Grasping Reality on TypePad, by Brad DeLong

Business Cycle Models

The key filing system and checklist to use in understanding how an economy subject to business cycle shocks behaves in both the flexible-price and sticky-price models is our IS Curve equation:

Where:

• $ Y^* $ is the level of potential output
• $ Y $ is the level of national income and product
• $ \mu $ is the so-called Keynesian multiplier
• $ c_o $ is our measure of consumer confidence
• $ I_o $ is our measure of corporate investment committe optimism—of business "animal spirits"
• $ G $ is the level of government purchases
• $x_f $ is the responsiveness of exports to a change in foreigners' incomes
• $ Y^f $ is national income and product abroad
• $ x_{\epsilon} $ is the responsiveness of exports to a shift in the exchange rate
• $ {\epsilon}_o $ is foreign exchange speculators' assessment of the long-term fundamental value of the exchange rate $ \epsilon $
• $ {\epsilon}_r $ is the responsiveness of the exchange rate to the differential between foreign and domestic interest rates
• $ r^f $ is the foreign interest rate
• $ I_r $ is the responsiveness of investment to the domestic real interest rate $ r $
• $ r $ is the long-term risky real domestic interest rate
• $ c_y $ is the marginal propensity to consume
• $ t $ is the marginal tax rate on income
• $ im_y $ is the marginal propensity to import

Plus the other equations and variables of the business-cycle model:

• $ \mu = \frac{1}{1-c_y(1-t)+im_y} $ is the Keynesian multiplier
• $ r = i - \pi + \rho $ is the interest-rate determination equation
• $ C = c_o + c_y(1-t)Y $ is the consumption function
• $ I = I_o - I_{r}r $ is the level of investment
• $ IM = im_y{Y} $ is the level of imports
• $ GX = x_fY^f + x_{\epsilon}{\epsilon} $ is the level of gross exports
• ε = ε_o + ε_r(r - r^f) is the exchange rate
• $ Y = C + I + G + (GX - IM) $ is national income and product circular-flow
• $ i $ is the short-term safe nominal interest rate that the central bank controls
• $ π $ is the inflation rate
• $ \rho $ is the risk and term premium financial markets charge for lending long to risky businesses rather than lending short to a stable government possessing exorbitant privilege

Recall that the key difference between the flexprice and the sticky-price versions of our business-cycle model is that:

• in the flexprice version national income and product Y is always in equilibrium equal to potential output Y^*: the real wage quickly adjusts to make production equal to potential; the price level quickly adjusts to make aggregate demand and spending equal to production; the real interest rate quickly adjusts to balance the flow-of-funds through financial markets.

• in the sticky-price version none of these three prices move quickly and substantially enough to play a significant role in pushing the economy to its short-run equilibrium. Instead, rising or falling inventories induce firms to fire or hire people which decreases or increases production and incomes until inventories are stable and aggregate demand equals national income and product. Potential output is nowheresville in the sticky-price version.

 

Part A: Multiple Choice Questions

2 mins each

1. Long-Run Capital-Worker Ratio

If the model parameters s, n, g, α, and δ remain constant at their initial time-0 values, toward what path is output-per-worker $ K_t/L_t $ converging in the long run? Write "A1:" followed by the letter of the correct answer in your bluebook:

A $ \lim\limits_{t\to\infty}\left(\frac{K_t}{L_t}\right) = \left(\frac{s}{n+g+\delta}\right)\left(E_0{e^{gt}}\right) $

B $ \lim\limits_{t\to\infty}\left(\frac{K_t}{L_t}\right) = \left(\frac{s}{n+g+\delta}\right)^{\frac{1}{1-\alpha}}\left(E_0{e^{gt}}\right) $

C $ \lim\limits_{t\to\infty}\left(\frac{K_t}{L_t}\right) = \left(\frac{s}{n+g+\delta}\right)^{\frac{\alpha}{1-\alpha}}\left(E_0{e^{gt}}\right) $

D $ \lim\limits_{t\to\infty}\left(\frac{K_t}{L_t}\right) = \left(\frac{s{\alpha}}{n+g+\delta}\right)\left(E_0{e^{gt}}\right) $

E none of the above/cannot be determined from the information given

 

2. An Increase in Population Growth and Savings

If, in a SGM economy, the savings rate s and the population growth rate n both increase by the same proportional amount, the steady-state balanced-growth path values of capital per worker will:

A. increase

B. decrease

C. stay the same

D. increase unless both (a) capital is completely durable and does not depreciate and (b) the efficiency-of-labor is absolutely constant

E. none of the above/cannot be determined from the information given

 

3. An Increase in the Production Function Parameter $ \alpha $

Comparing two SGM economies with otherwise-identical parameter values, the one with a higher value of the production function parameter $ \alpha $ will have:

A. faster convergence to its steady-state balanced-growth path and a higher level of output per worker along its steady-state balanced-growth path

B. faster convergence to its steady-state balanced-growth path and a lower level of output per worker along its steady-state balanced-growth path

C. the same speed of convergence to its steady-state balanced-growth path and a higher level of output per worker along its steady-state balanced-growth path

D. slower convergence to its steady-state balanced-growth path and a lower level of output per worker along its steady-state balanced-growth path

E. none of the above/cannot be determined from the information given

 

4. Malthusian Economies: $ \alpha $

Comparing two Malthusian economies that are both on their steady-state balanced-growth paths and that otherwise have identical parameter values and identical initial values of the efficiency-of-labor at time zero E_o, the one with a higher level of the parameter h describing how rapidly productive ideas are generated will:

A. have a faster-growing population but the same level of output-per-worker than the other

B. have a faster-growing population and a faster growing level of output-per worker than the other

C. have a faster-growing population and a slightly-higher level of output-per-worker than the other

D. have a faster-growing population and a much higher level of output-per-worker than the other

E. none of the above/cannot be determined from the information given

 

5. Malthusian Economies

The failure of all the useful and productive ideas developed by humanity over the course of the long Agrarian Age—from 5000 BC to 1800 AD—to produce any significant boost to the standard of living of the typical human is primarily due to:

A. the failure of legal systems to provide sufficient protection for patents and copyrights

B. the strong positive relationship between standards of living and population growth

C. wars that kept the capital stock from growing sufficiently rapidly

D. the fact that it was much more rewarding for smart and aggressive people with resources to focus their energy on the win-lose struggle to make domination more effective than on the win-win task of making production more efficient

E. none of the above/cannot be determined from the information given

 

6. Malthusian Economies

In a Malthusian economy in which the ideas stock H grows at a proportional rate h, in which the natural-resource stock N is constant, and in which the efficiency-of-labor E is given by:

$ E_t = \left(H_t\right)^\left(\frac{\gamma}{1+\gamma}\right)\left(\frac{N_t}{L_t}\right)^\left(\frac{1}{1+\gamma}\right) $

Suppose the economy is initially on its steady-state balanced-growth path, and then the rate of generation of ideas parameter h undergoes a sudden, immediate, discontinuous, and permanent upward jump. After this jump, along this economy's balanced-growth path:

A. the population growth rate n will be less than $ \gamma $ times the growth rate h of the ideas stock

B. the population growth rate n will be the same as $ \gamma $ times the growth rate h of the ideas stock

C. the population growth rate n will be more than $ \gamma $ times the growth rate h of the ideas stock

D. the population growth rate n will be greater than the growth rate h of the ideas stock if $ \gamma > 1 $, and the population growth rate n will be less than the growth rate h of the ideas stock if $ \gamma < 1 $

E. none of the above/cannot be determined from the information given

 

7. Divergence

Suppose that we are trying to account for the 49-to-1 divergence in productivity levels and living standards across the globe today using the SGM, and assuming that economies are on their steady-state balanced-growth paths. Assume also that divergence across economies in capital-output ratios is perfectly correlated with divergence in efficiency-of-labor levels, and that all countries now are equal in their labor force growth, efficiency-of-labor growth, and depreciation rates. If the capital share parameter in the production function $ \alpha = 1/3 $ and if the divergence in efficiency-of-labor levels across countries is 7-to-1, how large a divergence in savings rates would be needed to account for observed global cross-country inequality?

A. 7-to-1

B. 49-to-1

C. $ \sqrt{7}$-to-1

D. none: the divergence in efficiency of labor levels does the job

E. none of the above/cannot be determined from the information given

 

8. Divergence

Suppose again that we are trying to account for the 49-to-1 divergence in productivity levels and living standards across the globe today using the SGM, and assuming that economies are on their steady-state balanced-growth paths. Assume also that divergence across economies in capital-output ratios is perfectly correlated with divergence in efficiency of labor levels, and that all countries now are equal in their labor force growth, efficiency of labor growth, and depreciation rates. If the capital share parameter in the production function $ \alpha = 2/3 $ and if the divergence in capital-output levels across countries is 7-to-1, how large a divergence in efficiency-of-labor levels would be needed to account for observed global cross-country inequality?

