ECON 222, Spring 2009 Assignment #3, Answer Key

Transcription

1 ECON 222, Spring 2009 Assignment #3, Answer Ke Question (40 marks) a) TF growth rate is constant and equal to At A t a, and the population growth rate is Nt N t n: From the growth-accounting equation we know that: A t A t + + ( ) N t N t which boils down to: a + + ( ) n The output-per-worker growth rate is just the di erence between the output grow rate and the population growth rate: N t N t Which reduces to: a + + ( ) n n K t Kt a + n () Bhe same logic: N t N t + n n

2 Substituting this last result in () gets: kt a + + n n kt a + + n n kt a + as required. Thus, real GD per worker grows because of two reasons: the growth in the technological progress a and the growth of capital per worker. b) In our analsis in class, where the TF was xed, increases in k led to reductions in the average product of capital t At A t kt. Hence, in the long run, the econom approached a stead state in which the average product of capital was low enough, so that kt 0: Then, with a 0, we had 0: The di erence now is that each increase in A raises the average product of capital, for given k. Hence, the negative e ect of rising k on k is o set b a positive e ect from rising A. The econom will tend toward a situation in which these two forces balance. That is, k will increase in the long run at a constant rate, and k will be unchanging. We call this situation stead-state growth. Since the average product of capital k does not change during stead-state growth, the numerator and the denominator are going to grow at the same rate. Therefore we have: k k where the " " refers to values in stead-state growth. This equation tells us that output and capital per worker are going to grow, eventuall, at the same rate. Hence we can substitute this result in (2) to get: k a + k a + a This is the expression for the stead-state output growth. Notice that it is positive (and nite), since a > 0 and 0 < < : Moreover, this equation tells 2

3 us that the stead-state growth rate or real GD per capita,, is greater than the rate of technological progress a. c) The production function can be written in per e ective worker terms b dividing both sides b A t N t (A t N t ) (A t N t ) : A t N t k t : Kt (A t N t ) (A t N t ) (A t N t ) Since S t s ;for all t, and since we are in a closed econom, S t I t. Therefore, I t s sk t (A t N t ), and i t s sk t : d) Such a level of investment per e ective worker is required in the stead state so as to maintain the level of capital per e ective worker constant at k. This stems from the ver de nition of the stead state, which characterizes a balanced growth equilibrium where output and capital per e ective worker are constant, while output and capital grow at a constant rate. More precisel, this level of investment is required to compensate for depreciation (at rate d) of the capital stock per e ective worker, its decrease due growth in the labour force at rate n, and its decrease due to growth in productivit at rate g: From c) and, we have the following stead state condition: And therefore, s(k ) (d + g + n)k k s (d + g + n) (k ) s (d + g + n) c ( s) ( s) s (d + g + n) The graph is as follows. Notice the di erence in the investment schedule and that the x-axis is now in capital per e cienc units: 3

4 e) The golden rule level of k, k g is the point where consumption is maximized. Maximizing consumption wrt k (subject to the constraint that savings is equal to investment): max k c k (d + g + n)k dc dk k g k (d + g + n) 0 (d + g + n) whereas, from part c) we know that in stead state, k s (d+g+n) : So, at the golden rule, the saving rate is s g. If we are below this saving rate (i.e. s < s g ) standards of living can be raised b increase the current saving rate. The short term e ect will be to slightl decrease consumption, while it will increase in the long run. On the other hand, if s > s g, the standard of living could be increased b lowering savings. Question 2 (30 marks) a) i) Real mone demand is given b: M d :2Y 000i :2(000) 000(0:) 600: 4

