Columbia University Department of Economics Lecture 21 Economics - - PowerPoint PPT Presentation

Columbia University Department of Economics Lecture 21 Economics UN3213 Intermediate Macroeconomics Professor Mart ´ ın Uribe Spring 2019

Announcements Homework 6 due now Homework 7 will be posted today, due April 22 in class. Recitations: – review of hwk6 – Numerical example in lecture 20 (fiscal deficit and the inflation tax).

Intermediate Macro Ricardian Equivalence

• M. Uribe, 2019

Topics Today

• Fiscal Policy and Ricardian Equivalence

– Motivating Questions

• How does a tax cut affect consumption, the real interest rate, and

investment?

• Do tax-cut induced fiscal deficits drive up interest rates and crowd-
• ut investment?

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View I: Tax cuts generate fiscal deficits. More government bor- rowing drives up interest rates. This crowds out investment. And tax cuts by increasing households’ disposable income stimulate con- sumption spending. (r ↑, I ↓, C ↑.) View II: Because tax cuts lead to higher public debt, sooner or later, the government must increase taxes to repay that debt (including interest). Hence tax cuts today lead to tax increases in the future. Households understand this, so they do not spend the tax cut on consumption goods. Instead, they save the tax cut in the bank to be able to pay for the future expected increases in taxes. So current spending does not increase, the interest rate does not increase, and investment does not fall. (∆r = ∆C = ∆I = 0). This result is known as Ricardian Equivalence.

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Goal To evaluate views I and II, we will build a model of the equilibrium de- termination of consumption, investment, and the real interest rate. The model will be a two-period economy like in the lectures on the short-run effects of monetary policy. However, now, we abstract from nominal price rigidity. Instead we assume that prices are fully

• flexible. We will drop nominal prices all together and focus on the

determination of real quantities only.

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A Two-Period Model of Consumption, Investment, Savings and the Interest Rate

The economy lasts for only two periods, period 1 and period 2. Basic Units of the Model

• 1. The Government
• 2. Firms
• 3. Households

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• 1. The Government

T1 = taxes in period 1 T2 = taxes in period 2 The government chooses T1 and T2. Taxes are lump sum, in the sense that they do not depend on the household’s level of income, spending, or any other manifestation of wealth. G1 = government spending in period 1 G2 = government spending in period 2 G1, G2, are exogenously given.

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Government Debt, the Primary Deficit, and the Secondary Deficit Let Bt denote government debt (bonds) issued in period t and ma- turing in period t + 1, for t = 1, 2, r = denote the interest rate, The primary fiscal deficit is the difference between government spending and tax revenue, Primary Fiscal Deficit = Gt − Tt and the secondary fiscal deficit is the primary deficit plus interest payments on the public debt Secondary Fiscal Deficit = rBt−1 + Gt − Tt. The negative of the secondary fiscal deficit is known as the secondary fiscal surplus or government savings. Thus, Government Savings = Tt − rBt−1 − Gt. How do government spending, taxes, and the primary and secondary fiscal deficits look like in the real world? The next two figures displays U.S. data over the period 1929 to 2017.

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1930 1940 1950 1960 1970 1980 1990 2000 2010 10 15 20 25 30 35 Percent of GDP Taxes and Government Spending: U.S. 1929−2017 Tt/Yt Gt/Yt (Gt+r Bt−1)/Yt 9

Intermediate Macro Ricardian Equivalence

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1930 1940 1950 1960 1970 1980 1990 2000 2010 −4 −2 2 4 6 8 10 12 Primary and Secondary Fiscal Deficit, U.S. 1929−2017 Percent of GDP (Gt−Tt)/Yt (Gt+r Bt−1−Tt)/Yt 10

Intermediate Macro Ricardian Equivalence

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Observations on the Graphs – Steady increase in the shares of government spending and taxes in GDP since 1930, from around 10% to near 30%. – Large increases in government spending and fiscal deficits are

