U.S. Demand for Imported and Domestically-Produced Foods: An Investigation of Intertemporal and Cross Substitution

International Finance Discussion Papers Number 116

January 1978

U.S. DEMAND FOR IMPORTED AND DOMESTICALLY-PRODUCED FOODS: AN INVESTIGATION OF INTERTEMPORAL AND CORSS SUBSTITUTION

by

Peter Isard, Barbara Lowrey, and P.A.V.B. Swamy

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an

acknowledgment by a writer that has had access to unpublished material) should be cleared with the author or authors.

December 1977

U.S. Demands for Imported and Domestically-Produced Foods: An

Investigation of Intertemporal and Cross Substitution

by

Peter Isard, Barbara Lowrey, and P.A.V.B. Swamy*

I. Introduction

Attempts to explain the time-series behavior of consumer purchases traditionally rely on econometric specifications that are alleged to reflect a utility-maximizing theory of consumer behavior. Most explicit presentations of the underlying theory view the consumer to be maximizing utility over a single-period horizon. Yet many econometric specifications include alistof lagged price variables that have no place in the single-period model of consumer choice.

Several rationalizations can be provided for the introduction and apparent explanatory power of lagged price variables. In some contexts consumer purchases may be related to earlier orders, and hence to prices prevailing when orders were placed. Alternatively, given that consumers in reality make their spending decisions in a multi-period

context, lagged prices may contain information that helps consumers form

*/ Board of Governors of the Federal Reserve System. The analysis and conclusions of this paper should not be interpreted as reflecting the views

of the Federal Reserve System or anyone else on its staff. We are indebted to an anonymous referee for a number of constructive criticisms of our

earlier paper, and to Richard Berner and Peter Hooper for their comments on this revision.

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their subjective expectations of future prices.

This paper invokes the multi-period model of consumer choice to specify consumer demands as functions of permanent income, current prices and expected future prices. Proxy variables are introduced, expressing permanent income in terms of current and lagged values of observed income, and expressing expected future prices in terms of current and lagged values of observed prices. For purposes of empirical estimation we focus on consumer choice between imported foods and domestically-produced foods =! treating all other consumables as a composite numeraire. For each category of foods we estimate a nonlinear demand function using quarterly data Spanning the period from fourth-quarter 1958 through second-quarter 1976. Our findings confirm that both groups of foods are necessities; permanentincome elasticities are less than unity. Intratemporal own-price elasticities are negative and in both cases significant; intratemporal cross-price elasticities are positive and in one case significant; intertemporal own-price elasticities are positive (specifically, current purchases are positively correlated with the expected future rate of own-price inflation) ; and intertemporal cross-price elasticities are negative. Although intertemporal price elasticities are not significantly different from zero, Suppression of the influence of expected future prices on consumer demands raises the standard error of estimate substantially in the case of domestically-produced foods and slightly in the case of imported foods. 1/" Habit formation provides a third rationalization, with many variations, Habit formation is complicated to model, however, even under the simple assumption that current purchases repeat lagged purchases that were based on price information at some initial BP Baring due “data period imports of (a) meats,(b) fish, (c) coffee and (d) sugar each represented roughly 15 per cent of total food imports, while

imports of (e) fruits, nuts and vegetables and (f) alcoholic beverages each represented roughly 10 per cent.

II. The Model

Our multi-period model of consumer choice is developed in Isaic, Lowrey and Swamy (1975), where we spell out a methodology for aggregating individual demands in a world in which consumer tastes are neither individual-invariant nor time-invariant. Here we merely write down our expressions for aggregate consumer demands during the first period of a multi-period horizon. ‘These expressions are simplified by the assumption that consumers always expect prices to change over time at steady rates, though these expected rates will generally differ across commodities and can change from period to period as consumers revise their expectations on the basis of experience. Thus, expectations held currently about the price of a given commodity on some particular date in the future can be expressed in terms of two variables: the current price and the currently~expected value of the future rate of inflation for that commodity.

We use the following notation, where all variables are in

logrithmic form:

c_, € == consumer purchases of domestically-produced and

c t imported foods in period t yp -- permanent income in period t

Yy, ~~ current income in period t

pe, PE -- prices of domestically-produced and imported foods in period t

d m m2 TM) Tt period-t values of expected inflation factors

attached to domestically-produced and imported foods

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Permanent income and the current prices of foods are measured relative

to the current price of "all other consumable goods and services" (the numeriare), while expected future prices of foods are measured relative to expected contemporaneous future prices of all other consumables.

