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Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Macroeconomic Theory Lecture Notes

III. A Two-Period Model of the Macroeconomy

Chunzan Wu University of Miami

October 15, 2017

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Outline

The Representative Household

The Representative Firm

Competitive Equilibrium

Applying the Theory: Ricardian Equivalence

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

A Two-Period Model of the Macroeconomy

In this part of the course, we construct a two-period model of the market economy, which is an extension to the one-period model by including the dynamic decisions of economic agents. We refer to the first period as the current period, and the second period as the future period.

In each period, the structure of the economy is similar to the one-period model. There are three economic agents: a representative household, a representative firm, and a government. The household is the consumer, worker and owner of the firm. The firm hires the worker, produces and sells the output to the consumer, and transfers all the profits to its owner. The government taxes the household to finance government expenditures.

Between the current and future periods, the household and the government can transfer wealth through a credit market. That is, they can save or borrow in the current period by purchasing or issuing a risk-free bond in the credit market. The firm, on the other hand, can invest in the current period to increase its capital stock in the future period.

As before, we start with analyzing the household’s and the firm’s optimization problems. And then we define and characterize the competitive equilibrium. Finally, we discuss in our model a key result in macroeconomics called the Ricardian equivalence theorem.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Outline

The Representative Household

The Representative Firm

Competitive Equilibrium

Applying the Theory: Ricardian Equivalence

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Modeling the Representative Household

Previously, we have studied the consumption-leisure decision problem of households in the one-period model. Now in the two-period model, we study the consumption-savings decision problem of households. A household’s consumption-savings decision is a dynamic decision, in that it has implications over more than one period of time, as opposed to the household’s static consumption-leisure decision.

A household’s consumption-savings decision is fundamentally a decision involving a trade-off between current and future consumption. By saving, a household gives up consumption in exchange for assets in the present to consume more in the future. Alternatively, a household can dissave by borrowing in the present to gain more current consumption, thus sacrificing future consumption when the loan is repaid. Borrowing (or dissaving) is thus negative savings.

For simplicity, we abstract from the representative household’s consumption-leisure trade-off in the two-period model, so household income is not affected by the household’s own choices. Alternatively, we can think it as the household does not care about leisure, so it will always work the maximum amount of time possible.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

The Representative Household’s Preference

The representative household’s preferences over current and future consumption are represented by a utility function U(c,c′). That is, if the household consumes c ≥ 0 units of the consumption good in the current period and c′ ≥ 0 units of the consumption good in the future, then its utility is given by U(c,c′).

We assume that the utility function is monotonic and concave. • Monotonicity: For all (c2,c′2) and all (c1,c

′ 1) such that c2 ≥ c1, c′2 ≥ c′1, and either

c2 > c1 or c′2 > c ′ 1, we have

U(c2,c ′ 2) > U(c1,c

′ 1)

• Concavity: For all (c2,c′2), (c1,c ′ 1) and all 0 < α < 1, we have

U(αc1 + (1 −α)c2,αc′1 + (1 −α)c ′ 2) > αU(c1,c

′ 1) + (1 −α)U(c2,c

′ 2).

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Indifference Curves

The utility function can be represented graphically in the (c,c′) space by indifference curves. The indifference curves are downward sloping because the utility function is monotonic, and the indifference curves are convex because the utility function is concave.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Marginal Rate of Substitution

The marginal rate of substitution of consumption in the current period for consumption in the future period, or MRSc,c′ , is minus the slope of an indifference curve passing through (c,c′).

MRSc,c′ is the number of units of future consumption that just compensate the household for losing one unit of current consumption.

1. The change in the household’s utility from losing ∆c units of current consumption is

∆Uc = U(c− ∆c,c′) −U(c,c′) ≈−U1(c,c′)∆c;

2. The change in the household’s utility from increasing ∆c′ units of future consumption is

∆Uc′ = U(c,c ′ + ∆c

′ ) −U(c,c′) ≈ U2(c,c′)∆c′;

3. To keep the household’s utility unchanged, we must set ∆c and ∆c′ such that

∆Uc + ∆Uc′ = −U1(c,c′)∆c + U2(c,c′)∆c′ = 0

⇒ ∆c′

∆c = U1(c,c

′)

U2(c,c′) = MRSc,c′

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Set of Feasible Choices

We say that a bundle (c,c′) belongs to the set of feasible choices if the household can afford it given its endowment and the market prices.

