Riemannian ConsumersFrançois Gardes Université Paris I-Panthéon Sorbonne, CES-Cermsem, CREST-LSMgardes@univ-paris1.frpersonal page

The difference observed between the social distribution of consumer expenditures and their change over time is modelled using Riemannian geometry. Social distribution is measured along the geodesics of Riemannian surfaces, while changes over time correspond to movements along the tangents of these Riemannian surfaces. The Riemann curvature of the consumption space is shown to be non-null for the Polish consumers as surveyed in a four years Polish panel. This implies that usual econometric methods based on a unique metric over the whole consumption space are inadequate to estimate geodesics on the Riemannian surface. In order to propose an alternative, we define a synthetic time axis in the space of the variables which are observed in cross-section. Considering the relative position of two individuals along this time dimension allows us to estimate equations of geodesics. Also, an instrumentation using this synthetic time axis is proved to be very efficient compared to usual instrumentation for dynamic models on panel data.

Content 1. Cross-section and time-series estimates 2. Shadow prices, latent variables 3. Geometric representation of the surveys 4. Riemannian curvature of Polish surveys 5. Related econometric problems 6. Synthetic time on cross-section 7. Application to the estimation of ynamic

models 8. Estimation of geodesics on cross-section data 9.Conclusion

1. Cross-section and time-series estimates

Economic relationships appear sometimes to differ, either when considered in time dimension or as a-temporal relations, or in the short and in the long period. This last difference is generally related, either to biases in the estimation of long term relationships, or to specification biases due to the dynamic structure of the model. The difference between cross-section and time-series estimates, since recognized early, is not so well accepted by the profession, as it implies to refuse to cross-section estimates the ability to anticipate the effect of evolutions in the explanatory variables (differences between agents observed in the same period bear a different information than changes between two periods for an individual or a population).

Difference between two estimators

This difference has been advocated as coming from aggregation biases, at a time when no individual panel exist. Specification biases (whenever the estimates performed on surveys do not take into account dynamic behavior, such as habits or addiction in consumption functions) and different effects of errors in variables in the two dimensions are also considered to challenge this evidence. On panel, it is possible to take care of these difficulties and to show that differences still appear between estimations in the time and spatial dimensions.

Riemannian Geometry We propose to consider that the spatial relationships

correspond to geodesics in a Riemannian space, while time relationships are modelized as displacements on the tangent surfaces to the Riemannian space. Quite curiously, Riemannian geometry does not seem to have been applied to economic problems (it appears recently in theoretical statistics). Thus tensor algebra allows to analyze the difference between the estimations on both dimensions, and to compute the curvature of the Riemannian space. When this curvature is not nul (i.e. when integrability conditions, which make it possible to define a common euclidian metric for all points of the space, are not verified), the space is no more Euclidian and the shortest way between two point are not lines, but geodesics, which gives rise to new econometric problems.

Differences between cross-section and time-series

estimates of demand functions have been observed in recent empirical works: for instance, Gardes et al. (2002) analyse the bias to income and total expenditure food elasticities estimated on panel or pseudo-panel data caused by measurement error and unobserved heterogeneity. Our results suggest that unobserved heterogeneity imparts a downward bias to cross-section estimates of income elasticities of at-home food expenditures and an upward bias to estimates of income elasticities of away-from-home food expenditures. Moreover, the magnitude of the differences in elasticity estimates across methods of estimation is roughly similar in U.S. and Polish-based expenditure data.

Table 1 Relative Income Elasticity of Food

Expenditures

PSID (U.S.) Polish panel Period 1984-87 1987-90 N 2430 3630 Prices no by region and social category Income Elasticity CS TS CS TS Food at home 0.19 0.38 0.49 0.76 Food away 1.00 0.39 1.22 0.36 Direct Price Elasticity -0.5 -0.5 Elasticity of the Shadow (i) F.H. 0.38 0.54 Price Relative to Income(ii) F.A. –1.22 -1.72 Reference: Gardes, Duncan, Gaubert, Starzec, Journal of

Economic and Business Statistics, 2005, January

2. Measuring the Shadow Prices Suppose that monetary price pm and two shadow prices

corresponding respectively to non-monetary resources pnm and to constraints faced by the households pcs are combined together into a complete price. Expressed in logarithm form, we have:

pc = pm + with = pnm + pcs as it is not possible with usual data sets to distinguish between the two components of the shadow price.

Suppose that two estimations of the same equation :xiht = Zht.i + uiht (equation 1)

with uiht = ih + iht. As discussed by Mundlak (1970), the cross-section estimates can

be biased by a correlation between some explanatory variables Z1ht and the specific effect. Such a correlation is due to latent permanent variables :

ih = BZ1ht.i + ih will add to the parameter i of these variables in the time average estimation : Bxiht = BZ1ht.(i + i) + ih + Biht, so that the between estimates are biased. The difference between the cross-section and the time-series estimates amounts to i.

Measuring the Shadow Prices From the shadow price function

iht = fiht (Z1ht, Z2ht) = Ziht.1 + Z2ht.2 + ih + iht we define a vector of endogeneous variables Z1ht as a vector of all variables correlated with the specific effect ih . For instance, the relative income position of the household, supposed to be invariant, can determine its location, which is correlated with purchase constraints.

The marginal propensity to consume with respect to Z1ht, when considering the effect of the shadow prices jht on consumption, can be written:

dxiht/dZ1ht= dgi/dZ1ht + j (dgi/djht).(djht/dZ1ht). This allows to reveal djht/dZ1ht knowing dgi/djht and

dxiht/dZ1ht, dgi/dZ1ht We can also consider only the direct effect through the price of

good i : ii.di/dZ1 with dgi/dj = ij., so that: di/dZ1 = [i

c.s. -it.s]/ii

3. Geometric representation of the surveys

x=f(z, Z1, Z2) z= observed variables in the surveys (n1) Z1=Latent variables, permanent (education level, parental

influence…) (n2) Z2=Latent variables, changing (education, social

interactions, macro variables…) Survey= (x, z) -> surface in Rn1+n2

Comparison of two points (agents) in the survey/ two periods for the same agent: Changes of Z1, Z2 -> conditional demand functions f| Z1, Z2

Assumptions: homogenous population, constant demand function (constant preferences and choice sets): regular change of latent variables according to observed variables