Thus, prices represent current and expected future real terms of exchange between foods and all other consumables. Permanent income should ideally be measured to reflect initial wealth plus the present discounted value of the expected stream of future income, expressed in terms of real

current purchasing power over the numeriare. These definitions justify the specification d . d d p m d m - = tap + +ko)+ (p, tk) CF Uy FY, F app, F agPy >, La, (P, + km) + a5, (ptr) k=1

4 ™

where D- + kre and P,. + ko respectively represent (the logrithms of) t cL the prices of domestically-produced and imported foods that are currently

(in period t) expected to prevail in period t+k, and where T is the length

of the consumer horizon. Substituting

T SO =a + a 2 2 Ds 4k k=1 T = + Og 73 ae T Oy, => ka, =1

T Og “2 F5%

and adding an error term, Oz + u., yields

t?

d d (1) c Fa taytap tap ean tan t+eaz +u F 0 Fe 2t “3't “@e 5t 6tr t Similarly m d m d m 2) c, = 8+ B.3 + + (2) ep = Bot BH + BP + BBP + Bo + Bot Be + y

t

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The terms Uy and v. are stochastic disturbances, and Zz, is a dummy variable introduced to capture the influence of dock strikes on imports. Due to data limitations, cr is measured as (the logrithm of) purchases of foods by importers, based on international trade statistics, rather than final sales of imported foods 3 to consumers. The dummy variable Zips taken from Isard (1975), is constructed as (the logrithm of) the ratio of observed imports to an estimate of the level of imports that would have been recorded in the absence of dock strikes. Thus, we expect Be to be approximately equal to one. Since purchases of domestically- produced foods during the sample period were roughly 20 times as great as purchases of imported foods, we expect any impact of dock strikes on purchases of domestically-produced foods ( a6) to not be more negative than 4/ -.05. . sea P d mm. We require definitions of Yoo 7 and TT in terms of t

variables that can be observed. There is some precedent (see Nerlove, 1967, pp. 142-3) for using weighted averages of current and lagged levels of income, and of current and lagged observations of one-period inflation factors, with weights that decline geometrically as the , 37 The literature pays surprisingly little attention to the fact that final sales of imported goods to end users (the dependent variable. in most theoretical import-demand hypotheses) can differ substantially from recorded imports due to changes in the inventories of imported goods held by intermediaries, on which available data are sparse. Roughly alf of U.S. food imports are imperishable (e.g., coffee, sugar, alcoholic beverages), and consequently our dependent variable is influenced by the demand for food as a stock as well as by the demand for food as a flow.

4/ That is, a one per cent decline in imports of foods due to a dock

strike should be associated with no more than a one-twentieth of one per cent increase in purchases of domestically-produced foods.

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length of lag increases. Thus, we define

we ‘ P. (1. J i < (3) yP= a 4) DO) Yeo with OS), <1 d — j.d 4 = (le - ith O<) <1 @) m= DP 20a Cp cP rep) with OSAZ fee} Q 5) m= (1+. ) ayg -p- ) with OS) <1 ( Te m m Peek Pe okel m j=0

Substitution of (3) - (5) into (1), after appropriate Koyck

5 transformations and manipulation, yields the complicated expression~'

d . d d = « - - + +) + - + (6) et Gy (1 iO. Ag? An? Ory na Aw tet Org Nan ny Ene

d + ydahateeg Oy -ry, - Oy (1-2) gt Ap + a, -2,) Narn pao

+

d [oto (1-2,) Jee ~ Lan Crytaghh, tay = Ag) 14,42) Ie y

+

d + + + - ++ 7 Lag (hy hgthg hgh h, Ayrtay (Loa Ot Agta, A) IPE

[ory Ay Ag yt, (Lm Ag) Ay Ng Peg + Lastos (1-,,) Py

m + +h )ta.(1- 1+, + + ARAFAT Lo; CA, 4 we) ag ( A» y APs Lor, ( yy ra ry

+ - ++ Wyk . + -) ) m 5 (1-1) OSFAgHAL Ag) IPL» san a (I-A DAA, Peng

t + 2. - Ch #XY.#AR ) + \ +) +3 - a, Me MPN DZ poy TH OYA TAT my IZ, 9 a tu - (+h 4+) )u + FERRY = RRA Co OP Bet OLR GAY 7 A Ukes

— oor . id

5/ This can be verified by transpusing the terms in c,_, to the left-hand ‘side, substituting equation (1) -- appropriately dated -- fer each of these terms, and noting from (3) - (5) that the left-hand side terms in y (respectively, 7, mg, u) are equivalent to che right-ha:d side terms

. ws

in y(resp., p, ps #, U)-

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Similarly, based on conditions (2) - (5),

(7) c = same as (6) with c

tok replacing ct (k=1, 2,3)

8.

