1 1 Solution Set: Chapter 22 Chapter Review: 2. The consumption function The answer is (a) describes the relation between desired consumption expenditures and the factors that determine it, like real disposable income. This is a definition, so there s not much I can help you with. You can find it on page 510 of your text: Consumption function The relationship between desired consumption and all the variables that determine it; in the simplest case, the relationship between consumption expenditure and disposable income. One way of thinking about it is in terms of graphs. Every time you plot a (twodimensional) graph, you re actually plotting a relationship between the two things on the axes. On a diagram such as the demand/supply diagram that has Price on one axis and Output on the other, you re plotting the relationship between price and output, for example. On page 513 of your text, you find your first graph of the consumption function. Since the axes of the diagram are Desired Consumption Expenditure and Real Disposable Income, you re plotting a relationship between desired consumption and disposable income. 4. Which of the following are basic assumptions about the consumption function? The answer is (e) The MPC is greater than zero and less than one and the APC falls as income rises. Since these are assumptions, you just have to learn them. They re discussed briefly on page 512 of your text. What do they mean? If the marginal propensity to consume (MPC) is in between zero and one, that means that if you get an extra dollar, you re going to spend in between zero and one extra dollars (and save the rest). That seems like a pretty safe assumption. What about the APC (average propensity to consume)? If 0 < MPC < 1, then the APC will fall as income rises. This is seen most easily algebraically. Suppose that our consumption function is linear, so that it takes the form C = zy d + C a Here, z = MPC, Y d = real disposable income, C a = autonomous consumption and C = desired consumption. If you graphed the line out, z would be the slope of the line, and C a would be the intercept. Note that autonomous consumption is greater than zero, because even when you don t earn any income (Y d = 0), you still need to consume food and water to stay alive.
2 2 Now recall the definitions of MPC and APC. MPC is just the slope of the consumption function. Here, that s equal to z. Remember that by assumption, z is in between zero and one, and therefore positive. APC is the value of the consumption function, divided by Y d : APC = C/Y d. In plain English, APC is the average amount of stuff you want to consume for every dollar you have available for spending. APC = C/Y d = z + C a /Y d So, what happens to APC as Yd goes up? z stays the same, but Ca/Yd falls. Since z, C a and Y d are all greater than zero, APC is always positive, but falls as Yd rises, which was what I wanted to show. 7. An increase in households real wealth is predicted to The answer is (d) shift the consumption function upward. Your book discusses this in detail on page 514 in the section Wealth and the Consumption Function, but you shouldn t need to read that to know the answer. Pretend you hadn t studied any economics, and someone asked you what would happen to a household s consumption if they got wealthier. What would you answer? Probably that they d spend more. Knowing economics shouldn t change that. As for why it s a shift of the whole curve rather than just a movement along it, think of what s on the axes of the diagram. In order to move along the curve, you need to change something that s on the axes: in this case, that d be only real disposable income. Since wealth isn t on the axes, the only way left to get consumption to rise is to shift the whole curve upwards. 9. A reduction in real interest rates The answer is (a) normally reduces the mortgage payments a home buyer must make. The payments on a mortgage are interest payments. The amount of cash that a home buyer pays out each month is proportional to the nominal interest rate, and the nominal interest rate is given by Nominal interest rate = Real interest rate + Expected inflation As long as expected inflation doesn t go up too much at the same time, a fall in the real interest rate will cause a fall in the nominal interest rate, and the interest payments will be reduced. 12. The aggregate expenditure function is a relationship between
3 3 The answer is (c) desired real expenditure and nominal national income. Another definition. You can find it on page 518 of your text. 13. In a simple macroeconomic model, with a closed economy and no government, the aggregate expenditure function is the sum of The answer is (a) desired consumption and desired investment. By definition. Again, you may find this on page 518. By now, you should know that AE = (desired) C + I + G + (X-M). The question tells you that it s a closed economy and therefore has no international trade, so exports and imports are zero (X-M) equals zero. Likewise, G = 0 because there s no government. What s left? AE = C + I. 14. The marginal propensity to spend The answer is (c) is the slope of the aggregate expenditure function. Any slope is equal to rise over run. On the AE graph, rise is given by the change in (desired) spending, and run is given by the change in national income. (Just look at the axes). Thus, the slope of the AE function is the change in spending over change in income, or the amount of spending done when you get an extra dollar but this is just the definition of the marginal propensity to spend, therefore MPS = slope of AE. 15. In a macroeconomic model that includes consumption and investment, equilibrium real national income is attained when The answer is (e) all of them. Let s go step by step. (a) actual real output is equal to desired real aggregate expenditure. In other words, Y = AE. By now you should know that this is an equilibrium condition. (b) the aggregate expenditure function intersects the 45 degree line. In other words, equilibrium is at the point where desired expenditure, AE, also fulfils the condition that Y = AE, that is, it hits the 45 degree line. (c) the average propensity of desired spending is unity. Unity means one, so APC = 1. But we already know that APC = C/Y = AE/Y, so if APC = 1, this means that Y = AE. Which is an equilibrium condition. (d) there is no unintended inventory accumulation or reduction. In other words, there is no saving or borrowing. This only happens when Y=AE. See Figure 22-7 on page 523 of your text.
