Economics 2 Spring 2017 Professor Christina Romer Professor David Romer SUGGESTED ANSWERS TO PROBLEM SET 5 1. The tool we use to analyze the determination of the normal real interest rate and normal investment is the long-run saving and investment diagram. Investment demand is determined by firms decisions about how much new capital to buy, and is a decreasing function of the real interest rate. That is, I is a decreasing function of r*. The supply of saving, S, equals, Y* G* C*. Since a rise in r* increases the opportunity cost of current consumption in terms of future consumption, current consumption is a decreasing function of r* which means that S is an increasing function of r*. In the diagrams below, and are the initial investment demand and saving supply curves. a. As noted above, normal national saving (S*) is potential output (Y*) minus normal consumption (C*) and normal government purchases (G*): r* S* = Y* C* G*. If the government increases normal government spending (and leaves normal tax revenues unchanged), national saving will certainly decrease at every level of the real interest rate. This decrease in saving at every level of the real interest rate corresponds to a shift back in the saving function (from to ). The shift back in the saving function leads to an increase in the equilibrium real interest rate (from r 1 to r 2 ) and a decrease in normal investment (from to I 2 ). r 2 r 1 I 2 S*,I* b. Again, normal national saving equals potential output minus normal consumption and normal government purchases: S* = Y* C* G*. If a change in attitudes causes consumers to permanently consume more at a given level of the real interest rate, C* at a given r* will be higher than before. Thus, Y* C* G* at a given r* will be lower than before. As in part (a), there will therefore be a shift back in the saving function (from to ). The shift back in the saving function again leads to an increase in the equilibrium real interest rate (from r 1 to r 2 ) and a decrease in normal investment (from to I 2 ). r* r 2 r 1 I 2 S*,I*
2. We are thinking about the market for low-skilled workers in normal times (that is, in the long run). Therefore, we want to think about the supply and demand relationships between the normal employment of low-skilled workers (N*) and the normal real wage (). There is an upward-sloping labor supply curve () that comes from utility maximization on the part of households. There is a downward-sloping labor demand curve () that comes from profit maximization on the part of firms. The labor demand curve in this set-up reflects the real marginal revenue product of low-skilled workers (MRP L). Because there is a negotiated wage () that is above the equilibrium wage, the quantity of lowskilled labor employed in normal times (N D1 ) will be determined by the quantity of labor demanded at the negotiated wage. Since the quantity of labor supplied at the negotiated wage (N S1 ) is greater than the quantity demanded, there will be unemployment () in normal times. N D1 N S1 N* 2 a. If the quantity of capital in the sector using lowskilled labor declines, low-skilled workers will be less productive. Thus, the demand for low-skilled workers will shift to the left (from to D 2). Because the wage is determined by the negotiated wage, it will not change. The number of low-skilled workers who are employed will fall (from N D1 to N D2 ). The quantity of low-skilled labor supplied will not change, and so unemployment will rise (from to Unemployment 2). D 2 N D2 N D1 N S1 N* Unemployment 2 b. When the negotiated wage is above the level where the quantity of labor demanded and the quantity supplied are equal, employment is determined by the quantity of demanded at the negotiated wage. Thus if the negotiated wage rises (from 1 to 2 ), employment will fall (from N D1 to N D2 ). In addition, the increase in the wage causes the quantity of labor supplied to rise (from N S1 to N S2 ). Thus unemployment will rise (from to Unemployment 2). 2 1 N D2 N D1 N S1 N S2 N* Unemployment 2
c. A reduction in the number of workers who are willing to work at a given wage corresponds to a shift to the left of the labor supply curve. The left-hand diagram below shows the normal case: the fall in labor supply (from to ) is not so large as to push the wage where supply and demand are equal above the negotiated wage. In this case, the wage and employment are unchanged (at and N D1, respectively). Because the number of workers who want to work at the negotiated wage falls (from N S1 to N S2 ), unemployment falls but is not eliminated (from to Unemployment 2). The right-hand diagram shows the case of a very large decrease in labor supply. In this case, the wage is now determined by the intersection of supply and demand. It is higher than the negotiated wage. Thus the wage rises (from to w 2 ), employment falls (from N D1 to N 2 ), and unemployment is eliminated it falls from to zero. 3 w 2 N D1 N S2 N S1 N* N 2 N D1 N S1 N* Unemployment 2 3. Planned aggregate expenditure () for a country is the total amount of spending people plan to do. It is the sum of consumption (C), the total amount consumers want to spend; planned investment (I P ), the amount firms plan to invest; government purchases (G); and net exports (NX), the amount foreigners want to buy from us minus the amount Americans want to buy abroad. Thus, = C + I P + G + NX. The expenditure line (also identified as ) shows how planned spending varies systematically with total output. It is upward sloping because consumption rises with total output (which is the same as total income). Its slope is less than one because people typically save at least part of every extra dollar of income they receive. In the short run, equilibrium output is determined by the intersection of the expenditure line and the 45-degree line. The 45-degree line represents the equilibrium condition that total output must equal total spending (Y = ) for the economy to be in balance. This line also captures the behavioral assumption that firms change output in response to changes in planned spending in the short run.
