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OLG_RQ - Review Problems OLG Model Prof Lutz Hendricks...

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Review Problems: OLG Model Prof. Lutz Hendricks. August 6, 2009 1 A Savings Function Consider the standard two-period household problem. The household receives a wage w t when young and a rate of return R t +1 on savings. (a) Illustrate the household°s intertemporal budget constraint and the optimal choice of consumption in a diagram. Label it clearly. (b) If the wage rate rises, what happens to savings? What if the household re- ceives an additional endowment when old? Explain and illustrate in your diagram. No math please. (c) Derive the consumption function c ( w t ; R t +1 ) and the savings function s ( w t ; R t +1 ) for the utility function u ( c t ; x t +1 ) = c ° t + x ° t +1 where 0 < ° < 1 is a constant. (d) Do the same for the utility function u ( c t ; x t +1 ) = A ° ln( c ° t + x ° t +1 ) ; 0 < A < 1 What do you ±nd and why? 1.1 Answer: Saving Function (a) This is the standard diagram of indi/erence curve and budget line. (b) If both goods are normal, an increase in the wage rate or in the endowment when old leads to higher consumption at both ages. Therefore: w ") s " but y 2 ") s # (c) The household solves: max c ° + x ° 1
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subject to the budget constraint c + x=R = w . The ±rst-order condition is u c = Ru x or °c ° ° 1 = R°x ° ° 1 Thus x=R = cR °= (1 ° ° ) Substituting this into the budget constraint yields c = w= (1 + R °= (1 ° ° ) ) and s = w (1 ± 1 = [1 + R °= (1 ° ° ) ]) (d) The answer is exactly the same. The utility function in (c) is a monotone increasing transformation of the one in (d). 2 Log-utility Example Consider the standard two-period OLG model with log utility: U ( c yt ; c o;t +1 ) = ln c yt + ± ln c o;t +1 . 1. Solve for the household°s saving function. 2. Find a law of motion for k t = K t =L t . 3. Show that the economy has a unique steady state (not counting k = 0 ), if (i) the old do not work ( = 0) and (ii) the production function is Cobb-Douglas: F ( K t ; L t ) = K ° t L 1 ° ° t . 2.1 Answer: Log-utility example 1. Saving function The Euler equation becomes c o;t +1 c y;t = ± R t +1 (1) A solution to the household problem is a vector ( C y;t ; C o;t +1 ; A t +1 ) that satis±es the Euler equation and two budget constraints. 2
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The present value budget constraint is c y;t + c o;t +1 R t +1 = ° W t = W t + ‘ W t +1 R t +1 (2) Substitute the Euler equation into the budget constraint to obtain c y;t = ° W t 1 + ± (3) c o;t +1 = ± ° W t 1 + ± (4) A t +1 = s ( W t ; ‘W t +1 ; R t +1 ) = W t ± ° W t 1 + ± (5) 2. Law of motion. Capital market clearing requires K t +1 = N t A t +1 = N t ° ± 1 + ± W t ± ‘W t +1 (1 + ± ) R t +1 ± (6) From the ±rm°s problem we know that W t = W ( k t ) with W 0 > 0 and R t = R ( k t ) with R 0 < 0 . Dividing through by N t therefore yields a law of motion for k : k t +1 = K t +1 L t +1 = K t +1 L t (1 + n ) = K t +1 N t (1 + n ) N t L t = s ( W ( k t ) ; ‘W ( k t +1 ) ; R ( k t +1 )) N t =L t 1 + n where L t = N t + ‘N t ° 1 = N t (1 + ‘= [1 + n ]) is just a constant. 3. Unique steady state. With = 0 the saving function simpli±es greatly (b/c the interest rate does not a/ect lifetime earnings ° W ): A t +1 = ± 1 + ± W t (7) Moreover, N t = L t . The law of motion for k becomes k t +1 (1 + n ) = ± 1 + ± W ( k t ) (8) As long as the production function is such that W 0 > 0 and W 00 < 0 , there can only be one solution with k t +1 = k t > 0 .
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