EC2B3 solutions 7
.pdf
School
London School of Economics**We aren't endorsed by this school
Course
ECONOMICS EC2B3
Subject
Economics
Date
Dec 20, 2024
Pages
7
Uploaded by DeanTurkey2511
1 EC2B3 Macroeconomics II Solutions to Problem Set 7 Lent Term 2023 1.Stabilizing business-cycle fluctuations using monetary policy: Consider the New Keynesian model with completely sticky goods prices. Assume wages are flexible and adjust so demand equals supply in the labour market, which implies the real wage is equal to households’ marginal rate of substitution between leisure and consumption (࠵? = ࠵?࠵?࠵?!,#). Assume all households are alike with consumption demands depending negatively on real interest rate ࠵?. In equilibrium, households are savers. Firms must borrow to finance investment at interest rate ࠵?!, so investment is determined by ࠵?࠵?$!− ࠵? = ࠵?!, where ࠵?࠵?$!− ࠵?is the future marginal product of capital net of depreciation. There is a spread ࠵? = ࠵?!− ࠵?between the interest rate ࠵?!paid by borrowers and the interest rate ࠵?received by savers. Suppose a financial crisis leads lenders to expect more defaults in the future. This results in a higher interest-rate spread ࠵?. (a)What is the effect of a higher interest-rate spread ࠵?on the output demand curve ࠵?%? Assuming the stance of monetary policy remains unchanged (a horizontal ࠵?࠵?line with a constant nominal and real interest rate ࠵?&), what happens to real GDP ࠵?and employment ࠵?? Answer: A higher spread ࠵?raises the interest rate ࠵?!at which firms borrow for each interest rate ࠵?received by savers. Higher ࠵?!reduces investment demand, shifting the output demand curve to the left.
2 With no change in the stance of monetary policy, the real interest rate remains at ࠵?"(on the horizontal ࠵?࠵?line). Real GDP falls from ࠵?#to ࠵?$, which reduces firms’ need for workers given they cannot sell as much output with prices remaining fixed, so employment ࠵?will also fall. Suppose the central bank adjusts monetary policy to close the gap between actual GDP and the ‘natural level of output’ (GDP if prices were fully flexible) after the shock to ࠵?. (b)Show in a diagram what adjustment of the real interest rate ࠵?is required for this. After the increase in ࠵?and the monetary policy response, what are the overall effects on ࠵?, ࠵?!, consumption ࠵?, and investment ࠵?? Answer: The natural level of output ࠵?∗is found at the intersection of ࠵?&and the (hypothetical) output supply curve ࠵?’. Although ࠵?∗declines from ࠵?#∗to ࠵?$∗when the output demand curve shifts the left, actual real GDP ࠵?$with a constant real interest rate declines by even more, opening up an output gap between ࠵?$and ࠵?$∗. The central bank needs to lower the nominal and real interest rate from ࠵?#∗to ࠵?$∗, shifting down the ࠵?࠵?line, to close the output gap. Overall, after the increase in ࠵?and the monetary policy response, real GDP ࠵?still falls because the natural level of output declines. Consumption increases because ࠵?is lower. Investment must fall overall because it is known real GDP will be lower, and this means that ࠵?!remains higher overall, even after cutting the interest rate ࠵?. (c)Explain why the monetary policy change in part (b) benefits households in terms of getting closer to a level of employment where ࠵?࠵?’= ࠵?࠵?࠵?!,#. Answer: At the natural level of output, ࠵?࠵?࠵?(= ࠵?࠵?࠵?!,*, and ࠵?࠵?࠵?(< ࠵?࠵?(. For output and employment further below ࠵?∗, the marginal product of labour ࠵?࠵?(rises because of diminishing returns to labour, and the marginal rate of substitution ࠵?࠵?࠵?!,*falls because people have more leisure when
3 employment declines. Hence, ࠵? < ࠵?∗is a situation where the economy’s ability to produce output is worth more than the value people put on their time in units of consumption, so it is optimal to increase employment and output. The monetary policy response in part (b) moves the economy closer to a point where ࠵?࠵?(= ࠵?࠵?࠵?!,*, which is in the interests of households. 2.Short question: Consider the model with strategic complementarity across employment at different firms coming from positive spillover effects on productivity. Show using diagrams that the economy can have multiple levels of GDP consistent with market clearing if the spillovers are strong enough to make the aggregate labour demand curve upward sloping and steeper than the labour supply curve. Answer: Suppose the productivity spillover across firms is sufficiently strong that the aggregate labour demand curve is both upward sloping and steeper than the labour supply curve. In this case, it is possible to have the goods market in equilibrium at two different levels ࠵?#and ࠵?$of real GDP. This can happen when both the output demand curve ࠵?+and the output supply curve ࠵?’are downward sloping and intersect at two points. In this case, people’s degree of optimism or pessimism might determine which of the multiple equilibria prevails. To understand how this can happen, consider a fall in real GDP from ࠵?#to ࠵?$. To be consistent with goods-market equilibrium, both demand and supply must drop. Lower demand for goods occurs when the real interest rate rises from ࠵?#to ࠵?$. But why does a higher real interest rate not increase the supply of output as usual? The output supply curve depicts the level of firms’ aggregate production implied by labour-market equilibrium. Using the production function ࠵? = ࠵?(࠵?)࠵?(࠵?, ࠵?), a lower supply of output must come from lower employment. While the higher real interest rate encourages saving by trying to earn more as usual, so the labour supply curve shifts to the right from ࠵?’(࠵?#)to ࠵?’(࠵?$), this positive effect on employment is outweighed by the large drop in the marginal product of labour at each firm as aggregate employment declines. This is the consequence of the productivity spillover across firms: if other firms performing complementary activities are not employing workers and producing, it is
