# Practice_Problems_2_with_answers - Economics 102 Winter...

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Economics 102, Winter, 2010 Midterm Exam # 2 practice questions Professor Lee Ohanian (1) Show that in the Solow model in which population grows at rate n, that the Golden Rule requires that the marginal product of capital be equal to the sum of the depreciation rate and the growth rate of the population. Why is the golden rule MPK higher for a higher population growth rate? Let°s start with the law of motion of capital in aggregate terms K t +1 = I t + (1 ° ° ) K t in per-capita terms we have K t +1 L t = I t L t + (1 ° ° ) K t L t This equation can be re-writte as: K t +1 L t +1 L t +1 L t = I t L t + (1 ° ° ) K t L t in the new variables is: k t +1 (1 + n ) = i t + (1 ° ° ) k t In the steady-state (we drop the time subscripsts) k ( ° + n ) = i Investment per-capita be written as output minus consumption per-capita k ( ° + n ) = y ° c Hence c = y ° k ( ° + n ) The optimal condition for the golden rule is when we derive respect to capital @c @k = @y @k ° ( ° + n ) If we make the derivative equal to zero (which is the neccesary condition for a maximum) @y @k ° ( ° + n ) = 0 MPK = ( ° + n ) 1

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When n is higher, golden-MPK is higher due to shrinking capital stock per worker, because of higher amount of investment necessary to provide higher number of new workers with capital. k is spreaded thinly among a larger number of workers (2) Show that in the Solow model in which there is constant growth in worker e¢ ciency that a steady state exists for the variable Y= ( NE ) ; and that once the steady state is reached that per capita output grows at rate g: Start with the production function Y = K ° ( NE ) 1 ° ° Divide through NE Y NE = ° K NE ± ° ° NE NE ± 1 ° ° re-de±ne the variables y = k ° where y = Y NE and k = K NE : The law of motion of capital can be re-expressed as: ° k = i ° ( ° + n + g ) k hence, we can ±nd a steady state in the new variables de²ate by e¢ cient labor units. From here we know that y
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