ECON 101 Spring 2014 Lecture 5-6 Notes Comparative Statics and the Multiplier Suppose the consumption function is linear and it is given by: where C 0 is a constant and 0 < ci <1 is the marginal propensity to consume. Assume that the government expenditure, Taxes and the Investment are not fixed any more, so they can change value. The planned expenditure equation becomes PE=C 0 + ci(y-t) + G + I 2) In equilibrium we must have Y = PE, so we can write: Now we ask: how does total income change if we change all the variables on the righthand side of 3)? Meaning: what is AFif we change consumption by AC, taxes by A7, investment by A/ and government expenditure by AG? From equation 1) we have that AC = C/A7 CyAT" (Co is a constant so its change is zero by definition). From equation 3), written with changes in the variables, we have that total change in income is given by: A7=c,A7-c,Ar+AG + A/ 4) So the total change in income is equal to the sum of the changes in the variables on the right-hand side. a) The effect of a change in government expenditure We now ask the following question: what is the effect of the change in government expenditure AG on total income, everything else constant (meaning AJ 1 = A/ = 0 )? From equation 4) we have A7=c 7 A7+AG A7-c/A7=AG = AG - 5) AG 1-c Equation 5) says that if the government expenditure increases by $1, the total income will increase by $17(1 -Ci).. The term l/(l-ci) is called the government purchases Multiplier and it depends on the Marginal Propensity to Consume c\. Since ci < 1, an increase in government expenditure will have a more than proportional effect on the income level. Why is the multiplier greater than 1?
Answer: because of the effect through consumption. Suppose the government spends an additional $100 million on defense. Then, the revenues of defense firms increase by $100 million, all of which becomes income to somebody: some of it is paid to the workers and engineers and managers, the rest is profit paid as dividends to shareholders. Hence, income rises $100 million (AY = $100 million = A(7). The people whose income just rose by $100 million are also consumers, and they will spend the fraction c of this extra income. Suppose ci = 0.75 (meaning a multiplier - 4), so C rises by $75 million. To be concrete, suppose they buy $75 million worth of Ford Explorers. Then, Ford sees its revenues increase by $75 million, all of which becomes income to somebody - either Ford's workers, or its shareholders (AF= $75 million). And what do these folks do with this extra income? They spend the fraction c (0.75) of it, causing AC = $56.25 million (0.75 of $75 million). Suppose they spend all $56.25 million on Pepperidge Farms Mint Milano cookies. Then, Pepperidge Farms, or actually General Mills Corporation experiences a revenue increase of $56.25 million, which becomes income to somebody or other. (AF= $56.25 million). So far, the total impact on income is $100 million + $75 million + $56.25 million, which is much bigger than the government's initial increase in spending. But this process continues, and the final impact on Fis $400 million (because the multiplier is 4). Mathematically: First total income rises by AG, therefore: A7= AG ; Then, the first change in consumption is: c;ag The second change in consumption is: Ci(c^AG) = c 2 A(z And so on until the effect of AG terminates. So we can write the change in Y given by a change in G as: AF= (1 + c + c 2 + c 3 +... + C"+. The terms in the brackets form a geometric series with geometric ratio given by c and the first element given by 1. A geometric series of infinite elements with geometric ratio first element 1 tends to: l/(l-ci). The effect of an increase in government expenditure, everything else constant, can be seen graphically: and Y