4 a. Adding up the components of planned expenditure in this numerical example yields: = 500 + 1000 + 2000 + 500 + 0.6Y. Therefore, the equation for the expenditure line is: = 4000 + 0.6Y. 10000 Y = (45-degree Line) (Expenditure Line) The expenditure line has an intercept of 4000 and a slope of 0.6. The coefficient on output in the 4000 equation reflects the sensitivity of consumption to output. This coefficient is called the marginal propensity to consume (MPC). In this example, the MPC is 0.6, which means that if consumers get 10000 Y another dollar, they will spend 60 cents of it and save 40 cents. If you draw the expenditure line and the 45-degree line carefully, equilibrium output in the short run appears to be around 10,000. b. The two equations that determine equilibrium output are Y = and = 4000 + 0.6Y. Therefore, to solve for equilibrium output algebraically, all one does is substitute the second equation into the first. This yields: Y = 4000 + 0.6Y (1 0.6)Y = 4000 Y = 4000/0.4 Y = 10000 c. If government purchases decrease to 1000, this changes the equation for the expenditure line to: Y = 1 = 3000 + 0.6Y. Graphically, this is a shift down in the expenditure line by 1000 at each level of Y (from 1 to 2). The new level of equilibrium output looks to be about 7500 in the graph. Algebraically, the new level of equilibrium output is determined by calculating: 10000 8000 7000 4000 3000 2 Y = 3000 + 0.6Y (1 0.6)Y = 3000 Y = 3000/0.4 Y = 7500 5000 10000 Y 2 Y 1 Y d. Output decreases by more than the decrease in government purchases because of the multiplier effect. The fall in government purchases lowers output. The decrease in output reduces planned spending further because consumption depends on output. The further fall in planned spending decreases output further, and so on. The algebraic formula for the multiplier is 1/(1 marginal propensity to consume). So, in this example the multiplier is 1/(1 0.6), which is 2.5. Notice, the size of the multiplier effect depends positively on the size of the marginal propensity to consume. This should make sense the larger the marginal propensity to consume, the greater the feedback effect of output on consumption.