• bserved during World War II and during the great contraction of

2007-2009. – Protracted fiscal deficits during Reagan (1981-1989), Bush senior (1989-1993), and Bush junior (2001-2009). – Only time of secondary fiscal surpluses in recent history: Clinton (1993-2001). Let’s go back to our two-period model and see what we can say about the macroeconomic effects of changes in fiscal policy

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The Government Budget Constraint in Period 1 Let Sg

1 denote government savings in period 1. Then, the budget

constraint of the government in period 1 is Sg

1 = T1 − rB0 − G1

We assume that the government starts period 1 with no debt, B0 =

• 0. Thus, the government budget constraint in period 1 is given by

Sg

1 = Tt − Gt

(1) Government savings can be positive or negative. When Sg

1 is positive,

government debt goes down, and when Sg

1 is negative, it goes up,

that is, B1 = B0 − S1. Since we assume that B0 = 0, we have that the amount of debt outstanding at the beginning of period 2 is B1 = −Sg

1.

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The Government Budget Constraint in Period 2 In period 2 (the last period of this economy), the government uses its tax revenue, T2, plus all of its period-1 savings including interest, (1+ r)Sg

1, to purchase goods, G2. The government’s budget constraint

in period 2 is then given by G2 = T2 + (1 + r)Sg

1

(2) Notice that the government does not issue debt in period 2 (B2 = 0). This is because after period 2 the world ends, so no one would buy it.

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The Intertemporal Budget Constraint of the Government Combining the period-1 and period-2 government budget constraints yields the intertemporal or present-value budget constraint of the government:

G1 + G2 1 + r = T1 + T2 1 + r

(3) This constraint says that the present discounted value of tax rev- enues (given by the right-hand side) must equal the present dis- counted value of government expenditures (given by the left-hand side).

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Summary of Government: Period 1 budget constraint: G1 + Sg

1 = T1

(1) Period 2 budget constraint: G2 = T2 + (1 + r)Sg

1

(2) Present value government budget constraint: G1 + G2 1 + r = T1 + T2 1 + r (3)

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Tax Cuts Today Imply Tax Increases in the Future Consider now a tax cut in period 1, that is, ∆T1 < 0. Suppose that the government does not change government spending in either period (i.e., ∆G1 = ∆G2 = 0). Then, to satisfy the intertemporal budget constraint, the government must raise taxes in period 2 by ∆T2 = −(1 + r)∆T1 > 0 The interpretation is that if neither current nor future government spending changes, a tax cut now (that is, in period 1) leads to a tax increase in the future (that is, in period 2). There’s no such thing as a free lunch.

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• 2. Firms
• Suppose that in period 1 firms borrow I1 units of goods and convert

them into capital goods (machines, structures, etc.)

• In period 2, firms use the capital to produce final goods using the

technology f(I1). The production function f(·) is assumed to be positive, f(I1) > 0, increasing, f′(I1) > 0, and concave, f′′(I1) < 0. Here, f′ and f′′ denote, respectively, the first and second derivatives

• f the function f(·).
• In addition, in period 2 firms must repay their loans in the amount

(1 + r)I1.

• In period 1, firms choose the level of investment, I1, so as to

maximizes profits.

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The Firm’s Profit Maximization Problem We denote the profits of the firm in period 2 by Π. Profit is given by the difference between output, f(I1), and the cost of investment, including interest, (1 + r)I1. That is Π = f(I1) − (1 + r)I1 (∗) Firms choose investment so as to maximize profits. That is, the

• ptimization problem of the firm is

max

I1

[f(I1) − (1 + r)I1] We assume that each firm borrows an amount of funds that is too small to affect the interest rate. Therefore, it makes sense to assume that in choosing I1 firms take the interest rate, r, as exogenously given.

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The first-order condition associated with the firm’s optimization problem is obtained by taking the derivative of profits with respect to I1 and equating the result to zero. This yields f′(I1) = 1 + r The left-hand side of this expression is the marginal product of cap- ital, and measures the increase in output due to a unit increase in capital. The right-hand side of the firm’s optimality condition is the marginal cost of capital which measures the marginal increase in cost of production due to a unit increase in capital. In this case, we have that the marginal cost of capital equals (1 + r). This marginal cost is composed of the unit of good borrowed in period 1, and the interest or financial cost it generates, r. The figure on the next slide illustrates the firm’s optimal investment choice.