4 4 16. At a level of national income where aggregate desired expenditure falls short of actual national income, there will be a tendency for The answer is (b) national income to fall. In other words, if you have more money available than you actually have to spend, what will happen? Well, you ll put some into the bank. Your available income will fall. In terms of the macroeconomy: if there s more output available (Y) than people actually want to buy (AE), firms will notice that and cut production (thus lowering Y), since they don t want to have warehouses full of unsold goods. If you still have trouble with this, read the first two paragraphs of page 521 of your text. 19. On a graph depicting equilibrium national income (where desired aggregate expenditure equals actual national income) consider a level of national income where the vertical distance from the horizontal axis to the AE function is less than the vertical distance to the 45 degree line. At such an income level unintended inventories are, and so actual national income will tend to. The answer is (d) accumulating, fall. Let s draw the graph. E Y=AE B AE A 45 deg. Y e Y actual Y The distance from the horizontal axis (the Y-axis) to the AE function is equal to the segment A. The distance from the horizontal axis to the 45 degree line (the balanced budget, or Y = AE line) is equal to A + B. Since A+B > A, Y actual is the actual GDP required by the question. Note that all points fitting this requirement are to the right of AE. If the horizontal distance from the Y-axis to the AE line is greater than the horizontal distance from the Y = AE line, then by definition AE is less than Y, and we re to the right of Y e. This is exactly the same situation as in question 16, and the answer is the same. Since people wish to buy less stuff
5 5 than is currently available, unsold goods (inventory) are piling up, or accumulating, in stores. Firms will notice this and lower their production levels, so GDP will tend to fall. (An easy way to remember this last part is that since Y e is a stable equilibrium, whenever you re away from it there will be forces nudging you back towards it.) 20. On a graph depicting equilibrium national income (where desired savings equals desired investment) consider a level of national income where the vertical distance from the horizontal axis to the saving function is less than the vertical distance to the investment function. At such an income level, inventories are -, and so national income tends to. The answer is (c) being depleted, rise. This is very similar to the previous question. Again, let s draw the diagram. S,I S I B 0 A Ca/1-z Y actual Y e Y I ve based this on the diagram on page 523 of your text. The reason the I line is flat is that it assumes investment is autonomous, or constant with respect to Y. The point labeled C a / (1-z) doesn t have to be there, but I wanted to point out to you that the saving function is negative until Y = autonomous consumption times the multiplier. Why? Recall the definition of Saving is Income minus Consumption, and that consumption is given by aggregate expenditure. Also, let s assume aggregate expenditure takes the linear form we used a few problems ago: All together, this means S = Y C C = AE AE = zy + Ca
6 6 If S < 0, then (1-z)Y Ca < 0, or S = Y zy Ca = (1-z)Y Ca Y < Ca/(1-z) But by now you should be familiar with the formula for the multiplier: K = 1/(1-z) Thus, savings are negative until Y = Ca x K. Back to the question. The distance from the horizontal (Y) axis to S is equal to A, and the distance to the I function is equal to A+B. As required, A+B>A. Note that all the points for which this is true (such as our Y actual ) are to the left of, and less than, equilibrium national income, Y e. When actual GDP is less than equilibrium GDP, people want more goods than are actually available for consumption: there s a shortage of everything, and products fly off store shelves. The amount of unsold stock, or inventories, shrink that is, they re depleted. Firms notice that people want to buy more of their stuff, and crank up production at their factories. As a result, national income rises. 