Consumer Demand—Separability and Commodity Aggregation

Abstract

A significant problem to address prior to empirical implementation of the theory of consumer demand is the problem of degrees of freedom. In particular, because of the large number of goods purchased by the consumer, there are literally thousands of individual goods and, therefore prices of related goods, one would need to include in the analysis to make it complete. Even for a modest number of commodities, there are still a relatively large number of elasticities to estimate requiring a relatively large data sample. This chapter considers the alternative approaches to this problem, though restrictions on price movements and separability and two-stage budgeting. The focus is on two-stage budgeting with an effort to clarify understanding of application to systems of demand functions. The main result is that either strong separability or homogeneous separability, in addition to weak separability, is required for consistent estimation of two-stage demand functions. The chapter concludes with a discussion on how to empirically implement two-stage budgeting for a complete system of demand functions.

Problems

• 3.1 Show that if preferences are homothetic then homogeneous preferences (utility function homogeneous of degree 1) preserve the same rank ordering of preferences. Also, show that the expenditure elasticities are all unity with homothetic preferences.

• 3.2 Why is the Consistency Requirement for two-stage budgeting important? If it doesn’t hold, what can we say about the resulting system of demand functions?

• 3.3 Discuss the differences between first-stage and second-stage demand elasticities. Why would we expect the compensated conditional demand elasticities to be more meaningful than the uncompensated conditional demand elasticities?

• 3.4 Verify that Eq. (3.30) is true starting from the definition as the ratio of Slutsky substitution effects. What will the restrictions be if preferences are homothetic? What does this imply about the relationship among compensated cross-price elasticities between groups for a given commodity?

• 3.5 Show that point-wise strong separability (i.e., utility function additive in individual commodities) yields the so-called Frisch Relations: $e_{ij} = - e_{i} w_{j} \left( {1 + \phi e_{j} } \right),\forall i \ne j$ and $e_{ii} = \phi e_{i} - e_{i} w_{i} \left( {1 + \phi e_{i} } \right)$. Discuss how these conditions can generate an entire matrix of price elasticities with knowledge only of budget shares, income elasticities, and the money flexibility. What can we say about these elasticities as the number of commodities increases?

• 3.6 Suppose there are three commodities: food consumed at home (FAH), food consumed away from home (FAFH), and non-food (NF). Suppose also that you are interested in testing whether FAH and FAFH can be treated as weakly separable from NF, i.e., whether the utility function has the structure, u = v[v₁(q_fah, q_fafh), v₂(q_nf)].

  • a. If you were to reject the null hypothesis of weak separability, what does this mean from an economic point of view? Does this make sense? (Hint: see Eq. 3.2).

  • b. Suppose that you found that the above commodity partition was indeed weakly separable. Is this restriction enough by itself for consistency of the two-stage maximization procedure?