• The household receives income y in the current period and y′ in the future period. • The government taxes the household through a lump-sum tax t in the current period

and t′ in the future period. • The household can save (or borrow) in the current period through a credit market, and

the interest rate on savings (or loans) is r.

The household’s budget constraint in the current period is then

c + s ≤ y − t

where s is the amount of savings (or loans if s is negative).

The household’s budget constraint in the future period is

c ′ ≤ y′ − t′ + (1 + r)s

Therefore, a bundle (c,c′) is feasible iff there exists an s such that both the current and future budget constraints are satisfied.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

We can rewrite the current budget constraint as

y − t− c ≥ s,

and rewrite the future budget constraint as

s ≥ c′ − (y′ − t′)

1 + r .

Substitute the first constraint into the second, and we have the lifetime budget constraint

c + c′

1 + r ≤ y − t +

y′ − t′

1 + r .

Therefore, a bundle (c,c′) is feasible iff the lifetime budget constraint is satisfied.

Remarks: Note that 1 1+r

is the price of future consumption relative to current consumption. The left-hand side of the lifetime budget constraint is the cost of the bundle (c,c′) in terms of current consumption, i.e., the present value of lifetime consumption, and the right-hand side is the present value of the household’s lifetime disposable income. The lifetime budget constraint states that the present value of lifetime consumption must be less than or equal to the present value of lifetime disposable income.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

The set of feasible choices can be represented graphically in the (c,c′) space. • The bundle (c,c′) = (y − t,y′ − t′) is always on the budget line, i.e., the household

can always just consume its disposable income in each period without saving or borrowing, and we call it the endowment point.

• The slope of the budget line is −(1 + r). • The lifetime (disposable) income of the household is we = (y − t) + y

′−t′

1+r .

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Household Optimization

We assume the household chooses (c,c′) within the household’s set of feasible choices to maximizes its utility. Formally, the household’s choice (c∗,c′∗) is the solution to the following maximization problem

max (c,c′)

U(c,c ′ )

s.t. c + c′

1 + r ≤ y − t +

y′ − t′

1 + r ,

c ≥ 0, c′ ≥ 0.

Alternatively, it is the same as

max (c,c′,s)

U(c,c ′ )

s.t. c + s ≤ y − t, c ′ ≤ y′ − t′ + (1 + r)s,

c ≥ 0, c′ ≥ 0.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Graphical Solution to the Household’s Problem

The household’s optimal consumption bundle is determined by where an indifference curve is tangent to the budget constraint. The household is a lender if the optimal current consumption is lower than the current disposable income:

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

The household is a borrower if the optimal current consumption is higher than the current disposable income:

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Note that at such tangent point, i.e., the optimal consumption bundle (c∗,c′∗)), two optimality conditions must hold:

1. The lifetime budget constraint holds with equality:

c ∗

+ c′∗

1 + r = y − t +

y′ − t′

1 + r .

2. The marginal rate of substitution of current consumption for future consumption (the slope of the indifference curve in absolute value) must equal (1 + r) (the slope of the budget constraint in absolute value):

MRSc,c′(c ∗ ,c ′∗

) = 1 + r.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Analytic Solution to the Household’s Problem

max (c,c′)

U(c,c ′ )

s.t. c + c′

1 + r ≤ y − t +

y′ − t′

1 + r ,

c ≥ 0, c′ ≥ 0.

