M Problem Set 5 - Solution

M.1 Gregory N. Mankiw - NYT - Nov 30, 2008

  1. According to Gregory N. Mankiw, the factors contributing to hold back consumption are low consumer confidence and “wait and see” behavior caused by falling house price values, shrinking 401(k) balances (due to the fall of the stock market, my addition) and increased unemployment. Yes, these factors can be interpreted in terms of a fall in \(c_0\), since they reduce consumption for a given level of income \(Y\).

  2. Gregory N. Mankiw writes: “Keynesian theory suggests a”paradox of thrift." If all households try to save more, a short-run result could be lower aggregate demand and thus lower national income. Reduced incomes, in turn, could prevent households from reaching their new saving goals." This is exactly the type of phenomenon which we explained in lecture 8: if there is a fall in \(c_0\), then output falls and saving falls as a result. Because investment is fixed and equal to \(\bar{I}\), output decreases until saving equals investment again.

  3. The neoclassical comment of the article is: “In normal times, a fall in consumption could be met by an increase in investment, which includes spending by businesses on plant and equipment and by households on new homes.” This is the usual logic of the Solow (1956) model, for instance. What happens in this model is that saving in fact determines investment entirely. The way this happens through market mechanisms is that the interest rate falls to equate demand and supply of capital. The cost of capital falls down to the point where firms and households want to invest enough to make productive use of all this saving. This logic is somewhat contradictory with the “paradox of thrift” logic, according to which investment is fixed or even increasing with demand (and the interest rate does not clear markets).

  4. Gregory N. Mankiw is very concerned about “the long-term fiscal picture. Increased government spending may be a good short-run fix, but it would add to the budget deficit. The baby boomers are now starting to retire and claim Social Security and Medicare benefits. Any increase in the national debt will make fulfilling those unfunded promises harder in coming years.” We will talk about this potential issue during lecture 10.

M.2 Readings - Multiplier

  1. Q: Robert Barro. Keynes Is Still Dead. Wall Street Journal. October 29, 1992. Robert Barro compares the idea of the Keynesian multiplier to an oft-caricatured version of supply-side economics. What is this caricature? Under what conditions is a Keynesian multiplier “self-financing”? A: The caricature of supply-side economics which Robert Barro is alluding to is the idea that with a general cut in tax rates leads to a sufficient rise in economic activity, so that tax revenues increase. This statement is true only under the assumption that the economy finds itself on the right-side of the Laffer curve, where the rate of taxation is already so high, that further increases in taxes reduces work (through incentive effects) more than they increase government revenues. According to Keynesian economics, a Keynesian policy can be self-financing, if automatic stabilizers are sufficiently large, and the multiplier sufficiently strong. For example, for a rise in government spending \(\Delta G\) then if the multiplier is \(M\), the effect on the public deficit is: \[\Delta (T-G)= \Delta T - \Delta G = t_1 \Delta Y - \Delta G = \left(t_1 M - 1\right)\Delta G.\] You can see that the indirect effect on the deficit is greater than the direct effect if and only if: \[M \geq \frac{1}{t_1},\] implying a tax multiplier higher than \(4\) if \(t_1=1/4\).

  2. Q: Mankiw, N. Gregory. What would Keynes have done? New York Times, November 30, 2008. According to Greg Mankiw, which factors were contributing to hold back consumption in November 2008? Is Greg Mankiw mostly neoclassical or Keynesian in his approach to investment spending? Justify your answer. A: The factors contributing to hold back consumption in November 2008 were declining house values (limiting home equity extraction, or the ability to refinance mortgages), declining retirement saving (401(k) balances) coming from the fall in the stock market, a high unemployment rate which both reduces the disposable income of the unemployed (forgetting about automatic stabilizers), as well as reduces the consumption of the employed who precautionary save, fearing future unemployment. (the risk of unemployment has increased) Greg Mankiw is mostly neoclassical in his approach to investment spending, as he mentioned that the reason why corporations are not investing is the high cost of capital (“interest rates on corporate bonds are up”) or the inavailability of funds because the banking sector is not doing well. In contrast, a Keynesian approach to investment spending would be to say that investment was low in 2007-2009 because aggregate demand was low, and firms were not selling so had to cut back on investment spending too.

  3. Q: Robert J. Barro, Government Spending Is No Free Lunch, Wall Street Journal, January 22, 2009. What is Robert J. Barro’s interpretation of Keynesian economics? Even if one agrees with Robert Barro’s view of government spending, what would be an alternative way to boost aggregate demand? A: According to Robert Barro, Keynesian economics is about the fact that the government is better than the private market at marshalnig idle resources to produce useful stuff: there is something wrong with the price system. An alternative way to boost aggregate demand, which Robert Barro does not mention however, would be to cut taxes, and let the market supply the corresponding demand. The market would not require further fixing: the only problem is that saving is too high, so all it takes is to get people to spend more. Robert Barro would however not believe this stimulates the economy because of “Ricardian equivalence”: people save in anticipation of future taxes to come (to reduce government debt). Robert Barro has used “Ricardian equivalence” to explain the absence of the crowding-out effects of government debt, particularly following the Reagan tax cuts. Of course, as we saw, an alternative explanation is that investment is not sensitive to the cost of capital, and mainly determined by aggregate demand.

  4. Q: Bruce Bartlett. Keynes Was Really A Conservative. Forbes. August 14, 2009. In what sense was J.M. Keynes a conservative? A: Keynes really was a conservative because he was trying to maintain the liberal capitalist order. J.M. Keynes was trying to make sure that the capitalist system would survive. As Keynes himself explained: “the class war will find me on the side of the educated bourgeoisie.” He even expressed contempt for the British Labor Party, who responded to “anti-communist rubbish with anti-capitalist rubbish.”