4 harder for other firms profitably to employ workers. The large downward shift of an individual firm’s marginal product of labour ࠵?࠵?(!implies that employment falls overall, which is the case where aggregate labour demand is steeper than the labour supply curve. 3.For discussion: What factors might affect whether monetary or fiscal policy is a more effective tool in stabilizing an economy hit by demand shocks? Some points to consider: •According to the New Keynesian model with sticky prices, monetary policy is the most direct tool to address the problem of a shortage of aggregate demand following a shock. In this view of how the economy works, the reason for an output gap is that the real interest rate does not automatically adjust to achieve demand equal to supply in the goods market. Since sticky prices also mean that the central bank can set the real interest rate through its control over nominal interest rates, it is natural to think that the job of the central bank is to manage the level of private demand (consumption and investment) by changing interest rates. •Nominal interest rates are constrained by a lower bound, so there is a limit to far the central bank can reduce the real interest rate. It is possible that the real interest rate that eliminates an output gap sometimes requires a nominal interest rate below the lower bound, so monetary policy cannot close the output gap. •There are alternatives to conventional monetary policy based on adjusting short-term nominal interest rates, such as quantitative easing and forward guidance. However, there is a debate about whether changing the quantity of money is effective, and whether forward-guidance announcements are credible. Direct purchases of risky assets may be effective, though the transmission mechanism is different, working through risk premia rather than the level of risk-free interest rates, and such purchases expose the central bank to the risk of losses. •The real effects of fiscal policy are more direct and can operate even if prices are not sticky. This is because fiscal policy can directly affect aggregate demand itself through public expenditure, and change households’ disposable income through taxes or transfers. •However, the government budget constraint puts limits on the effectiveness of fiscal policy. If Ricardian equivalence holds, changes in taxes and transfers are completely ineffective because households fully save any tax cuts. More generally, even when the government directly increases public expenditure, the higher anticipated tax burden can ‘crowd out’ private consumption, offsetting the effect on aggregate demand. •Higher government expenditure might raise real interest rates (if these go to their market-clearing level, or the central bank raises interest rates when GDP rises), which also leads to crowding out of private demand, lessening the effectiveness of
5 fiscal policy. This effect is absent when the central bank holds interest rates constant (perhaps because of the interest-rate lower bound). •Borrowing constraints on the private sector can increase the effectiveness of fiscal policy. When some households face a binding credit limit, they spend all of an increase in disposable income. This gives rise to a multiplier effect of fiscal policy on output, where higher public expenditure raises GDP, boosts disposable income (assuming no increase in current taxes), and hence has a further positive effect on consumption, aggregate demand, and output, and so on. Note that the current consumption of credit constrained households is not affected by an increase in future taxes. 4.The financial accelerator and macroprudential policy: Capital is purchased either by investors who can make a profit of ࠵?per unit of capital, but whose own financial resources limit how much capital they can purchase, or by deep-pocketed investors who face no such limits, but who can only make a profit ࠵? < ࠵?on the marginal unit of capital they purchase (because they are less skilled in putting capital to use). The first group’s capital purchases ࠵?are subject to the budget constraint ࠵?࠵? = ࠵? + ࠵?and the binding collateral constraint ࠵? = ࠵?(࠵?/(1 + ࠵?), where ࠵?is the price of capital, ࠵?(is the future capital price, ࠵?is borrowing, ࠵?is the net worth or equity of this group, and ࠵?is the real interest rate. Since the collateral constraint is binding, the second group of investors is the marginal purchaser of capital and the capital price is ࠵? =࠵?(1 + ࠵?+࠵?(࠵?)1 + ࠵?where ࠵?(࠵?)is the second group’s marginal product, an increasing function of ࠵?(the more capital the first group holds, the less need for the second group to buy capital). (a)Show that ࠵?(࠵?)࠵? = (1 + ࠵?)࠵?and deduce that the first group’s capital purchases ࠵?are positively related to their equity ࠵?. Hence explain why higher equity ࠵?causes ࠵?(࠵?)and the price of capital ࠵?to rise. Answer: Substituting the equation for the capital price ࠵?and the binding collateral constraint on ࠵?into the budget constraint of the most effective group of investors: 4࠵?,1 + ࠵?+࠵?(࠵?)1 + ࠵?8 ࠵? =࠵?,࠵?1 + ࠵?+ ࠵?Cancelling the term in ࠵?′from both sides and multiplying by 1 + ࠵?implies ࠵?(࠵?)࠵? = (1 + ࠵?)࠵?. This equation says that each unit of equity/net worth/own financial resources of the first group of investors can be transformed into the ability to hold ࠵?(࠵?)/(1 + ࠵?)units of capital ࠵?through borrowing against the future value ࠵?′࠵?of the capital.
