Exercício para o Modelo de NK com governo lista 7 - o autor da lista tem exercícios bons

# Monetary economics problem set 2 where ε w 1 is the

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Monetary Economics, Problem Set 2 where ε w > 1 is the elasticity of substitution between labor inputs in firms’ production function. L d t is is aggregate labor demand and W t = ˆ 1 0 W j t 1 - ε w d j 1 1 - εω is an index of the wages prevailing in the economy at time t . Given the wage W j fixed by union j , agents stand ready to supply as many hours on labor market j , L j t , as required by firms. Note that we assume that all households supply all labor types, that is, households are distributed uniformly across unions. This means that each household i supplies all the types of labour j and aggregate demand of labor type j is spread uniformly between all households. It follows that the individual quantity of hours worked, L t ( i ) = ˆ 1 0 L j t d j is common across households and we can denote it with L t . Households thus will have also a common labour input equal to ˆ 1 0 W j t L j t d j = L d t ˆ 1 0 W j t W j t W t - ε w d j. The labor market structure rules out differences in labor income between households without the need to resort to contingent markets for hours. Hence, under these assumptions all household are identical and we just have a single representative household-union (which comprises all the unions). Finally, assume nominal wage rigidities according to the Calvo (1983) mechanism. In each period a union j faces a constant probability 1 - θ w of being able to reoptimize the nominal wage to maximize the representative households lifetime utility. Question 5.1. Write down the problem of the representative household, assuming the period utility function has the following separable form, common across households: U t = C t ( i ) 1 - σ 1 - σ - L t ( i ) 1+ φ 1 + φ where C t ( i ) is agent’s i ’s consumption and L t ( i ) are hours worked. Solution: The period by period budget constrain could be written P t C t + Q t,t +1 B t + D t = L d t ˆ 1 0 W j t W j t W t - ε w dj + B t - 1 where D t is dividend income. Maximizing the utility function over consumption yields the usual Euler Equation Euler equation : 1 C σ t = β E t P t P t +1 (1 + i t ) 1 C σ t +1 . Note that the consumption problem does not need to take into account the Calvo framework. The problem of the household-union j is max W j * t E t s =0 ( θ w β ) s C t + s ( i ) 1 - σ 1 - σ - ´ 1 0 L j t + s dj 1+ φ 1 + φ 20

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M.Phil. 2015-2016, Macroeconomics Oleg I. Kitov Monetary Economics, Problem Set 2 subject to L j t + s = W j t W t + s - ε w L d t + s and the budget constraint above. Question 5.2. Solve for the optimal path of consumption and for the wage setting problem of the representative household-union j . Solution: The FOC over the optimal reset wage is E t s =0 ( θ w β ) s - ´ 1 0 L j t + s dj φ ( - ε w ) W j * t W t + s - ε w - 1 L d t + s W t + s + λ t + s (1 - ε w ) L d t + s W j * ( - ε w ) t W ε w t + s = 0 where λ t + s = u ( C t + s ) /P t + s = is the Lagrangian multiplier of the budget constraint and equals the marginal utility of nominal wealth. Now observe that ´ 1 0 L j t + s dj φ = - u ( L t + s ) , then one can write E t s =0 ( θ w β ) s u ( L t + s ) ε w W j * t W t + s - ε w L d t + s W j * ( - 1) t - λ t + s ( ε w - 1) W j t W t + s - ε w L d t + s = 0 and since: 1) λ t + s u ( L t + s ) = mrs t + s the marginal rate of substitution between consumption and labour is

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