a) Present the Lagrangian (constrained maximisation) problem for the household under this modified specification and derive the first order conditions in this case. Hint: the household chooses c1, c2, ℓ1 and ℓ2. b) Use the first order conditions for ℓ1 and ℓ2 to derive an expression for the relative amount of leisure time chosen by the household over the two periods, i.e. derive an expression for (1–ℓ1)/(1–ℓ2). Explain how an increase in the relative wage (w2/w1) affects the household’s decision about how much leisure to enjoy in each period. c) Calculate the intertemporal elasticity of substitution between period 1 and period 2 leisure time in this case. Explain how the magnitude of this elasticity influences the household’s decision-making in the model. d) Use the first order conditions for c1 and c2 to derive an expression for the relative amount of consumption chosen by the household over the two time periods, i.e. derive an expression for c2/c1. Provide an economic interpretation to accompany your answer.
a) Present the Lagrangian (constrained maximisation) problem for the household under this modified specification and derive the first order conditions in this case. Hint: the household chooses c1, c2, ℓ1 and ℓ2.
b) Use the first order conditions for ℓ1 and ℓ2 to derive an expression for the relative amount of leisure time chosen by the household over the two periods, i.e. derive an expression for (1–ℓ1)/(1–ℓ2). Explain how an increase in the relative wage (w2/w1) affects the household’s decision about how much leisure to enjoy in each period.
c) Calculate the intertemporal elasticity of substitution between period 1 and period 2 leisure time in this case. Explain how the magnitude of this elasticity influences the household’s decision-making in the model.
d) Use the first order conditions for c1 and c2 to derive an expression for the relative amount of consumption chosen by the household over the two time periods, i.e. derive an expression for c2/c1. Provide an economic interpretation to accompany your answer.
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