Assuming an interior solution, we ignore the nonnegative constraints and define the Lagrangian function as

L = U(c,c ′ ) −λ

[ c +

c′

1 + r − (y − t) −

y′ − t′

1 + r

] The first-order conditions are:

∂L

∂c = U1(c,c

′ ) −λ = 0, (1.1)

∂L

∂c′ = U2(c,c

′ ) −

1 + r = 0, (1.2)

∂L

∂λ = c +

c′

1 + r − (y − t) −

y′ − t′

1 + r = 0. (1.3)

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Equation (1.3) is just the lifetime budget constraint. Combining Equation (1.1) and (1.2), we have

U1(c,c ′ ) = (1 + r)U2(c,c

′ ). (1.4)

Alternatively, using our previous formula for MRSc,c′, we can rewrite Equation (1.4) as

MRSc,c′ = U1(c,c

′)

U2(c,c′) = 1 + r.

We can find a (potential) solution (c∗,c′∗) to the household’s optimization problem by searching for the value of (c,c′) that satisfies Equation (1.3) and (1.4) simultaneously.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Comparative Statics

Question 1: What happens when the current income of the household increases?

When current income increases from y1 to y2, lifetime income we increases.

∆we = we2 −we1 = y2 −y1 = ∆y.

Hence the lifetime budget constraint shifts out, and the slope of the constraint remains unchanged.

Assuming both the current and future consumption are normal goods, consumption increases in both periods, i.e, ∆c = c2 − c1 > 0, and ∆c′ = c′2 − c′1 > 0. The lifetime budget constraint implies

∆c + ∆c′

1 + r = ∆we = ∆y,

⇒ ∆y > ∆c, ⇒ ∆s = s2 −s1 = (y2 − t− c2) − (y1 − t− c1) = ∆y − ∆c > 0,

i.e., the increase in current consumption is smaller than the increase in current income, and savings increase.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Question 2: What happens when the future income of the household increases?

When future income increases from y′1 to y ′ 2, lifetime income we increases.

∆we = we2 −we1 = y′2 −y′1 1 + r

= ∆y′

1 + r .

Hence the lifetime budget constraint shifts out, and the slope of the constraint remains unchanged.

Assuming both the current and future consumption are normal goods, consumption increases in both periods, i.e, ∆c = c2 − c1 > 0, and ∆c′ = c′2 − c′1 > 0. The change of savings in the current period is

∆s = s2 −s1 = (y − t− c2) − (y − t− c1) = −∆c < 0,

i.e., savings decrease. The lifetime budget constraint implies

∆c + ∆c′

1 + r = ∆we =

∆y′

1 + r ,

⇒ ∆y′ > ∆c′

i.e., the increase in future consumption is smaller than the increase in future income.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Our comparative static results about current and future income changes reflect the household’s desire for consumption smoothing. In other words, the household wants to spread its extra income into its entire life span rather than consuming all of it at one time. Similarly, when hit by a negative income change, the household wants to spread the loss into multiple periods such that consumption in each period does not fall a lot. The household achieves this goal by adjusting its savings.

The permanent income hypothesis: Milton Friedman argued that a primary determinant of a household’s current consumption is its permanent income (e.g., the lifetime income we in our model). Changes in income that are temporary yield small changes in permanent income, which have small effects on current consumption, whereas changes in income that are permanent have large effects on permanent income and current consumption.

In our model, we can show the effects of temporary versus permanent changes in income by examining an increase in income that occurs only in the current period versus an increase in income occurring in the current period and the future period.

• A temporary income change: ∆y = ∆, ∆y′ = 0; • A permanent income change: ∆y = ∆y′ = ∆.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

The permanent income hypothesis is qualitatively consistent with the data, i.e, household consumption responds more to permanent income changes than to temporary ones. However, quantitatively, consumption responses to temporary income changes are larger than what the theory predicts. One potential explanation is that some people are credit constrained: they would like to borrow but they cannot and when they get an increase in income they spend it all.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Question 3: What happens when the interest rate increases?

An increase in the real interest rate causes the lifetime budget line of the household to become steeper and to pivot around the endowment point.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Substitution effect: An increase in r causes future consumption to become cheaper relative to current consumption, and induces the household to consume more in the future and less in the current period.

