# Version "A"

## Multiple Choice Questions

### A.1: Solow Growth Model

We have the Solow Growth Model (SGM) system of equations:

$ \frac{d\left(L_t\right)}{dt} = nL_t $ :: labor force growth equation

$ \frac{d\left(E_t\right)}{dt} = gE_t $ :: efficiency of labor growth equation

$ \frac{d\left(K_t\right)}{dt} = sY_t - \delta{K_t} $ :: capital stock growth equation

$ Y_t = \left(K_t\right)^{\alpha}\left(L_tE_t\right)^{1-\alpha} $ :: production function

#### A.1.1: Long-Run Capital-Output Ratio

If s, n, g, α, and δ remain constant at their initial values, toward what value is the capital-output ratio $ K_t/Y_t $ converging in the long run? Set the value of the variable in the code cell immediately below to the number of the correct answer:

$ \lim\limits_{t\to\infty}\left(\frac{K_t}{Y_t}\right) = \frac{s}{n+g+\delta} $

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

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

none of the above

#### A.1.2: Long-Run Output-per-Worker Path

If s, n, g, α, and δ remain constant at their initial values, toward what path is the output-per-worker ratio $ Y_t/L_t $ converging in the long run? Set the value of the variable in the code cell immediately below to the number of the correct answer:

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

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

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

none of the above

#### A.1.3: Long-Run Capital-per-Worker Path

If s, n, g, α, and δ remain constant at their initial values, toward what path is the capital-per-worker ratio $ K_t/L_t $ converging in the long run? Set the value of the variable in the code cell immediately below to the number of the correct answer:

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

$ \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) $

$ \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) $

none of the above

#### A.1.4: An Increase in Savings: Output

If the savings rate s in a SGM economy doubles, then the level of output per worker along the economy's steady-state growth path will:

double

halve

increase by an amount that will be more than a doubling if the production function parameter $ \alpha > \frac{1}{2} $, but increase by an amount that will be less than a doubling if the production function parameter $ \alpha < \frac{1}{2} $.

none of the above

#### A.1.5: An Increase in Savings: Capital

If the savings rate s in a SGM economy doubles, then the level of the capital-output ratio along the economy's steady-state growth path will:

double

halve

increase by an amount that will be more than a doubling if the production function parameter $ \alpha > \frac{1}{2} $, but increase by an amount that will be less than a doubling if the production function parameter $ \alpha < \frac{1}{2} $.

none of the above

#### A.1.6: An Increase in the Production Function Parameter $ \alpha $

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

faster convergence to its steady-state balanced-growth path

slower convergence to its steady-state balanced-growth path

the same speed of convergence to its steady-state balanced-growth path

none of the above

#### A.1.7: An Increase in the Production Function Parameter $ \alpha $

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

a larger difference in output per worker levels along their steady-state balanced-growth paths

a smaller difference in output per worker levels along their steady-state balanced-growth paths

there is not enough information to determine what will be the difference in output per worker levels along their steady-state balanced-growth paths

none of the above

#### A.1.8: An Increase in the Production Function Parameter $ \alpha $

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

a larger difference in capital per worker levels along their steady-state balanced-growth paths

a smaller difference in capital per worker levels along their steady-state balanced-growth paths

there is not enough information to determine what will be the difference in capital per worker levels along their steady-state balanced-growth paths

none of the above

#### A.1.9: Malthusian Economies

The slow pace of increase of average output per capita in the world economy over the long Agrarian Age—from 5000 BC to 1800—is best accounted for by:

the absence of intellectual property institutions that allowed inventors to profit mightily from their inventions

population growth that generated increasing resource scarcity that offset what would otherwise have been the positive effects of inventions and innovations on the efficiency of labor E

wars that kept the capital stock from growing rapidly

none of the above

#### A.1.10: Post-Malthusian Economies

The rapid pace of increase of average output per capita in the world economy since the end of the long Agrarian Age—since about 1800—is best accounted for by:

the presence of intellectual property institutions that allowed inventors to profit mightily from their inventions

population growth that has been slower than before and thus did not generate resource scarcity to offset what would otherwise have been the positive effects of inventions and innovations on the efficiency of labor E

the absence of wars and thus the ability of the capital stock to grow rapidly

none of the above

#### A.1.11: Divergence

Since 1800 the global distribution of income and wealth has become:

much more unequal as imperialism and capitalism have absolutely impoverished the global south

much more equal as the knowledge gained since the start of the Industrial Revolution diffuses around the world in a heartbeat due to our advanced information and communication technologies

much more unequal as productive use of knowledge about modern machine technologies has required effective communities of engineering practice, and those have proven difficult to construct outside of the global north

none of the above

#### A.1.12: Convergence

Since 1970 the global distribution of income and wealth has become:

somewhat more equal as China (and to a lesser extent India) have had two miraculously good generations of growth

somewhat more unequal as the end of the age of colonialism has been followed by an age of neocolonialism

somewhat more unequal as the growth of global value chains have allowed the global north to increase the rate of exploitation in the global south

none of the above

#### A.1.13: In the Future...

We are likely to see the global distribution of income and wealth:

remain the same as corruption and other forms of rent extraction hobble attempts at development in the global south

become much more equal as the knowledge gained since the start of the Industrial Revolution diffuses around the world in a heartbeat due to our advanced information and communication technologies

become more unequal as further technological progress competes withe and reduces the value of the "unskilled" human brain while the concentration of physical capital and intellectual property wealth increases

any of the above is not an implausible scenario—and their are other plausible scenarios too

#### A.1.14: Long Run Growth

The most important determinant of the long-run rate of growth of an economy is:

the presence of intellectual property institutions that allowed inventors to profit mightily from their inventions

its rate of growth of the efficiency of labor E

its steady-state balanced-growth capital-output ratio (K/Y)*

none of the above

#### A.1.15: Demographic Transition

The principal reason that the world is now closing in on zero population growth and a stable population of about 10 billion rather than seeing population double every generation or two is:

that with greatly reduced child mortality families no longer believe they need to have as many children as possible in the hope that at least one will outlive them, and the power of literate and relatively prosperous women to control their own fertility

that resource scarcity in the form of limited farmland makes large families unaffordable in the global south, and resource scarcity in the form of a very high price of education makes large families unaffordable in the global north

the development of Social Security and other government-sponsored social insurance schemes has greatly lessened the perceived need people see to have children who will outlive them

none of the above

### A.1: Business Cycle Models

#### A.1.16: IS Curve

Shifting gears from long-run growth to short-run business cycles, a key tool in both the flexible-price and the stick-price models is the so-called IS Curve equation, in its two forms:

with the Keynesian multiplier:

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

Of these two forms, (A) and (B):

the sticky-price model uses form (A) and the flexible-price model uses form (B)

the flexible-price model uses form (A), but (B) is not the form used in the sticky-price model

the sticky-price model uses form (B) and the flexible-price model uses form (A)

none of the above

#### A.1.17: Exchange Rate

In the so-called IS Curve equation:

with the Keynesian multiplier:

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

an increase in the value of the dollar—of the home currency—that makes home-produced goods less attractive to foreigners and so discourages exports is modeled by:

an increase in the value of the exchange rate variable $ \epsilon $

a decrease in the value of the exchange rate variable $ \epsilon $

a decrease in the value not of the exchange rate variable $ \epsilon $ but of the parameter $ {\epsilon}_o $

none of the above

#### A.1.18: Determinants of the Multiplier

In the so-called IS Curve equation:

with the Keynesian multiplier:

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

comparing two otherwise identical sticky-price economies, the one with the higher value of the marginal propensity to consume c_y will have a:

higher value of the Keynesian multiplier $ \mu $ and thus a larger positive impact on the level of real national income and product of an increase $ {\Delta}G $ government purchases

lower value of the Keynesian multiplier $ \mu $ and thus a larger positive impact on the level of real national income and product of an increase $ {\Delta}G $ government purchases

higher value of the Keynesian multiplier $ \mu $ and thus a smaller positive impact on the level of real national income and product of an increase $ {\Delta}G $ government purchases

none of the above

#### A.1.19: Determinants of the Multiplier

In the so-called IS Curve equation:

with the Keynesian multiplier:

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

an increase in the share of income that households spend on imports—an increase in the parameter $ im_y $ is associated with a:

higher value of the Keynesian multiplier $ \mu $ and thus a larger positive impact on the level of real national income and product of an increase $ {\Delta}G $ government purchases

lower value of the Keynesian multiplier $ \mu $ and thus a larger positive impact on the level of real national income and product of an increase $ {\Delta}G $ government purchases

lower value of the Keynesian multiplier $ \mu $ and thus a smaller positive impact on the level of real national income and product of an increase $ {\Delta}G $ government purchases

none of the above

#### A.1.20: Investment and Interest Rates

In the so-called IS Curve equation:

with the Keynesian multiplier:

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

in the flexible-price model, 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 rise in the domestic long-term risky real interest rate r

a decline in the domestic long-term risky real interest rate r

a rise in the value of the foreign long-term risky real interest rate $ r^f $

none of the above

#### A.1.21: Investment and Interest Rates

In the so-called IS Curve equation:

with the Keynesian multiplier:

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

in the sticky-price model, 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 decline in the domestic long-term risky real interest rate r

a decline in the level of national income and product Y

a rise in the value of the foreign long-term risky real interest rate $ r^f $

none of the above

#### A.1.22: Investment and Interest Rates

In the so-called IS Curve equation:

with the Keynesian multiplier:

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

in the sticky-price model, 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 induce a central bank that is trying to keep national income and product Y from changing either up or down to:

take steps to boost the short-term safe nominal policy interest rate i it controls, and so induce a rise in the domestic long-term risky real interest rate r

take steps to cut the short-term safe nominal policy interest rate i it controls, and so induce a fall in the domestic long-term risky real interest rate r

take steps to boost the short-term safe nominal policy interest rate i it controls, and so induce foreign central banks to react and thus trigger a rise in the value of the foreign long-term risky real interest rate $ r^f $

none of the above

#### A.1.23: 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 fall and stay below its NAIRU for a long period of time will probably:

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

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

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

none of the above

#### A.1.24: 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 rational:

$ {\pi_t}^e = \pi_t $

a central bank that attempts to push or lets the unemployment rate fall and stay below its NAIRU for a long period of time will probably:

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

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

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

none of the above

#### A.1.25: 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} $

you can derive the following equation for how inflation evolves over time:

$ {\pi_t} = \pi_{t-1} - \beta\left(u_t - u^*\right) $

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

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

none of the above

#### A.1.26: 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 a hybrid of static and adaptive:

$ {\pi_t}^e = \lambda\pi^* + (1 - \lambda)\pi_{t-1} $

you can derive the following equation for how inflation evolves over time:

$ {\pi_t} = \pi_{t-1} - \frac{\beta\left(u_t - u^*\right)}{1-\lambda} $

$ {\pi_t} = (1 - \lambda)\pi_{t-1} - \beta\left(u_t - u^*\right) $

$ {\pi_t} = \pi_{t} - \frac{\beta\left(u_t - u^*\right)}{1-\lambda} $

none of the above

#### A.1.27: Monetary Policy

Consider an economy where the long-term risky real interest rate r is equal to the short-term safe nominal policy rate i controled by the central bank, minus the inflation rate $ π $, plus the spread $ \rho $ between the interest rate at which businesses can borrow long-term and the interest rate at which the government can borrow short term. If there is thought to be only a negligible chance that the central bank will ever wish that it could lower its policy rate i below zero, a prudent central bank should probably:

focus on keeping inflation at a moderate level so that it can reduce r to negative values on occasion

