Over the last several years, a number of municipalities in California have adopted restrictions on the bags that grocers are allowed to give to customers. The nature of the restrictions varies by municipality, but generally a customer must either provide his or her own bags, or the store is required to charge some minimum price for bags. A recent paper finds that one of the consequences of these policies is an increase of roughly one minute in the time required for “checkout” of each customer. Suppose that consumers value both consumption of groceries (c), leisure (l), and other goods (z) according to an ordinal utility function
U(c, l, z) = α log c + β log l + γ log z.
All consumers have 24 hours in a day, which they can devote to leisure, to work, or to shopping. For each hour a consumer works he or she earns a wage w. For each bag of groceries purchased the consumer must spend 10 minutes in the absence of bag restrictions, and pay a price p per bag. Take “other goods” z to be num ´eraire (i.e., the price of z is one). Suppose that in addition to labor earnings the consumer also has non-labor income x.
a) Formulate the problem facing the consumer; in particular, how are constraints on time and money related?.
b) Derive an expression for the marginal rate of substitution between groceries and leisure.
c) Calculate Marshallian demand functions for groceries, leisure, and other goods. How do these depend on bag restrictions?
d) Suppose that consumers are all identical, but that rich consumers have twice as much non-labor income as poor consumers. Do the bag restrictions affect rich and poor differently? (Explain precisely, using the Marshallian demand functions you’ve derived).