In the model discussed in Lecture Note 1, we assumed that utility was sepa- rable in the public good. In this
question, we will explore a different scenario. In particular, suppose instead that we can write u (c, l, g) = U (c, g) + V (l). That is, utility is now separable in leisure, not the public good. Assume that Uc, Ug, V ′ > 0 and Ucc, V ′′ < 0.
NOTE: Assume throughout this question that none of the NNCs ever bind.
(a) (5 marks) Set up the Lagrangian for the household's problem and use it to obtain the household FOC governing its consumption-leisure choice (i.e., analogous to equation (9) on p.11 of Lecture Note 1). Give an economic interpretation of the condition you found.
(b) (4 marks) For the special case of π = τ, draw a graph showing the determination of the household's optimal choice of consumption and leisure. Be sure to label carefully all components of the graph, including the axes, any curves or lines, and any intercept values.
(c) (12 marks) Suppose g increases, but τ doesn't change (i.e., we continue to assume as in part (b) that τ = π). Under what conditions on the utility function will c in- crease/decrease? What about l? Is it possible for both to increase or both to decrease? Explain intuitively your answers. In a graph similar to the one from part (b), illustrate the case where c increases.
(d) (8 marks) Suppose again that g increases, but assume this time that g = τ, so that τ increases by the same amount as g. Thus, we can now write the budget constraint as c = w(1 − l) + π − g. Under what conditions will c increase/decrease? What about l? Is it possible for both to increase or both to decrease? Explain intuitively your answers.
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