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The Firm’s Optimal Investment Choice

f ′(I1) 1 + r I1 I1

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Now note that because the production function is assumed to be strictly concave, the marginal product of capital, f′(I1), is decreasing in the level of investment. If the marginal cost of capital increases, i.e., if the interest rate, r, increases, then in order for the optimality condition of the firm to hold investment must fall. This effect is illustrated in the next slide. In the figure, the interest rate increases from r to r′, causing a fall in investment from I1 to I′

1.

It follows that there exists a downward sloping relationship between r and I1, which we refer to as the investment schedule: I1 = I(r), with I′(r) < 0, where I′(r) denotes the derivative of I(r) with respect to r.

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Increase in the Inteest Rate and Optimal Investment

f ′(I1) 1 + r 1 + r′ I1 I1 I′

1

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Intermediate Macro Ricardian Equivalence

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The Investment Schedule It follows that there exists a downward sloping relationship between r and I1, which we refer to as the investment schedule: I1 = I(r), with I′(r) < 0, where I′(r) denotes the derivative of I(r) with respect to r. The figure displays the investment schedule in the space (I1, r).

I(r) I1 r

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How does the investment schedule I(r) change in response to a tax cut in period 1, ∆T1 < 0, coupled with a tax increase in period 2 of magnitude ∆T2 = −(1 + r)T1 > 0? The answer is that the investment schedule does not change at all. The reason is that neither T1 nor T2 appear in the firms problem.(As we will see shortly, both T1and T2 are taxes imposed on consumers, not on firms.)

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How do profits depend on the interest rate and on taxes? At the op- timum level of investment, profits are given by the following function

• f the interest rate:

Π(r) ≡ f(I(r)) − (1 + r)I(r). Now take derivative with respect to r to obtain Π′(r) = f′(I(r))I′(r) − I(r) − (1 + r)I′(r). Recall that, by the first-order condition of the profit maximization problem we have that f′(I1) = 1 + r. This means that the first and last terms on the right hand side of the above expression cancel each

• ther.

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Then we have that Π′(r) = −I(r) < 0 This expression says that when the interest rate rises, profits go

• down. This makes sense, because an increase in r raises the financial

cost of investment, thereby reducing the profitability of the firm. As we will see below firms are owned by households. Thus profits will be part of the second-period income of households. So when the interest rate increases (r ↑), that part of second-period income falls (Π ↓).

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Summary of Firm Behavior Investment Demand Schedule: I1 = I(r) with I′(r) < 0 Profits: Π(r) with Π′(r) < 0

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An Example

Suppose that the production function is of the form f(I1) = 2

• I1.

This production function is positive, strictly increasing, and concave. The marginal product of capital in this case, is given by MPK ≡ f′(I1) =

• 1

I1 . The marginal product of capital is decreasing in I1. Or, in other words, the production technology displays diminishing returns to

• scale. Equating MPK to MCK, we obtain
• 1

I1 = 1 + r.

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Solving this optimality condition for I1 yields the investment schedule I(r) =

• 1

1 + r

2

This expression says that the optimal level of investment is a strictly decreasing function of the real interest rate. We can also obtain the

• ptimal level of profits as a function of the real interest rate. To this

end, start with the definition of profits and then replace investment for its optimal value: Π(r) ≡ f(I(r)) − (1 + r)I(r) = 2

• 1

1 + r

2

− (1 + r)

• 1

1 + r

2

= 1 1 + r According to this expression, the optimal level of profits is a de- creasing function of the real interest rate, which is in line with our previous result.

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Next lecture we will model the consumption and savings behavior

• f the households. Once we have that, we can answer the question
• f what happens if the government cuts taxes in period 1, will it

increase consumption, drive up rates, and crowd-out investment or not.

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