Suppose that G increases from Gi to G2. The initial equilibrium is Yi. This change in G will increase the planned expenditure by the same amount, therefore, the line describing E will shift upwards by the amount AG. Given the multiplier effect, there will an increase in total income (larger than AG), and therefore, the new equilibrium will be 2. b) The effect of a change in Taxes We now ask the following question: what is the effect of the change in the taxes A!/ on total income, everything else constant (meaning AG = A/ = 0 )? From equation 4) we have: Therefore: If taxes increases by 1%, total income decreases by -ci/(l-ci) % The term -ci/(l-ci) is called the Tax Multiplier. It is negative because an increase in taxes (AT positive) will decrease disposable income. This will decrease consumption and therefore, since Y = C +G + I, total income will decrease as well. On the other hand a decrease in taxes (AT negative) will have the opposite effect. The main features of the Tax Multiplier are: It is negative: a tax increase reduces C, which reduces income. It is greater than one (in absolute value) if the marginal propensity to consume is greater than 0.5: a change in taxes can have a multiplier effect on income. It is always smaller than the govt. spending multiplier in absolute value: for example if MPC=0.75, the government expenditure multiplier is 4, while the tax multiplier is 3 (in absolute value). Let's do an example to illustrate the difference for the economy of assuming that total output is fixed by supply side factors, versus relaxing that assumption. Suppose Y = 6000 C = 500 + O.S(Y-T), I = 750-50r, G = 1000, T = 1000, Y = 6000. Here Y is fixed at 6000, so C is determined by plugging in the values for Y and T, giving C = 4500. Since Y = C + I + G, this gives I = 6000-4500 - 1000 = 500. Or, thinking about it from the loanable funds side, private saving is Y - T - C = 6000-1000 - 4500 = 500, while public saving is 1000-1000 =0, so S( = private + public saving) = 500 + 0 = 500. To find r we use the Investment equation: 500 = 750-50 r, which gives r = 5%. Let's now look at the effect of increasing G by 200 to 1200. Crucially, since we are assuming that Y is fixed by the supply side of the economy, the total size of the pie does not change, so if one component
of demand increases, another must decrease. Which one? Well, since Y and T do not change, by our given consumption function, C does not change. That leaves I. What is the mechanism? An increase in G reduces savings, specifically the public component of savings, T - G, here by 200 to -200. This means that total savings, S is reduced from 500 to 500-200 = 300. With less loanable funds available to satisfy the demand for loanable funds ( = investment), the cost of those funds (the real interest rate) increases, so equilibrium investment falls. Here, 300 = 750-50r gives r = 9% and equilibrium investment falls to 300. Now, let us see how things change when we no longer assume that factors of production are fully employed, that is, we are dropping the assumption that we are constrained by the supply side of the economy. We now solve for output/income - it is now an endogenous variable in our model. In the Keynesian Cross model, we take investment as exogenous, so let us amend the example by taking away the Y = 6000, and setting I = 500. Our equilibrium condition is Actual Expenditure - Planned Expenditure * Y=C+I+G Y = 500 + 0.8(Y - 1000) + 500 + 1000. Now we want to solve for Y. Y - 0.8Y = 500-800 + 500 + 1000 = 1200. (1-0.8)Y = 1200 Y = 1/(1-0.8)1200-6000. Here the multiplier = 1/(1-0.8) = 1/.2 = 5, so a $1 change in one of the expenditure components will cause a $5 increase in Y. Note that I set Investment such that we get the same Y as before. The interesting thing is to see what happens now when we change G. As we saw above, when Y is fixed, investment had to fall when G increased. Now, however, Y is not fixed; in fact, we see that Y increases by $5 for every $1 increase in G, so Y will increase by $1000 when G increases by $200. Which component of Y changes other than G? Consumption (the story being as told above - increased income leads to increased consumption leads to further increases in income etc.). (Note, you can find the change in Y either by the quick method using the multiplier directly as we did above, or plugging in the new value of G in the equilibrium condition, and re-solving, then subtracting off the original Y.) Next, let us see what happens if taxes increase by $200. We saw above that the tax multiplier is -Ci/(l-Ci). Let's work through the example to confirm that.
