(2) Over the last several years, a number of municipalities in California have adopted restrictions on the bags that grocers are allowed to give to...

(2) Over the last several years, a number of municipalities in California have adoptedrestrictions on the bags that grocers are allowed to give to customers. The nature ofthe restrictions varies by municipality, but generally a customer must either providehis 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 ofroughly one minute in the time required for “checkout” of each customer. Supposethat consumers value both consumption of groceries (c), leisure (`), and other goods(z) according to an ordinal utility functionU (c, `, z) = α log c + β log ` + γ log z.All consumers have 24 hours in a day, which they can devote to leisure, to work, orto shopping. For each hour a consumer works he or she earns a wage w. For eachbag of groceries purchased the consumer must spend 10 minutes in the absence ofbag 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 consumeralso has non-labor income x.a) Formulate the problem facing the consumer; in particular, how are constraintson time and money related?.b) Derive an expression for the marginal rate of substitution between groceries andleisure.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 asmuch non-labor income as poor consumers. Do the bag restrictions affect richand poor differently? (Explain precisely, using the Marshallian demand functionsyou’ve derived).