A. 7-to-1

B. 49-to-1

C. $ \sqrt{7}$-to-1

D. none: the divergence in capital-output levels does the job

E. none of the above/cannot be determined from the information given

 

9. Divergence...

What economist Richard Baldwin calls the "Ancient Seven" economies—China; what are now India and Pakistan and Bangladesh; what are now Iraq and Syria; Iran; what is now Turkey; what are now Italy and Greece; and Egypt—dominated the pre-Industrial Revolution world in terms of their shares of total world product predominantly because:

A. they had uniquely advantageous positions located on major world trade routes, so they had higher levels of the efficiency-of-labor

B. they had more and more fertile agricultural land—much more ample stocks of natural resources—than anywhere else

C. their high civilizations were more sophisticated and generated more ideas useful for production than anyplace else, so they had higher levels of the efficiency-of-labor

D. all of the above are not implausible possible scenarios

E. none of the above/cannot be determined from the information given

 

10. An Adverse Supply Shock

Suppose, in the flexible-price business cycle model, an adverse supply shock reduces potential output by an amount $ {\Delta}Y^* $. Suppose nothing else in the economic environment changes. Suppose that there are no changes in fiscal policy to government purchases or the tax rate.

Which of the following happens to the economy?

A. Consumption spending goes down by $ {\Delta}C = (c_y - im_y){\Delta}Y^* $

B. Investment spending goes down by $ {\Delta}I = $ $ -\frac{I_r}{I_r + x_{\epsilon}{\epsilon}_r}{\Delta}Y^* $

and the interest rate goes up by: $ {\Delta}r = \frac{{\Delta}Y^*}{I_r + x_{\epsilon}{\epsilon}_r} $

C. The fall in national income forces a reduction in government purchases of $ {\Delta}G = - {\Delta}Y^* $

D. The interest rate goes up by $ {\Delta}r = \frac{{\Delta}Y^*}{\mu(I_r + x_{\epsilon}{\epsilon}_r)} $

E. none of the above/cannot be determined from the information given

 

11. A Decline in Business Animal Spirits

In the flexprice IS Curve equation:

with the Keynesian multiplier:

$ \mu = \frac{1}{1 - (1-t)c_y + im_y} $

holding other things equal, a decrease in investor "animal spirits"—a fall in the parameter $ I_o $ in the equation determining the level of business investment spending—will cause:

A. a fall in investment spending I, no change in the domestic interest rate, no change in exports or government purchases, but a fall in imports and consumption spending

B. a fall in investment spending I, a rise in the domestic interest rate r to generate the fall in investment spending, a fall in the value of the dollar—the home currency—a rise in exports, and no change in government purchases, imports, or consumption spending

C. a fall in investment spending I, a fall in the domestic interest rate r, a fall in the value of the dollar—the home currency—a rise in exports, and no change in government purchases, imports, or consumption spending

D. a fall in investment spending I, a rise in the domestic interest rate r to generate the fall in investment spending, a rise in the value of the dollar—the home currency—a fall in exports, and no change in government purchases, imports, or consumption spending

E. none of the above/cannot be determined from the information given

 

12. The "Neutral" Rate of Interest

As Federal Reserve Bank of New York President John Williams defines it, the "neutral" rate of interest is the rate of interest at which total spending—aggregate demand—is equal to potential output. It is thus the same as the equilibrium interest rate in the flexible price model. Which of the following statements about influences on the neutral interest rate is correct?

A. a rise in animal spirits, a rise in foreign interest rates, and a rise in confidence in the long-run fundamental value of the domestic currency—the dollar—will all raise the neutral interest rate

B. a rise in animal spirits, a rise in foreign interest rates, and a rise in the value of the exchange rate defined as the value of foreign currency $ \epsilon $ will all raise the neutral interest rate

C. a rise in animal spirits, a rise in foreign interest rates, and a rise in confidence in the long-run fundamental value of the domestic currency—the dollar—will all lower the neutral interest rate

D. a rise in animal spirits, a rise in foreign interest rates, and a rise in the value of the exchange rate $ \epsilon $ will all lower the neutral interest rate

E. none of the above/cannot be determined from the information given

 

13. Offsetting Fiscal Contraction

In one of his early weekly lunches in 1993 with Federal Reserve Chair Alan Greenspan, the newly installed Treasury Secretary, Lloyd Bentsen, expressed concern that the 2% of GDP fiscal contraction planned by President Clinton in order to try to to balance the budget might send the economy into recession.