Income effect: An increase in r may have positive or negative income effect depending on whether the household is a lender (saver) or borrower in the current period.

• If the household is a lender, the income effect is positive, and it increases both current and future consumption. Combining the substitution effect with the income effect, we know future consumption must increase, but current consumption could either increase or decrease depending on the relative strength of the income and substitution effects. Consequently, savings could either decrease or increase.

• If the household is a borrower, the income effect is negative, and it decreases both current and future consumption. Combining the substitution effect with the income effect, we know current consumption must decrease, and hence savings must increase. Future consumption could either increase or decrease depending on the relative strength of the income and substitution effects.

Remarks: Note that we are assuming that the household will remain a lender or borrower after the interest rate change when discussing the sign of the income effect.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

(a) Household is a lender. (b) Household is a borrower.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Outline

The Representative Household

The Representative Firm

Competitive Equilibrium

Applying the Theory: Ricardian Equivalence

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

The Representative Firm’s Technology

The representative firm, as in the one-period model, produces goods using inputs of labor and capital. The key differences here are that output is produced in both the current and future periods, and that the firm can invest in the current period by accumulating capital so as to expand the capacity to produce future output. The production function in the current period is

Y = zF(K,N d ),

where Y is current output, z is total factor productivity, F is the production function, K is current capital, and Nd is current labor input. Similarly, the production function in the future period is

Y ′

= z ′ F(K

′ ,N

d′ ),

where Y ′, z′, K′ and Nd′ are the corresponding variables for the future period.

We suppose that some of the capital stock wears out through use in each period. That is, there is depreciation, and we assume that the depreciation rate is a constant d, where 0 ≤ d ≤ 1. Hence if the capital stock is K at the beginning of a period, only (1 −d)K capital stock will be left after use in production.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

For simplicity, we assume that the firm can convert one unit of the consumption good in the current period into one unit of capital in the future period. Letting I denote the quantity of current investment measured in units of current consumption, we have

K ′

= (1 −d)K + I.

This often referred to as the law of motion for capital. Because the future period is the last period, it would not be useful for the representative firm to retain any capital after production in the future period, and so the firm liquidates it. We suppose that the firm can covert the capital left at the end of the future period, (1 −d)K′, one-for-one back into the consumption good, which it can then sell.

Therefore, the current profits in units of current consumption are

π = Y −wNd − I,

and the future profits in units of future consumption are

π ′

= Y ′ −w′Nd′ + (1 −d)K′.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Profit Maximization Problem of the Representative Firm

Profits earned by the firm in the current and future periods are paid out to the shareholders of the firm as dividend income in each period. There is one shareholder in this economy, the representative household, and the firm acts in the interests of this shareholder. This implies that the firm maximizes the present value of the household’s dividend income, i.e.,

V = π + π′

1 + r .

The firm’s problem is

max (Nd,Nd

′ ,I,K′)

[ zF(K,N

d ) −wNd − I

] +

[ z′F(K′,Nd

′ ) −w′Nd′ + (1 −d)K′

] 1 + r

s.t. K′ = (1 −d)K + I.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Define the Lagrangian function as

L = [ zF(K,N

d ) −wNd − I

] +

[ z′F(K′,Nd

′ ) −w′Nd′ + (1 −d)K′

] 1 + r

−λ [K′ − (1 −d)K − I] .

The first-order conditions (FOC’s) are

∂L

∂Nd = zF2(K,N

d ) −w = 0,

∂L

∂Nd′ = (1 + r)

−1 [ z ′ F2(K

′ ,N

d′ ) −w′

] = 0,

∂L

∂I = −1 + λ = 0,

∂L

∂K′ = (1 + r)

−1 [ z ′ F1(K

′ ,N

d′ ) + (1 −d)

] −λ = 0,

∂L

∂λ = − [K′ − (1 −d)K − I] = 0.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

From the first two FOC’s, we have the firm’s optimality conditions about labor demand:

MPN = zF2(K,N d ) = w, (2.1)

MP ′ N = zF2(K

′ ,N

d′ ) = w

′ . (2.2)

where MPN and MP ′N are the current and future marginal product of labor. These are the same optimality conditions we have for the representative firm in the one-period model.