focus on keeping inflation low and stable so as not to make economic calculation more unpredictable

ignore inflation and focus on keeping unemployment from rising above the natural rate

none of the above

#### A.1.28: Monetary Policy

Consider an economy where the long-term risky real interest rate r is equal to the short-term safe nominal policy rate i controlled by the central bank, minus the inflation rate $ π $, plus the spread $ \rho $ between the interest rate at which businesses can borrow long-term and the interest rate at which the government can borrow short term. If there is thought to be a substantial chance that the central bank will ever wish that it could lower its policy rate i below zero, a prudent central bank should probably:

focus on keeping inflation at a moderate level so that it can reduce r to negative values on occasion

focus on keeping inflation low and stable so as not to make economic calculation more unpredictable

ignore inflation and focus on keeping unemployment from rising above the natural rate

none of the above

#### A.1.29: Monetary Policy

The principal reason that then-Federal Reserve Chair Alan Greenspan was largely unworried about speculation in housing and the rise in housing prices in the mid-2000s was probably:

he believed the Federal Reserve had the tools to prevent even a large financial crisis in housing finance from having large effects raising the interest rate spread $ \rho $ and lowering business "animal spirits" $ I_o $

he believed that a large financial crisis in housing finance having large effects raising the interest rate spread $ \rho $ and lowering business "animal spirits" $ I_o $ would be ultimately beneficial because it would see irrational speculators with bad judgment replaced by prudent managers, and their would be long-term gain for short-run pain

he believed that there were very large benefits from a large expansion of America's housing stock, even if those who financed that expansion wound up bankrupt

none of the above

#### A.1.30: Monetary Policy

From 1979 to 1982 the principal reason that then-Federal Reserve Chair Paul Volcker raised interest rates high and kept them high was probably because:

he believed that inflation expectations were rational and thus that he could reduce inflation without seeing a significant or long lasting rise in unemployment

he believed that inflation of between 5% and 10% per year had a serious effect reducing investment and thus slowing long run growth

he believed it was very important to make inflation expectations a hybrid of static and adaptive rather than see them become a hybrid of adaptive and rational because the second would produce a very unstable economy

none of the above

### A.2: Calculations

#### A.2.1: Solow Growth Model

On December 19, 2017, economist Robert Barro of Harvard University wrote https://tinyurl.com/dl20180428d that the effects on economic growth of Donald Trump and the Republicans' TCJA would raise real GDP Y in the long run by 7% and over ten years by 2.8% because:

[with] a capital share of income of 40%, the convergence rate in the neoclassical growth model would be around 5% per year... [with] the long-run 7% change... after ten years, the level of real per capita GDP is higher by 2.8%.

Let's suppose that we have a SGM economy with a savings share of 24%, an efficiency of labor growth rate of 1.5% per year, a population growth rate of 1% per year, and a depreciation rate of 3.5% per year. With a capital share production function parameter $ \alpha $ of 40%, what is the rate of annual convergence to the steady-state balanced-growth path in the SGM? How high would the sum of the efficiency of labor growth rate of 1.5% per year, the population growth rate of 1% per year, and the depreciation rate have to be to generate a rate of annual convergence of 5% per year? What does Barro's claim of a 7% long run boost to output per worker along the steady-state balanced-growth path require for the steady-state balanced-growth path capital-output ratio? How big a jump in the savings rate s would that require?

In the code cell below, set the variables:

```
convergence_rate
required_sum
required_KoverY
required_savings
```

to the answers to these four corresponding questions.