Greenspan expressed a willingness to keep interest rates lower than he would have done in the baseline, no-deficit reduction scenario in order to avoid this risk.

Suppose we model this as a reduction in G by an amount $ {\Delta}G = - 2 $ (% of GDP). Suppose that the Keynesian multiplier $ \mu = 3 $, and that the parameters are:

• $ I_r = \frac{2}{3}, $
• $ {\epsilon}_r = \frac{1}{3}, $
• $ x_{\epsilon} = 1 $.

By how much would Alan Greenspan have had to take steps to lower r below its baseline path in order to have kept his promise?

A. $ \frac{2}{3} $%-points

B. 6%-points

C. 2%-points

D. 3%-points

E. none of the above/cannot be determined from the information given

 

14. Foreign Monetary Policy

If foreign governments reduce their interest rates and by so doing induce a boom abroad that raises total incomes abroad, the consequence for the domestic economy in the sticky-price model is likely to be:

A. a decrease in the value of the dollar and an increase in exports, consumption spending, and national product

B. an increase in the value of the dollar and a reduction in exports, but an increase in investment spending and no change in the level of consumption spneding and national product

C. an increase in the value of the dollar and a reduction in exports, consumption spending, and national product

D. a decrease in the value of the dollar and an increase in exports, but a decrease in investment spending and no change in the level of consumption spneding and national product

E. none of the above/cannot be determined from the information given

 

15. Phillips Curve

In an economy described by the so-called Phillips Curve equation:

       $ {\pi_t} = {\pi_t}^e - \beta\left(u_t - u^*\right) $

where inflation expectations are adaptive:

       $ {\pi_t}^e = \pi_{t-1} $

a central bank that pushes or lets the unemployment rate rise and stay above its NAIRU for a long period of time will probably:

A. see a higher rate of inflation at the end of the period than at the start

B. see a lower rate of inflation at the end of the period than at the start

C. see about the same rate of inflation at the end of the period than at the start

D. none of the above/cannot be determined from the information given

 

Part B: Short Calculation Questions

Five minutes each

1. Long-Run Capital-Output Ratio

Suppose that in a Solow growth model economy the values of s, n, g, α, and δ are 0.18, 0.01, 0.02, 0.5, and 0.03 and remain constant, and suppose further that in year 0 $ E_o = 1, L_o = 1, K_o = 9 $. What is the value of the capital-output ratio $ K_t/Y_t $ in year 100? Write "B1:" and then the correct value in your bluebook.

 

2. An Increase in Savings

Suppose that in a SGM economy the production function parameter $ \alpha = 0.25 $. If the savings rate s quadruples, by what factor will the level of output per worker along the steady-state balanced-growth path be? Write "B2:" and then the correct value in your bluebook.

 

3. Sticky-Price Monetary Offset

Suppose that in a sticky-price economy the multiplier is two, a one-percentage point increase in the domestic long-term risky real interest rate discourages 200 billion of annual investment, and that foreigners raise and lower their interest rates in tandem with the domestic central bank. Suppose that the President and the Congress raise annual government purchases by 400 billion dollars. By what amount would the central bank have to change the long-term risky real interest rate to keep that expansion of government purchases from changing the level of national income and product? Write "B3:" and then the correct value in your bluebook.

 

4. Sticky-Price Financial Crisis

Between the start of 2007 and the end of 2008 the wedge between the policy interest rate the Federal Reserve controls and the long-term risky real interest rate that matters in the IS equation rose by 15 percentage points. Suppose that similar effects were happening abroad so that the interest rate differential between home and abroad did not change. Suppose further that a 1%-point rise in the interest rate discourages 100 billion of annual investment, that a 1%-point shift in the domestic-foreign interest rate differential causes a 5% shift in the value of foreign currency, and that a 1% increase in the value of foreign currency raises annual exports by 30 billion. Suppose that the multiplier was 2.5.

If the U.S. government had reacted to 2007-8 the way it reacted to 1929-1933—that is, had it done nothing—what would have been the effect, in billions, on annual national income and product?