Now, T = 1200. We have: Y = 500 + 0.8(Y -1200) + 500 + 1000. Y = 500 + 0.8Y - 960 + 500 + 1000 Y - 0.8Y = 1040 Y = 5 x 1040 = 5200. In terms of changes, we have AY = MPC(AY - AT) = 0.8(AY - 200) (1-0.8) AY = -0.8AT * AY = -0.8/(1-0.8)200 = -4 x 200 = -800 So Y = 6000-800 = 5200. So in our example, the expenditure multiplier is 5 and the tax multiplier is -4. What happens if we increase G and also increase T to pay for the increase in G? That is, we do not worsen the budget deficit (or surplus). We saw that increasing G by 200 resulted in an increase in Y of 1000, while increasing T resulted in a decrease in Y of 800. Therefore, doing both together means there is a net change in Y of 1000-800 = 200, or the amount of the increase in G. We can combine the multipliers together to get what we call the "Balanced Budget Multiplier" 1/(1 - Cl) + - Cl /(l - Cl) = (1 - Cl)/ (1 - d) = 1 That is, just as we found above, increasing G increases Y by the amount of the increase in G even when we pay for the increase through increased taxes. That is a pretty compelling argument to increase government expenditure during a recession. Changes in other exogenous Variables We have looked at what happens to Y if we change G or T. What else might change exogenously? Well, wealth or consumer confidence or business confidence are some examples. How would these show up in our model? A fall in wealth or consumer confidence can be represented as a fall in C 0, the exogenous component of consumption, that is, the part that does not depend on current disposable income. A fall in business confidence can be represented as a fall in I. Let's put the model together: Y=C+I+G Y = C 0 + Ci(Y-T) + I + G Solving for Y, Y - c x Y = C 0 + I + G - cj 1)*[C 0 + I + G-C 1 T] r
So we see that a change in C 0 or G has the same effect on Y as a change in G, so we can call l/(l-ci) the autonomous expenditure multiplier, not just the government purchases multiplier. Extensions 1) Investment is a function of Y If investment also depends on output, we would replace the constant term for investment with a function, e.g. Y = b 0 + biy. This would seem to make sense, as firms might want to increase their purchase of capital goods when demand and therefore Y are high, and reduce it when demand and Y are low. Let us see how that affects the multiplier. Now Y = C 0 + Ci(Y-T) + b 0 + biy + G So Y(l- Ci - bi ) = C 0 + b 0 + G - cj i - bi)*[ C 0 + b 0 + G - cj] Now, the expenditure multiplier contains a b : term also. We see that the new expenditure multiplier is larger than the simple one we had before. This makes sense, as there are now 2 channels for the multiplying effect of a change in one of the components of autonomous expenditure, such as G, C 0 or b 0. The initial increase in, say, G increases Y which leads to both more consumption and more investment, which in turn increase Y further, etc. This also works in reverse. An initial decrease will lead to a bigger decline in aggregate economic activity - there is more volatility. Let's continue our example from earlier, replacing the I term with I = 200 + 0.05Y Now, Y = 500 +.8Y - 800 + 200 + 0.05Y + 1000 (1-0.8-0.05)Y = 900 Y = 900/(l-.85) = 6000 as before, but the value of the expenditure multiplier is now 1/0.15 = 6.67, larger than in the simpler case (where it was 5). Now, suppose b 0 increases by 100 (there is a rise in business confidence or expected profitability). What is the effect on Y? AY=l/(l-d-bi)*Abo AY = 6.67 x 100 = 667. Or suppose G increases by 200, as in our example earlier. AY = 6.67 x 200 = 1334, that is an increase in G has a larger effect on Y than we found earlier when only consumption depended on Y, because the effect now works through both consumption and investment.
2) Taxes depend on Y. Another important extension to the basic Keynesian Cross model is that taxes are usually structured to depend on aggregate income, through income tax, profits tax etc. Let's introduce a function for total taxes T to replace the constant we used earlier. T = t 0 + tiy The constant term t 0 could be -ve reflecting the role of transfers, or positive to indicate that some taxes are less tied to aggregate economic activity. Let's solve for Y now: Y = C 0 + Ci(Y-( t 0 + tiy)) + b 0 + biy + G = C 0 + Cl(Y-t 0 - tiy) + b 0 + biy + G Y = C 0 + dy - Cit 0 - CitiY + b 0 + biy + G Y(l - Ci - bi + Citi) = C 0 + b 0 + G - Cit 0 Y - 1/(1 - ci - bi + dt!)*[ C 0 + b 0 + G - What is the size of this multiplier relative to the previous ones? Since the Citi term is positive in the denominator, it reduces the size of the multiplier. This makes sense, as tax revenues taken out of the system reduce the multiplying effects of increases in consumption and investment. We call a tax system like this an automatic stabilizer, since it automatically reduces Y as Y increases, and decreases the reduction in Y as Y falls - thus it helps to make recessions less severe, without action having to be undertaken by policy makers. Continuing with our earlier example, let us introduce the tax function T - 400 + 0.1Y Now we have Y = 500 + 0.8Y - 320-0.08Y + 200 + 0.05Y + 1000 (1-0.8-0.05 + 0.08)Y = 1380 Y=(l/0.23)*1380 = 6000 Here the value of the multiplier is 1/0.23 = 4.348 What is the effect of an increase in G of 200 on Y now? AY = 4.348 x 200 = 839.60