F replacing ow. (j=0,...,6)

and Veok replacing Uk (k=0,...,3) Each of these consumer demand equations is overidentified, with a total of 19 coefficients determined by 10 basic parameters. Consequently, we must estimate the 10 parameters under 9 nonlinear constraints In doing so we make the simplifying assumption that the u_ and VE follow third-order autoregressive processes such that the combined stochastic terms in (6) and (7) have the conventionally-assumed properties of being independently distributed with zero means and constant variances, III. The Data

Our estimates of equations (6) and (7) are based on imperfect measures of the dependent variables, because data on final sales to consumers are not available. For of we used seasonally-adjusted data

on manufacturers’ shipments of food and kindred products, as published

6/ We have used a nonlinear estimation program that incorporates Marquardt's (1973) iterative procedure,

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by the U.S, Department of Commerce, Bureau of the Census, For en we used seasonally-adjusted data on end-use imports of foods, feeds and beverages, also published by the Bureau of the Census. Both cd and c"™ are deflated, as discussed below. For y we used seasonallyadjusted personal disposable income, as published by the U.S, Department of Commerce, Bureau of Economic Analvsi:, For p" we constructed a fixed-weighted average of unit value indexes for imports of crude foods and manufactured foods, which we then adjusted to include tariff rates; the unit value indexes are computed by the Bureau of the Census. A moving-weighted average of the same tariff-adjusted unit value indexes was constructedas a deflator for c™, with the moving weights appropriately reflecting changes in the composition of imports. fFor pe we used the wholesale price index for consumer foods, published by the Department of Labor, Bureau of Labor Statistics; and (because we could not easily construct a moving-weighted price index) we also used this fixed-weighted index as a deflator for of, Our choice of a wholesale price index for pt, rather than a consumer price index, was based partly on the fact that the former assigns a smaller weight to import prices, and partly on a desire to measure pt and p" at similar stages of marketing.

To conform with the discussion following our notational uci:uitions above, we expressed y, pt and p™ in terms of purchasing power over"all other consumables", the numeraire, The deflator for

personal disposable income was chosen to represent the price of "all other

onsumabies”,

Before estimation all variables were transformed into their natural logrithms. Quarterly data were used, with t running from fourth-quarter 1958 through second-quarter 1976.

IV. Empirical Results

Tables 1 and 2 present our empirical results. For each category of foods, unconstrained estimates of the 10 parameters are shown as case 1, and 5 sets of constrained estimates are shown as cases 2-6, Constraints are described in the left-most column of each table. The @ and 8 parameters are defined in the equations at the tops of the tables, corresponding to equations (1) and (2) of the model, and the \ parameters are defined in equations (3) - (5).

For each category of foods, cases 1-4 suggest that: (i) the data do not pin down hy within the interval between 0 and 1; and (ii) the data Support the hypothesis that A= Ag=Os or that expectations of future inflation are based solely on the most recent observation of actual inflation, The first of these conclusions is strengthened by unreported regressions in which Ay was respectively constrained to equal 0, .25, .5, and .75 (under the additional constraint that Apt Ag=0) - Although estimates of 4y are significantly different from zero in some cases, the unreported regressions fail to confirm that permanent income--as defined by the general form of equation (3) -- contains significantly more explanatory power than current income,

Cases 1-4 establish that the model fits the data well, The Standard errors of estimate are roughly 0.2 percent of the sample mean for domestically=produced foods and 0.9 percent for imported foods, Estimates of a and 8 parameters are similar in all four cases, The estimated magnitudes

of a, and By confirm that both domestically-produced and imported foods are

necessities;

10 -

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income elasticities are significantly greater than zero and significantly less than one. Insofar as a < Be the estimates

Suggest that as income grows, ceteris paribus, consumption of imported foods increases relative to consumption of domestically-produced foods. Cases 1-4 also provide estimates of the dock-strike parameters, C7, and Bes that support our prior expectations (recall the discussion following the presentation of equations (1) and (2) above). In particular, the estimates of 36 are not significantly different from one.

