PS1solutions

# PS1solutions - Economics 1011b Problem Set 1 Professor Aleh...

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Economics 1011b Problem Set 1 Professor Aleh Tsyvinski The problem set is due next Tuesday, February 20, by 5 pm in your TF’s mailbox in Littauer. Late problem sets will not be accepted. You can work in groups (discuss the solutions, etc). However, you must write the solutions by yourself. Please write down all derivations/explanations. Exercise 1 Consumption, Work, and Production We introduced in class a simple economy and the decision problem faced by the household. Here is one example. Assume that the utility function is u ( c,l ) = c 1 - γ - 1 1 - γ + ( T - l ) 1 - σ - 1 1 - σ , γ,σ > 0, where l is labor, and T is the total time endowment. The production function is f ( l ) = Al α . 1. What condition on α must be satisﬁed in order to have decreasing marginal productivity of labor? Marginal productivity of labor is f 0 ( l ) = αAl α - 1 . Therefore f 00 ( l ) = α ( α - 1) Al α - 2 . This is negative when 0 < α < 1. 2. What happens to the marginal utility of consumption (leisure) as con- sumption (leisure) goes to zero? What is the intuition for this property? Does it make sense? Discuss. These are also called ”Inada conditions”. u ( c,l ) = c 1 - γ - 1 1 - γ + ( T - l ) 1 - σ - 1 1 - σ ,γ,σ > 0 u c = c - γ = lim c 0 c - γ = u T - l = ( T - l ) - σ = lim T - l 0 ( T - l ) - σ = As leisure ( T - l ) goes to zero, the marginal utility of an extra unit of leisure goes to inﬁnity. The intuition is that you value a unit of leisure more the less that you have. Leisure has diminishing marginal utility. 3. Represent the decision problem graphically in a consumption/labor di- agram (i.e. show the production possibility frontier and indiﬀerence curves). How is the Inada condition on leisure represented by the indiﬀerence curves? 1

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PPF Indifference curve c Asymptotes toward dashed line T l 4. Write the ﬁrst order conditions for the maximization problem. The Lagrangian is: L = c 1 - γ - 1 1 - γ + ( T - l ) 1 - σ - 1 1 - σ + λ ( c - Al α ) The ﬁrst order conditions for this are: dL dc = c - γ + λ = 0 dL dl = - l - σ + λ ( - αAl α - 1 ) = 0 5. Fix c,l and compute the limit of u ( c,l ) when γ,σ 1 (Hint, you will need to use l’Hopital’s rule and if f ( γ ) = c 1 - γ - 1, then f 0 ( γ ) = - c 1 - γ log( c )). u
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PS1solutions - Economics 1011b Problem Set 1 Professor Aleh...

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