21. Which of the following will shift the AE function upward in a parallel fashion? The answer is (c) The real rate of interest falls. If interest falls, then investment goes up. If investment goes up, that means AE goes up. Since the interest rate is not on the axes of the AE diagram, the entire curve shifts up. 25. If expenditure in the economy did not depend on the level of real national income, the value of the simple multiplier would be The answer is (b) unity. Remember that unity means one. If spending is independent of income, as in this question, then all we re left with is autonomous spending: the slope of the AE line is zero. (A slope of zero just means that AE doesn t change when you change Y. The rise in AE for a run along Y is zero.) As we know well, the simple multiplier is given by K = 1/(1-z) where z = the slope of the AE function. If AE is zero, as in this case, then K = 1/(1-0) = 1 = unity
7 7 26. The multiplier is larger The answer is (b) the steeper the slope of the AE function. This follows from the previous question. K = 1/(1-z), and z is the slope of the AE function. The steeper AE is, the higher z is, and the higher z is, the higher K is. For those of you who know calculus, d dz 1 1 z = 1 ( 1) = 0 ( ) ( ) 1 z 1 > z for 0<z<1 27. Assuming constant prices and a marginal propensity to spend of 0.75, an increase in autonomous investment of $1 million will increase equilibrium real national income by The answer is (b) $ 4 million. This is identical to the example I used to illustrate the multiplier in a previous handout. Recall that K = Y e AE 1 = 1 z You re given that z = 0.75 = ¾, and that I rises by $1 million. Since AE = C + I + G + NX, that means that AE also rises by 1 million. You re asked to find. Y e Solving the previous equation, Y e 1 1 = AE = 1 z 1 3 $1,000,000 4 = 1 $1,000, = $4,000,000 (By definition, you multiply the change in AE by the multiplier to get the resulting change in equilibrium Y, hence the name. In this case, the multiplier is equal to 4.)
8 8 Exercises 1. This question tests your knowledge of a Keynesian consumption function. For this example, the consumption function is given by the expression C = Yd. The first two columns of the following schedule depict this functional form for various levels of disposable income. Y d C S Y d C MPC APC n.a. n.a. n.a. n.a To make things easier, let s get a bit ahead of ourselves and graph the data in the first two columns Desired Consumption (C) Real Disposable Income (Yd) As usual in your text, the consumption function turns out to be a straight line. (a) Fill in the missing values for the change in real disposable income ( Y d ). The change in Yd is just the present value minus the previous one. For the first blank, Y d = = 40. For the second, Y d = = 200.
9 9 (b) Using the definition for the average propensity to consume, fill in the missing values for APC. What did you notice happened to the value of APC as the level of Y d increased? We ve done all this before. APC = C/Yd. For the first blank, APC = 180/200 = For the second, APC = 455/750 = As disposable income goes up, APC goes down. The reason for this was covered in a previous question. (c) Fill in the missing values for C. Recall that means change in. As before, just subtract the previous value from the current one. For the first blank, C = = 100. For the second, C = = 175. (d) Using the definition for MPC, calculate it for the income change from 400 to 750. There s two ways to do this. The first is to note that MPC = slope of C = rise/run = C / Y d = 175 / 350 = 0.50 The second is to realize that since MPC = slope of C, and C is a straight line, the slope is constant : since it s been 0.50 before, it s 0.50 now. (You can also pull the slope right out of the expression you re given for C: C = Y d.) (e) Using the definition for saving S = Y d C, fill in the missing values in the table. Using the formula S/ Y d, prove that the marginal propensity to save is constant and equal to 0.5. Using S = Y d C, for the first blank we have S = = 0, and for the second, S = = 295. Because S = Y d C, S = Y d C. We re given that MPSave = S/ Y d. Plugging in our previous result, we obtain MPSave = ( Y d C) / Yd = 1 - C / Y d But we know that MPC = C / Y d, so this means that MPSave = 1 MPC. Since MPC is constant and equal to 0.50, that means MPSave is constant and equal to = 0.50.