 

Part C: Model Applications and Extensions

10 minutes each

1. A "Strong Dollar"

My boss's boss back in 1995, Clinton Administration Treasury Secretary Robert E. Rubin liked to say "a 'strong dollar' is in America's interest". In our business-cycle framework, a strong dollar is a low value of the exchange-rate-value-of-foreign-currency parameter $\epsilon $, which is itself determined by:

A "strong dollar" can thus be generated by any of:

1. A low value of $ {\epsilon}_o $, when foreign exchange speculators are confident about the fundamental value of the home currency
2. A low level of $ r^f $, interest rates abroad
3. A high level of $ r $, interest rates at home

(a) In the flexprice model, what are the consequences of a "strong dollar" generated through channel (1) or channel (2) on the economy? Use the symbols for parameters (i.e., $ I_r $), to specify what are the effects on economic variables of interest of a dollar stronger via a $ {\Delta}{\epsilon}_o = - 0.1 $ and via a $ {\Delta}r^f = - 0.02 $. (Recall that in the flexible price model, r is not something that can vary except as a result of some other change in the economic environment or in the fiscal-policy variables G and t.)

(b) In the sticky-price model, what are the consequences of a "strong dollar" generated through channel (1), (2), or (3) on the economy? Use the symbols for parameters (i.e., $ I_r $), to specify what are the effects on economic variables of interest of a dollar stronger via a $ {\Delta}{\epsilon}_o = - 0.1 $, via a $ {\Delta}r^f = - 0.02 $, or via a $ {\Delta}r = +0.02 $.

(c) Finally, what adjustments/additions/corrections to the models of this course occur to you that might make Bob Rubin's "strong dollar" mantra a more correct belief than your analyses in (a) and (b) suggest?

 

2: Solow Growth Model

In 1977 China's level of real national product per capita was 909 dollars per person a year. Forty years later, in 2017, it was 15200 dollars per person a year.

1. By what factor was China's level of output per capita greater in 2017 than it had been in 1977?

2. If the production function parameter $ \alpha = 1/3 $ happened to be true for China over this period, how much of a multiplicative increase in the capital output ratio would have been needed to drive such growth?

3. If the production function parameter $ \alpha = 1/2 $ happened to be true for China over this period, how much of a multiplicative increase in the capital output ratio would have been needed to drive such growth?

4. If the production function parameter $ \alpha = 2/3 $ happened to be true for China over this period, how much of a multiplicative increase in the capital output ratio would have been needed to drive such growth?

5. In fact, China's capital output ratio doubled between 1977 and 2017. If the production function parameter $ \alpha = 2/3 $ happened to be true for China over this period, how fast did the efficiency of labor grow on average in China from 1977 to 2017?

6. Suppose you believe that China will no longer be able to raise its capital-output ratio, but that it will still be able to increase its efficiency of labor as rapidly as it has over the past forty years. What would you forecast for the growth rate of output per capita in China over the next generation?

7. Suppose you believe that China will no longer be able to raise its capital-output ratio, and that—because China is now a middle income country—it will only be able to exceed the 1.6% per year growth rate of the efficiency of labor in the global north by half as much in the future as it has over the past forty years. What would you forecast for the growth rate of output per capita in China over the next generation?

8. Can you think of something to say about this that it is important for Xi Jinping to either learn (if he does not know it already) or keep constantly in the front of his mind (if he does know it already)?

 

Part D: Essays

10 minutes each; 150 words maximum each

1. The Current American Macroeconomy

Suppose you are Kevin Hassett, Chair of the President's Council of Economic Advisers. President Donald Trump comes to you and asks if you think he should fire Federal Reserve Chair Jay Powell because interest rates are going up, the value of foreign currencies like the renminbi, the yen, the euro, and the pound are falling, and the trade deficit is increasing. Using the ideas you have learned in this course, write what you would say to him to try to guide him to an accurate understanding of the current state of the American macroeconomy. We take the word limit seriously. So think, write that many words, and then stop.

 

2. Italy Today

Earlier this week, the extremely sharp Adam Tooze wrote:

More than 32 percent of Italy’s young people are unemployed. The gloom, disappointment and frustration are undeniable. For the commission to declare that this is a time for austerity flies in the face of a reality that for many Italians is closer to a personal and national emergency.... The Bank of Italy and the Peterson Institute of International Economics, warn that Italy is caught in a trap: Anxieties about debt sustainability mean that any stimulus has the perverse effect of driving up interest rates, squeezing bank lending and reducing growth...

Think in terms of our sticky-price model. What do you believe would have to be true about the interrelationships of the economic variables in the model for the BoI and the PIIE's worries to be correct?

We take the word limit seriously. So think, write that many words, and then stop.

 

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