4--into implied changes in total lifetime utility of total consumption across both grocery and non-grocery spending categories. Estimates assume fixed lifetime resources, with expenditures not increasing to offset the price shocks, and an elasticity of intertemporal
substitution
$$\theta = 0.25$$. The exercise requires an assumption about how cost of living paths respond to inflation shocks. Baseline results assume households' costs of living take linear paths back to the average cost of living
over the specified duration.
The figure shows six densities summarizing the distribution of implied changes to lifetime utility of consumption based on households' realized inflation rates and an assumption that expenditures remain constant. Four of these densities reflect baseline assumptions applied to inflation experiences in different years: 2019 (dotted green), 2021 (dashed blue), 2022 (solid red) and 2023 (dot-dashed yellow). That baseline assumption is that households' grocery baskets take three years to rever to average prices after an inflation shock. Two more densities reflect apply, to the 2023 inflation distribution, alternative assumptions that it takes 2 years (light red) or 5 years (brown) for grocery bakets to return to average prices. Three of these densities are located at the right end of the plot. Those for 2019, 2023, and the 2023 density under an assumption of a 2 year return to average inflation have nearly all of their mass between -0.1 and 0. Using utility loss bins with a width of 0.025, these densities peak between 30 and 37 percent of households. The 2021 density is slightly wider and falls to the left, between values of -0.15 and -0.05, and peaks at a value of about 28. The remaining two densities are considerably wider. In solid red, the baseline assumption for 2023 implies a distribution of lifetime utility losses ranging from about -0.35 to -0.1. That distribution peaks at -0.225 with a density of 15 percent of households. The brown line has the widest density, intersecting the boundary of the plot at a lifetime utility loss of -0.5 with a density of about 2 percent and a similar support in the distribution stretching to -0.2 percent. It peaks at a utility loss of about -0.35 with a density of about 10 percent of households.