10 10 (f) Prove that the algebraic expression for the saving function is S = Y d. By definition, the slope of the saving function is equal to the marginal propensity to save. We know that MPSave = 0.50, so the slope of the saving function will always be equal to A function with a constant slope is a line. The only other bit of information we need to get the formula is the intercept of the line. If the line is written in the form S = my d + S a, we have m, and only need S a. The intercept of the saving function is just the value of the function when Y d is equal to zero. But we have that from the table, when Y d = 0, S = -80. Thus, the intercept is equal to 80, and the equation of the saving function is S = Y d. Another way to do it is by using the definitions of S and C. S = Y d C C = Y d This implies that S = Y d ( Y d ) = (1-0.5)Y d 80 = 0.5Y d 80 (g) What is the break-even level of real disposable income? What is the amount of saving at this level of Y d? When you break even, Y d = C. From the table, this happens when Y d = 160. You can also find it from the algebraic equations: Y d = C Y d = Y d (1-0.5)Yd = Y d = 80 Y d = 80/0.5 = 2 x 80 = 160 At the break-even point, S = 0. You can look this up in the table, but it should be obvious: S = Y d C, and if Y d = C, then S = C C = 0. (h) Plot both the desired consumption (as line C) and desired saving functions (as line S) in figure In addition, draw the 45-degree line and prove that this line intersects the consumption function at a level of Yd for which S = 0. First, a note on plotting lines: all you need to plot a line is two points, and you can get these from the slope and the intercept. The intercept gives you the height of the line when the value on the horizontal axis is equal to zero. Since
11 11 the slope is equal to rise/run, you can get your second point by using it in combination with the intercept. For example, to graph the consumption function C = Y d The intercept is 80, so when Y d = 0, C = 80. The slope is 0.5 = ½, so let s pick a nice even number that doesn t give us any decimals when we multiply by ½, say 500. When Y d = 500, C = x 500 = = 330. Now you have two points: (0,80) and (500,330). To graph the consumption function, just connect the dots. The end graph should look like the one below: C,S C S 45-degree Real Disposable Income (Yd) Now we have to prove that the 45-degree line crosses the C line when S = 0. It certainly looks like it s true, just by glancing at the graph. Let s check it with algebra. The 45-degree line is the line along which Y d = C. This equation will be true at all points along that line. In particular, it will be true when the C line crosses it. But the definition of S is S = Y d C, so if Y d = C, then S = C C = 0, which was what we wanted to show. There is no borrowing or saving along the 45-degree line: what you spend is exactly equal to your income.
12 12 (i) Is desired consumption expenditure both autonomous and induced? Explain. In algebraic terms: Is C autonomous? This is asking, is the intercept of the equation for C nonzero?, or equivalently, when Y d is zero, is C non-zero?. The answer to all these questions is yes. In fact, autonomous consumption is equal to 80. Is C induced? This is asking, is the slope of the equation for C non-zero?, or equivalently, when you change Y d, does C change?. Again, the answer is yes: the slope of the C function (the MPC) is equal to ½. So yes, C is both autonomous and induced. 5. The following table summarizes desired levels of aggregate expenditure for an economy where the price level is assumed constant. Y C S I AE Before we start, let s graph the information we have, just to make things easier to see. 400 C,S,I,AE C S I AE Y
13 13 (a) Fill in the missing entries in the consumption, saving, and aggregate expenditure columns, assuming that the marginal propensity to consume out of disposable income remains constant. Since the MPC is constant, the MPSave = 1 MPC is also constant. This means that both the C and S functions will be straight lines. We already know that in order to plot straight lines, we only need the slope and the intercept. From the table, the intercept of C (the value of C when Y = 0) is 80. The intercept of S is 80. Note that they re negatives of each other. This will always be true: when you have no income, you must borrow in order to consume, and saving is just negative borrowing. What about the slopes? Slope = rise/run. For C, rise = change in C, and for S, rise = change in S. For both, run = change in Y. So MPC = slope of C = rise/run = change in C/change in Y Since it s constant, we only need to calculate it for one point. Let s take the first two values in the table. Rise: C2 C1 = = 60 Run: Y2 Y1 = = 100 This implies that MPC = slope of C = 60/100 = 3/5 Since we have the slope and intercept of C, we can write its equation: C = intercept + slope x Y = 80 + (3/5) x Y In order to get the values of C for the table, just plug in the appropriate values of Y into the equation above. For example, when Y = 300, C = 80 + (3/5)x300 = 260 Similarly, we can get the equation for S: Rise = S2 S1 = -40 (-80) = = 40 Run = Y2 Y1 = = 100 This means that MPSave = slope of S = 40/100 = 2/5 (Note that, as expected, MPC + MPSave = 1. We could have just solved for MPSave by using this, but I wanted to show you the long way to do it.) The equation for S is
14 14 S = (2/5) x Y Again, you can find the appropriate values for S by plugging Y into the equation. We don t need to solve for I, since all the values are given. It s always 120. What about AE? Remember that AE = C + I + G + NX. This economy is closed (no NX) and has no government (no G). Thus, AE = C + I = C You can find the values of AE just by adding 120 to the values you find for C. The finished table is as follows: Y C S I AE Graphing it all out, 700 C,S,I,AE C S I AE Y (b) What is the equilibrium level of national income in this economy? At equilibrium, Y = AE. Equivalently, S = I. From the table or the graph, we find this happens when Y = 500.