M.3 Procyclical Government Spending

  1. We write that Output = Demand: \[ \begin{aligned} Y=Z &=C+\bar{I}+G\\ Y &=c_{0}+c_{1}\left(Y-T\right)+ \bar{I} + g_{0}+g_{1}Y\\ Y &=\left(c_{0}-c_{1}T+g_{0}+\bar{I}\right)+\left(c_{1}+g_{1}\right)Y \\ & \Rightarrow \quad \boxed{Y=\frac{1}{1-c_{1}-g_{1}}\left(c_{0}-c_{1}T+g_{0}+\bar{I}\right)}. \end{aligned} \]

  2. The tax multiplier is found by computing the change \(\Delta Y\) in output corresponding to a given change \(\Delta T\) in taxes: \[ \begin{aligned} \Delta Y = -\frac{c_1}{1-c_1-g_1}\Delta T \end{aligned} \] Therefore, if \(\Delta T = -\$ 1\), the change in output is \(\frac{c_1}{1-c_1-g_1}\). Therefore: \[ \begin{aligned} \boxed{\text{Tax Multiplier} = \frac{c_1}{1-c_1-g_1}}. \end{aligned} \] The multiplier is higher in this economy than when government spending does not depend on GDP since: \[ \begin{aligned} \frac{c_1}{1-c_1-g_1}>\frac{c_1}{1-c_1}. \end{aligned} \] The intuition is that government spending automatically increases when GDP increases, which increases demand further. Thus, the multiplier is higher.

  3. This policy appears to be the opposite of what automatic stabilizers are doing, which is to stabilize the economy. The government is having a very procyclical policy, which means that when things go well, it is spending more and therefore helping things go even better; while when things go wrong, it is making it worse by cutting spending. This is clearly not a good policy ! Note however that this is the policy you end up having if you follow a fixed deficit rule, for instance. With a fixed deficit rule, \(T-G\) needs to be constant. If \(T\) depend on GDP through automatic stabilizers, through \(T=t_0+t_1 Y\) then by construction \(G\) needs to respond as well, and it needs to be that \(g_1 = t_1\).

  4. The (ZZ) curve in this problem has a slope equal to \(c_1+g_1\). The impulse to autonomous spending is given by \(\$ c_1\), since one dollar of decreased taxes leads to an increase in consumption equal \(c_1\). This increase leads to a second round of increased consumption and investment \(c_1(c_1+g_1)\), and so on: \[ \begin{aligned} \text{Tax Multiplier} &= c_1 + c_1(c_1+g_1) + c_1(c_1+g_1)^2 + ... + c_1(c_1+g_1)^n + ...\\ &= c_1\left(1 + (c_1+g_1) + (c_1+g_1)^2 + ... + (c_1+g_1)^n + ...\right) \\ \text{Tax Multiplier} &=c_1 \sum_{i=0}^{+\infty}(c_1+g_1)^i = \frac{c_1}{1-c_1-g_1} \end{aligned} \] A graphical justification for the multiplier is given below. The initial impulse is given by \(c_1\). A fraction \(c_1+g_1\) of the additional income injected in the economy is being consumed, so that adds \(c_1(c_1+g_1)\). This repeats itself in a loop, and all of these effects are summed up, so that the total effect is: \[c_1 + c_1(c_1+g_1)\]

  1. If \(g_1+c_1>1\), then each new round of spending leads to an even greater new round of new income and new spending. Therefore, the above infinite sum is then infinite, and the tax multiplier is infinite: \[ \begin{aligned} \text{Tax Multiplier} =c_1 \sum_{i=0}^{+\infty}(c_1+g_1)^i = +\infty. \end{aligned} \] Of course, this is not possible. Therefore, this just means that output cannot be demand-determined and that it is always constrained by supply (the amount of inputs there is in the economy).

M.4 Accelerator and Automatic Stabilizer

  1. We have both an automatic stabilizer as well as an accelerator effect of investment. We write, again, the total demand for goods \(Z\) and then equate demand to output so that \(Y=Z\): \[ \begin{aligned} Y=Z &=C+\bar{I}+G\\ Y &=c_{0}+c_{1} Y - c_1 t_0 - c_1 t_1 Y+ b_0+ b_{1}Y + G\\ Y &=\left(c_{0}-c_{1}t_0+b_{0}+G\right)+\left(c_{1}(1-t_1)+b_1\right)Y \\ & \Rightarrow \quad \boxed{Y=\frac{1}{1-b_{1}-c_{1}(1-t_1)}\left(c_{0}-c_{1}t_0+b_{0}+G\right)}. \end{aligned} \]

  2. The condition is now given by: \[b_1 + c_1(1-t_1)<1.\]

  3. If this condition isn’t satisfied, then the multiplier becomes infinite: one dollar of stimulus leads to more than 1 dollar in additional spending, etc. This leads to an infinite sum. This does not, of course, mean that GDP becomes infinite. It just means that output cannot be constrained by demand, and that the Keynesian model no longer applies. Such an economy should always be neoclassical, in other words GDP should always be limited by available factors of production (capital and labor).

  4. The graph is very similar to the usual one, only that the slope is \(b_1+c_1(1-t_1)\). (if asked for it in an exam, you need to provide it) The geometric sum is: \[1+ \left(b_1+c_1(1-t_1)\right) + \left(b_1+c_1(1-t_1)\right)^2 + ... = \frac{1}{1-b_1-c_1(1-t_1)}.\] (again, you will need to provide the multiplier intuition when asked for it in an exam)