6 Since ࠵?(࠵?)is increasing in ࠵?(because less capital is held by low-marginal-product investors when better investors hold more), the left-hand side of the equation ࠵?(࠵?)࠵? = (1 + ࠵?)࠵?is increasing in ࠵?. This confirms there is a positive relationship between the amount of capital ࠵?purchased by the group of more effective investors and their equity ࠵?. Moreover, this means that ࠵?(࠵?)will be high when equity ࠵?is high, and the capital price ࠵?depends positively on ࠵?(࠵?), so this also rises as equity ࠵?increases. Intuitively, if better investors have more equity, they can hold more capital, which reduces the need for low-marginal-product investors to hold it, so the marginal buyer has a higher marginal product of capital, pushing up the willingness to pay for capital and its price ࠵?. Initial equity is ࠵? = (࠵?࠵? + ࠵?)࠵?<− (1 + ࠵?)࠵?=, where ࠵?<and ࠵?=are past capital purchases and borrowing, and ࠵?is the fraction of profits that can be retained to build up equity. Given past capital, debt, and interest rates, higher capital prices ࠵?boost equity ࠵?. (b)Explain why the model features a ‘financial accelerator’, a positive feedback loop between the price of capital ࠵?and equity ࠵?of the most effective group of investors. Answer: Taking past choices of ࠵?<and ࠵?=as given, an unexpected decrease in the capital price ࠵?has a negative effect on equity ࠵?. This is because existing debt obligations (1 + ࠵?)࠵?=are fixed even though the capital price unexpectedly changes the value of assets held. The answer to part (a) explains why falling ࠵?reduces ࠵?, ࠵?(࠵?), and the capital price ࠵?. A further reduction in the capital price leads to another fall in equity ࠵?, and a further fall in the capital price ࠵?, and so on. This feedback loop is the financial accelerator. The financial accelerator magnifies the effects of a shock to equity ࠵?on investment. To avoid this amplifying business-cycle fluctuations, suppose regulators use macroprudential policy to limit borrowing to ࠵? = (1 − ࠵?)࠵?(࠵?/(1 + ࠵?)with ࠵? < 1. (c)Find the leverage ratio ࠵?࠵?/࠵?in a steady state where ࠵?(= ࠵?and ࠵? = ࠵?′. Answer: In a steady state with no surprise changes, the future price ࠵?′is consistent with the collateral constraint (1 + ࠵?)࠵? = (1 − ࠵?)࠵?′࠵?. Multiplying both sides of the budget constraint ࠵?࠵? = ࠵? + ࠵?by 1 + ࠵?and substituting the collateral constraint with ࠵?,= ࠵?in steady state: (1 + ࠵?)࠵?࠵? = (1 − ࠵?)࠵?࠵? + (1 + ࠵?)࠵?This implies (࠵? + ࠵?)࠵?࠵? = (1 + ࠵?)࠵?, so the steady-state leverage ratio is ࠵?࠵?࠵?=1 + ࠵?࠵? + ࠵?This is consistent with leverage ratio (1 + ࠵?)/࠵?in the special case of no macroprudential policy (࠵? = 0).
7 (d)With reference to your answer to (c) and the elasticity )*+*+)of equity with respect to the capital price, explain intuitively why macroprudential policy can reduce the strength of the financial accelerator. Answer: The formula for equity ࠵? = (࠵?࠵? + ࠵?)࠵?<− (1 + ࠵?)࠵?=implies ࠵?࠵?/࠵?࠵? = ࠵?<, and hence the elasticity of equity with respect to the capital price in a steady state (࠵? = ࠵?<) is ࠵?࠵?࠵?࠵?࠵?࠵?=࠵?࠵?<࠵?=࠵?࠵?࠵?=1 + ࠵?࠵? + ࠵?The elasticity is equal to the leverage ratio, so a 1%change in the capital price is multiplied by the leverage ratio to find the percentage change in equity ࠵?. Macroprudential policy with ࠵? > 0implies a lower leverage ratio and thus a lower capital-price elasticity of equity. This weakens the financial accelerator feedback loop. In the diagram depicting the financial accelerator, the effect of lower leverage is a flatter “Balance sheet” line representing the effect of the capital price ࠵?on equity ࠵?. (This question does not consider any changes to the other line, “Determination of capital price”.) Hence, a shock now has a smaller effect on the capital price and investors’ equity.