5 4. As discussed in the answer to question 1, we use the long-run saving and investment diagram to analyze the behavior of the normal real interest rate and normal investment. This problem describes two developments. (1) First, firms believe reduced regulations will cause the marginal revenue product of capital to be permanently higher. This change in firms expectations raises the present value of the stream of expected future MRP K s at a given interest rate, and so causes firms to buy more new capital at a given interest rate. By doing so, they move down the declining marginal revenue product of capital curve and restore the profit-maximization condition to equality. This corresponds to a shift r* r 2b r 2a r 1 I 2a I 1 I 2b out in the investment demand curve. The diagram shows two possible shifts a small one (from to I 2a) and a large one (from to I 2b). (2) Second, the government permanently cuts taxes. Long-run saving is Y* C* G*. The cut in taxes does not affect Y* or G*. But it does affect C*: consumers disposable income when output is at potential, Y* T, is higher; as a result, this causes consumption at a given interest rate to rise. Thus, the supply of saving at a given interest rate, Y* C* G*, falls. This corresponds to a shift back in the supply of saving (from to ). As the diagram shows, each of the changes tends to increase the economy s normal real interest rate. Thus, the normal real interest rate unambiguously rises. However, the first change operates in the direction of raising the normal amount of investment while the second change operates in the opposite direction. The problem does not provide enough information to tell which effect is larger. For example, in the diagram, normal investment falls with the small shift in investment demand but rises with the large shift in investment demand. Thus, normal investment can rise, fall, or stay the same. 5.a. The line shows planned aggregate expenditure as a function of output (or, equivalently, as a function of income). Normally, we assume that only one of the components of (consumption) varies as output changes, and that the other three components (investment, government purchases, and net exports) are not functions of output. In this case, the line slopes up because as output rises, consumption rises. Thus, the slope of the line is given by the MPC: when Y rises by 1 unit, C rises by the MPC and the other components of are unchanged, and so rises by the MPC. With the new policy, now two components of vary with Y: C rises as Y rises (as before), but now G falls as Y rises. As a result, when Y rises by 1 unit, rises by less than the MPC: C rises by the MPC, but G falls (and I and NX do not change), and so the total rise in is less than the MPC. Thus, the change reduces the slope of the line. I 2a S*,I* I 2b The problem says economy starts with output equal to its normal or potential level, and that the government does not change what purchases are when output is equal to potential. Thus we can be more specific about how the line changes. First, the information given implies that the initial line intersects the 45-degree line at Y = Y*. And second, it implies that at Y = Y* is the same as before, and thus that the new line also intersects the 45 degree line at Y = Y*. Thus, the 2 1 Y = Y* Y
line rotates clockwise around the point on the 45-degree line where Y = Y* from 1 to 2. (In the case shown, the line becomes flatter but still slopes up. However, there is nothing in the problem to rule out the possibility that the slope becomes negative. If the amount that the government reduces its purchases when Y rises by 1 unit is greater than the MPC, then total will actually fall rather than rise when Y rises. In this case, the slope of the line becomes negative.) If we want to (although this certainly isn t necessary), we can express these ideas with an equation. Usually, G is just an exogenous variable that doesn t depend on Y: G = G. But the problem asks you to consider the case where G falls when Y rises, and rises when Y falls. In addition, the problem says that when output equals potential, the government s purchases are the same as before. An equation that captures these ideas is: G = G g (Y Y*), where g is a positive parameter. (If you want to give g a name, think of it as the government s marginal propensity to stabilize. ) If we add up all the components of planned expenditure, we get: = C + I p + G + NX = C + c (Y T) + G g (Y Y*) + NX = [C ct + G + gy* + NX] + (c g)y Thus, the slope of the line is no longer c. Instead it is c g, which is smaller. (Notice also that there is now another term in the intercept term, gy*. Thus the line crosses the vertical axis at a higher point than before. The old and new lines cross at Y = Y*.) 6 b. The fall in autonomous consumption will shift the line down by a certain vertical distance. (To see this, note that the only thing that changes at a given Y is that autonomous consumption, C, is lower. This means that at any Y is lower by the amount that C has fallen that is, the amount of the vertical downward shift is the same at every level of Y.) 2 1 2B Y = The diagram shows the effects of the downward shift of the line in the two cases. In the case where G is not a function of Y, the line shifts from 1 to 1A, and output falls from Y* to Y A. In the case where G is lower when Y is higher, 1A Y A Y B Y* Y the line shifts from 2 to 2B, and output falls from Y* to Y B. As the diagram shows, the fall in output is smaller under the policy of cutting G when Y rises and raising G when Y falls than it is in the usual case of constant G. This finding makes sense. Recall that the multiplier arises from the fact that planned spending is a function of output (see part (d) of Problem 3). Under the policy of cutting G when Y rises and raising G when Y falls, planned spending changes less when output changes than it does in the usual case of constant G. As a result, the impact of the fall in autonomous consumption on output is magnified less by the multiplier effect than it is in the usual case, and so the fall in output is smaller.