15 15 You should also practice finding this from the algebraic equations for C and S. (c) Explain why an output level Y of 400 is not an equilibrium situation. When Y = 400, Y is not equal to AE and S is not equal to I. Thus, it is not an equilibrium. (d) What is the marginal propensity not to spend on domestic goods? What is the simple multiplier for this economy? Whatever you don t spend, you save. Thus, marginal propensity not to spend just means marginal propensity to save. We already know that this is equal to 2/5 (or 0.4, if you prefer decimals). The simple multiplier is equal to 1/(1-z), where z = MPC. Here, then, K 1 = 1 z 1 = = = = 2.5 (e) If investment were to increase by 60, what would the new equilibrium level of national income become? Show your answer in two ways, by using the table above and by applying the simple income multiplier. Using the table if investment increases by 60, to 180, then AE also increases by 60: Y C S I AE Now, Y = AE and S = I somewhere in between Y = 600 and Y = 700. To find the exact value, we d need to use the equations. Let s use S = I. S = (2/5) x Y I = 180 Hence, S = I implies that (2/5) x Y = 180
16 16 (2/5) x Y = 260 Y = (5/2) x 260 = 650 The new equilibrium Y is equal to 650. As expected from the table, this is between 600 and 700. What about using the multiplier? From the definition of the multiplier, K Ye AE Ye = K AE We re given that investment has gone up by 60. Since AE = C + I (in this case), that means that AE has also gone up by 60: AE = 60. We found earlier that the multiplier K = 2.5. Using the formula above, Ye = K x AE = 2.5 x 60 = 150 The change in Y is 150, which means that equilibrium Y goes up by 150 from its previous level. Hence, the new Ye = (old Ye) = = 650, as expected. (f) If investment is 120, what is the level of Y at which S = I? We solved for this back in part (b). S = I is one of the equilibrium conditions, so S = I when Y = Ye, and Ye = 500 for I = Indicate whether or not the following separate event is expected to increase the equilibrium level of national income. Explain how the event would be depicted in an AE diagram for which the marginal propensity to spend is positive, but less than one. Assume the economy is originally at equilibrium national income. What the above tells you is, pretend that you have a Y = AE diagram just like all the others you ve seen before, and that originally, Y = Ye.
17 17 E Y = AE AE = C + I + G Ye Y Two things you should know: first, moving something that s ON the axes but not AE (in this case, Y) will cause a move along the AE curve. Moving anything that matters to AE but isn t on the axes, that is, isn t Y, will cause a shift of the whole AE curve. Common sense should tell you what direction the moves and shifts will be in. (a) An increase in actual national income. Y increases. Y is on the axes, so we get a movement upwards (to the right) along the AE curve. (b) A decrease in autonomous consumption. A change in anything autonomous will shift AE, but apart from that A fall in C will lower AE, since AE = C + I + G + NX. Autonomous C is independent of Y, and so is not on the axes. (If C depended on Y, then it d be as if Y itself had moved.) Since it s not on the axes, the entire AE curve shifts down. (c) A decrease in the real interest rate. A fall in real interest rates causes a rise in I, which means a rise in AE. Since real interest rates are not on the axes, the entire AE curve shifts up. (d) A downward shift in the saving function by the same amount at every level of national income. S = Y C. If S falls by the same amount at every level of national income, that must mean that (autonomous) C has gone up. A rise in autonomous C means a rise in AE, and since C isn t on the axes, the entire AE curve shifts up.