Estimates of the intratemporal own=price and cross-=price elasticities all have correct signs in cases 1-4, The intratemporal own-price elasticities, oy) and B3, are both significantly different from zero, as is the elasticity of demand for domestic foods with respect to the contemporaneous price of imports, O36 In addition, for each group of foods demand is estimated to be more sensitive to current own price than to current cross price; that is Q exceeds @, in absolute vaiue, and B3 exceeds Bo -

The most distinguishing feature of our model is the assumption that expected future prices play an explicit role in determining current demands, The data support this assumption, although our estimates of relevant parameters cannot be distinguished from zero with much statistical confidence, For each group of foods, current demand is positively related to the expected future rate of own-price inflation Coy, and Bs are positive in cases 1-4), confirming our prior beliefs

about the sign of intertemporal own-price elasticities, Elasticities

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of demand with respect to expected future rates of cross-price inflation (ar, and Bb are estimated to be negative. This is consistent with our estimates of the intertemporal own-price effects, insofar as the increase in current purchases of imported foods (respectively, domestically=produced foods) that is stimulated by an increase in the expected future rate of import-price inflation (resp., domestic price inflation) is likely to be partially offset by a decline in current purchases of domestically=produced foods (resp., imported foods). In this connection, since purchases of domestically-produced foods were roughly 20 times larger than purchases of imported foods during our sample period, it is appropriate that Be is less than 20 times as large as @, in absolute value, while it is inappropriate that all but the case2 estimates of @, are greater than one-twentieth as large as B. in absolute value.

Cases 5 and 6 provide additional support for the assumption that expected future prices influence current demands, particularly in the case of domestically-produced foods. When the role of expected future prices is suppressed (under the constraint =a=B =B5=0), Standard

4

errors of estimate increase, roughly doubling in the case of domestically-

i produced foods .~

J/ In addition, estimates of the dock-strike parameter,a@., become implausibly negative in the case of domestically-produced foods.

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V. Summary and Conclusions

By conventional standards our model fits the data well. For domestically-produced foods the standard error of estimate is roughly 0.2 percent of the sample mean, and for imported foods roughly 0.9 percent. Moreover, the estimated income and price elasticities have

correct signs and plausible magnitudes.

The distinguishing features of our model derive from the view that the problem of consumer choice should be posed in a multi-period context. In this context current consumer purchases depend on both current and expected future prices, as well as on initial wealth and the expected stream of future income, which Friedman (1957) has combined into the summary concept of permanent income.

The explicit introduction of expected future prices is a novel feature of consumer-demand estimation. We assume that expectations of future prices are based on observations of current and lagged prices, and for empirical purposes we assume a conveniently-simplified relationship between expected future prices and historically-observed prices. Nonlinear estimation techniques are used to explain quarterly data from 1958 through

mid-1976, a period during which food prices exhibited moderate cyclical variation relative to prices of other consumables.

| The form in which our model is estimated relates current purchases of foods to current and lagged price variables. In this sense our model is similar to conventional models of consumer demand. Unlike conventional models, however, our relationship between current purchases

and historically-observed prices involves structural parameters that

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describe the relationship between current purchases and expected future

prices. We interpret our empirical results as weak confirmation that current purchases do depend on expected future prices. Stronger confirmation should be pursued in other empirical investigations®/--not because there is much doubt of the theoretical presumption, but rather because of the importance

of relating current purchases to lagged prices in a manner that is

consistant with the underlying theory. References

Friedman, M. (1957): A Theory of the Consumption: Function, Princeton,

Princeton University Press.

Isard, P. (1975): "Dock Strike Adjustment Factors for Major Categories of U.S. Imports and Exports, 1958-1974," Federal Reserve Board, International Finance Discussion Paper No. 60.

> B. Lowrey and P.A.V.B. Swamy (1975): "Theory and Estimation of the Demand for Imports of Consumer Goods,'' Federal Reserve Board, International Finance Discussion Paper No. 61.

Marquardt, D.W. (1963): "An Algorithm for Least-Squares Estimation of

Nonlinear Parameters," Journal of the Society for Industrial and Applied Mathematics, 11; pp. 431-4.

Nerlove, M. (1967): "Distributed Lags and Unobserved Components in Economic Time Series," pp. 127-69 in W. Fellner et al.,

Ten Economic Studies in the Tradition of Irving Fisher, New York, John Wiley and Sons.

8/ Ideally our hypothesis should be compared and/or integrated with a model of order-delivery lags and, perhaps more importantly, with a model that recognizes that consumers may be slow to adjust their habits following changes in prices or other behavior-determining variables,