5 Nominal mone demand is therefore M d ( M d ) : Recall, velocit can be calculated (in equilibrium) b: V Y :67: M d ii) When the price level doubles, real mone demand does not change since neither Y or i have changed. Nominal mone demand, however, now becomes M d ( M d ) new : Velocit is also unchanged since V Y Y and neither Y nor real mone demand has changed. M d ( Md ) iii) From part ii), we know that we can write velocit as V Y M d Y Y 500+0:2Y 000i ( Md ) : Then, the e ect of an increase in real income (when i 0:0): dv dy V Y :2Y 400 ( :2Y ) 2 > 0: So, when Y increases, V increases. The e ect of an increase in the nominal interest rate (when Y 000): V dv di i ( i) 2 > 0: So, when i increases, V increases. As for the e ect of an increase in the price level, note that we can write velocit as just a function of Y and i, so an increase in the price level has no e ect on velocit. The reason is that nominal mone demand changes proportionall with the price level, so that real mone demand and hence velocit, is unchanged. b) i) If the nominal mone suppl is expected to grow at a rate of 0% per ear, then e M M 0:0: Bhe Fisher equation, i r+e 0:05+0:0 0:5:In equilibrium, the real mone suppl is then given b: M L 0:0 50 0:5 0: 5

6 And, thus the price level can be found: M L : ii) Now, e M M 0:05, and i r + e 0:05 + 0:05 0:0: It follows that M L 0:0 50 0:0 5: M L : The slowdown in mone growth reduces expected in ation, increasing real mone demand and thus lowering the price level. Question 3 (30 marks) a)to nd the IS curve, start from the de nition of desired saving and using T 50 and G 00: S d Y C d G Y 300 0:7(Y 50) + 75r 00 Setting S d I d, we get: :3Y + 75r 85 75r r 3:2 0:002Y (IS): Solving the mone demand equation in terms of r, ields: 00r 0:6Y (M ) 00 e r 0:006Y 0:0(M ) e (LM) Finall, setting the IS and LM curves equal ields: Y :25(M ) + 25 e (AD) The AD curve slopes downward in terms of Y and, because a fall in the price level shifts the LM curve down and to the right, lowering interest rates and increasing output, holding other variables constant. 6

7 b) If r 0:05, then use the IS curve to solve for Y : 0:05 3:2 0:002Y Y 575 Given the interest rate and output, and assuming e 0, we can use the LM curve to get the level of M (since ): 0:05 0:006Y 0:0M M 940 From the consumption and investment functions we have: C d :7(Y T ) 75r :7(575 50) 75(0:05) 293:75 I d 85 75r 85 75(0:05) 8:25 c) In the short run, the rise in e, with the mone suppl constant, shifts the LM curve to the right and causes the real interest rate to fall. With the IS curve una ected, the new, short-run equilibrium is found where these two curves intersect. Thus we can use the AD curve to get the new level of output: Y :25(M ) + 25 e :25(940) + 25(0:02) 577:5 Now that Y is known, we can use the IS relation to get the short-run level of r: r 3:2 0:002Y 3:2 0:002(577:5) 0:045 Since the nominal mone suppl is unchanged, and given that e is transitor, the old equilibrium is restored. The price level will not be a ected in long-run equilibrium. 7

8 Show the usual graph, LM shifting down and then back. d) Lowering taxes b 0 will a ect the IS curve. Starting from the relationship for private saving, we have: S d Y C d G Y 300 0:7(Y 40) + 75r :3Y + 75r which, given the investment equation above (I d 85 75r), ields the IS curve: r 3: :002Y: Given an unchanged LM curve, the new AD relationship is: Y 405: :25(M ) 405:8333 :25(940) 580:8333 Given this value of Y and the new IS curve, we get r 3: :002Y 3: :002(580:8333) 0:085: These levels of Y(580.83) and r(0.085) are the new short-run equilibrium levels. Show the usual graph with onlhe IS curve shifted. In the long run, the econom returns baco its equilibrium level because the LM curve will shift (from short-run equilibrium, Y > Y F E, so the price level will start to rise). With taxes now lower, the government sector is making a larger claim on resources and this has a ected the IS curve. We can use it to solve for the new interest rate, and use it to nd the new level of desired consumption and investment: r 3: :002Y 3: :002(575) 0:0966 C d :7(Y T ) 75r :7(575 40) 75(0:09666) 297:25 I d 85 75r 85 75(0:09666) 77:75: Both consumption and investment have fallen compared with the original levels ( and 8.25, respectivel). Regarding investment, there has been some crowding out (a fall of 3.5) but it is not complete as consumption made of the rest of the di erence. Show IS curve shifting up and then LM shifting back as price level rises to a new equilibrium with a higher interest rate. 8