From the third FOC, we have λ = 1. Substitute λ for one in the fourth FOC, we have the optimality condition about investment:

MP ′ K = z

′ F1(K

′ ,N

d′ ) = r + d. (2.3)

where MP ′K is the future marginal product of capital. The fifth FOC is just the constraint, i.e., the law of motion for capital:

K ′

= (1 −d)K + I. (2.4)

We can solve the 4 × 4 system of Equation (2.1) to (2.4) in (Nd,Nd′,I,K′) for the solution to the firm’s problem.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

To understand the optimality condition about investment, i.e, Equation (2.3), we can use the cost-benefit analysis:

1. If the firm invests in one unit of capital in the current period, the firm gives up one unit of current profits. So the marginal cost of investment for the firm is 1 unit of current profits.

2. If the firm invests in one unit of capital in the current period: (1) the firm will have additional output produced equal to the firm’s future marginal product of capital, MP ′K ; and (2) there will be an additional 1 −d units of capital remaining at the end of the future period, which can be liquidated. So the marginal benefit of investment is MP ′K + (1 −d) units of future profits, and (1 + r)−1 [MP ′K + (1 −d)] in present value.

3. At the optimal, the marginal cost must equals the marginal benefit so the firm has no incentive to adjust investment.

1 = MP ′K + (1 −d)

1 + r ,

⇒ MP ′K = r + d.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Investment and Interest Rate

Our analysis tells us that, given the interest rate r, the representative firm optimally invest such that MP ′K = r + d, or MP

′ K −d = r. Given our assumptions on the production

function, MP ′K should be decreasing in K ′ holding other things constant, and because K′

is increasing in investment I (given K), there is a negative relationship between the firm’s investment and interest rate.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Outline

The Representative Household

The Representative Firm

Competitive Equilibrium

Applying the Theory: Ricardian Equivalence

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Government

In the one-period model, the government can only finance its expenditures through taxes in the same period. In the two-period model, however, the government has another method to pay for its expenditures in the current period. That is, the government can borrow in the current period by issuing bonds and pay back in the future period.

Suppose the government wishes to purchase G units of the consumption good in the current period and G′ units in the future period, with these quantities of government purchases given exogenously. The government’s current-period budget constraint is then

G = t + B,

where t is the lump-sum tax revenue collected from the household in the current period, and B is the quantity of government bonds issued in the current period.

In the future period, the government needs to pay back both the principal and interest rate on government bonds issued, and we assume government bonds bear the same interest rate r as private bonds (e.g., household debts). So the government’s budget constraint is

G ′ + (1 + r)B = t

′ ,

where t′ is the tax revenue in the future period.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

As what we did for the representative household, we can combine the government’s current and future budget constraints into its present-value budget constraint by eliminating government debt B:

G + G′

1 + r = t +

t′

1 + r .

An interpretation of the government’s present-value budget constraint is that the government must eventually pay off all of its debt by taxing its citizens.

Note that given government expenditures G and G′, once the government decides on the current tax t, the quantity of government debt B and the future tax t′ are automatically pinned down by the government budget constraint. In particular,

B = G− t, t ′

= (1 + r)(G− t) + G′.

Therefore, government policy in this economy can be represented by a combination of G, G′ and t, and we will treat them as exogenous when defining the competitive equilibrium. This also means that government tax policy and government debt policy are the same thing because one implies the other, holding government expenditures constant.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Competitive Equilibrium

For simplicity, we assume the representative household has current and future time endowment h and h′ available for work, and the household bears no disutility from working. As a result, the household will always sell all of its time endowment in the labor market, and both the current and future labor supply curves are vertical.

Let the current and future real wage be w and w′, the household’s current and future labor income are then wh and w′h′. In addition, the household also receives current and future dividend income from the representative firm, π and π′. So the before-tax income of the household is y = wh + π for the current period and y′ = w′h′ + π′ for the future period.