**Relevant equations from the SGM framework are**:

$ \left(\frac{K}{Y}\right)^* = \frac{s}{n+g+\delta} $ :: steady-state balanced-growth capital-output ratio

$ \frac{Y_t}{L_t} = \left(\frac{K_t}{Y_t}\right)^\left(\frac{\alpha}{1-\alpha}\right)\left({E_o}e^{gt}\right) $ :: output per worker as a function of the capital-output ratio

$ \frac{K_t}{Y_t} = \left(1- e^{-(1-\alpha)(n+g+\delta)t}\right)\left(\frac{K}{Y}\right)^* + \left(e^{-(1-\alpha)(n+g+\delta)t}\right)\left(\frac{K_o}{Y_o}\right) $ :: convergence to the steady-state balanced-growth capital-output ratio

#### A.2.2: Greece

Countries sometimes choose to use monetary policy to fix their exchange rate at some value $ \epsilon^* $. in the case of Greece in this millennium, it has chosen to be part of the eurozone and so has given up all independent freedom of action to alter the long-term real risky interest rate r in Greece's IS Curve equation:

Instead of choosing r, the requirement that the exchange rate be $ \epsilon^* $ forces the interest rate to be:

$r = r^f - \frac{\epsilon^* - \epsilon_o}{\epsilon_r} $

The higher are foreign interest rates $ r^f $, the higher the domestic interest rate has to be; the higher are foreign exchange rate speculators' beliefs $ \epsilon_o $ about the fundamental value of foreign currency, the igher the domestic interest rate has to be; and the higher is the value of foreign currency that the country has set its currency peg to, the lower can the domestic interest rate r be. If you substitute this exchange peg maintaing value of r into the IS Curve equation, you get:

$ Y = E = \mu\left(c_o + I_o + G\right) + {\mu}x_f{Y^f} + {\mu}\left(x_{\epsilon} + \frac{I_r}{\epsilon_r}\right){\epsilon}^* - \frac{{\mu}{I_r}{\epsilon_o}}{\epsilon_r} - {\mu}I_r{r^f}$

The first half of this equation—through the $ {\mu}x_f{Y^f} $ term—is the same as the IS Curve equation. But the second half of the equation tells us that:

- The more valuable is the domestic currency at the peg—the lower is $ \epsilon^* $—the lower is national income and product Y. Why? Because a higher value of domestic currency at the peg discourages exports directly, and also requires a higher domestic real interest rate to balance supply and demand for foreign exchange at that peg, and that higher rate discourages investment.
- The less confident are foreign exchange speculators in the domestic currency—the higher is $ \epsilon_o $—the lower is Y. Why? Because foreign exchange speculators less confident about the peg also requires a higher domestic real interest rate to balance supply and demand for foreign exchange at that peg, and that higher rate discourages investment.
- The higher are interest rates abroad $ r^f $, the lower is Y. Why? Because the domestic interest rate must rise and fall with foreign interest rates in order to maintain the currency peg, and so a higher value of interest rates abroad requires a higher domestic real interest rate to balance supply and demand for foreign exchange at that peg, and that higher rate discourages investment.

Why did Greece not expand government purchases—not raise G—when the second wave of the financial crisis hit it in 2010? Because—accurate—beliefs that Greek fiscal policy was unsustainable led foreign exchange speculators to tie their beliefs about $ \epsilon_o $ to the level of government purchases and taxes. Let's write down an equation for these beliefs:

$ \epsilon_o = \gamma_o - \gamma_GG $

If $ I_r =10 $ and if $ \epsilon_r = 4 $, what is the value of the parameter $ \gamma_G $ at which Greece would have been correct not to increase government purchases—at which increases in Greek government purchases in 2010 and after would have not increased Y, and above which increases in government purchases would have reduced Greek national income and product Y? Set the variable

```
PS13A22-critical_gamma_g
```

in the code cell below equal to that value.

(All this, of course, assumes that Greece is going to—for whatever reason—stay in the eurozone and thus maintain its currency peg. Which it did.)

### A.3: Essay

#### A.3.1: U.S. Policy Dilemmas Today

The major question in American macroeconomic policy today is whether the United States should raise its inflation target above 2%. What do you think about this issue? Use material taught in this course to buttress your argument where appropriate. Alternatively, attack material taught in this course as unhelpful or misleading in terms of thinking about this problem.

We will read at most 400 words. So think, write that many, and then stop.

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