The idea of competitive equilibrium is the same as in the one-period model: • Households make choices to maximize utility subject to their lifetime budget

constraints taking prices as given. • Firms make choices to maximize the present value of profits subject to their

technology constraints taking prices as given. • Government collects taxes and purchases goods subject to the government

present-value budget constraint taking prices as given. • Prices are such that all the markets clear, i.e., demand equals supply in all markets.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Definition of Competitive Equilibrium for the Two-Period Economy

Given the fundamentals of the economy, i.e. utility function U, production function F , and exogenous variables G and G′ (government spending), t (current lump-sum tax), z and z′

(total factor productivity), K (initial capital stock), and h and h′ (time endowment), a competitive equilibrium is a set of the representative household’s consumption choice (c∗,c′∗), the representative firm’s labor demands Nd∗ and Nd′∗, investment I∗, future capital stock K′∗ and profits π∗ and π′∗, future lump-sum tax t′∗, the real wage w∗ and w′∗, and the real interest rate r∗ such that

1. Given the real wage w∗, w′∗ and the real interest rate r∗, profits π∗ and π′∗, and lump-sum taxes t and t′∗, (c∗,c′∗) solves the representative household’s problem:

max (c,c′)

U(c,c ′ )

s.t. c + c′

1 + r ≤ w∗h + π∗ − t +

w′∗h′ + π′∗ − t′∗

1 + r∗

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

2. Given the real wage w∗, w′∗ and the real interest rate r∗, (Nd∗,Nd′∗,I∗,K′∗) solve the representative firm’s problem:

max (Nd,Nd

′ ,I,K′)

[ zF(K,N

d ) −w∗Nd − I

] +

[ z′F(K′,Nd

′ ) −w′∗Nd′ + (1 −d)K′

] 1 + r∗

s.t. K′ = (1 −d)K + I.

and

π ∗

= zF(K,N d∗

) −w∗Nd∗ − I∗;

π ′∗

= z ′ F(K

′∗ ,N

d′∗ ) −w′∗Nd′∗ + (1 −d)K′∗.

3. Government budget constraint holds:

G + G′

1 + r∗ = t +

t′∗

1 + r∗ .

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

4. Markets clear: • The current labor market:

h = Nd ∗ ;

• The future labor market: h′ = Nd

′∗ ;

• The current consumption good market:

zF(K, Nd ∗ ) = c∗ + I∗ + G;

• The future consumption good market:

zF(K′∗, Nd ′∗ ) + (1 − d)K′∗ = c′∗ + G′;

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Solving for the Competitive Equilibrium

Given the definition of competitive equilibrium, solving for the competitive equilibrium is equivalent to finding the values of (c,c′,Nd,Nd′,I,K′,π,π′, t′,w,w′,r) that satisfy all the equilibrium conditions.

From our previous analysis of the representative household’s problem, we have the following optimality conditions:

MRSc,c′ = U1(c,c

′)

U2(c,c′) = 1 + r, (3.1)

c + c′

1 + r = wh + π − t +

w′h′ + π′ − t′

1 + r . (3.2)

Similarly, from the representative firm’s problem, the optimality conditions are

MPN = zF2(K,N d ) = w, (3.3)

MP ′ N = zF2(K

′ ,N

d′ ) = w

′ , (3.4)

MP ′ K = z

′ F1(K

′ ,N

d′ ) = r + d, (3.5)

K ′

= (1 −d)K + I. (3.6)

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

The definitions of profits:

π = zF(K,N d ) −wNd − I, (3.7)

π ′

= z ′ F(K

′ ,N

d′ ) −w′Nd′ + (1 −d)K′. (3.8)

The government budget constraint:

G + G′

1 + r = t +

t′

1 + r . (3.9)

The market clearing conditions:

h = N d , (3.10)

h ′

= N d′ , (3.11)

zF(K,N d ) = c + I + G, (3.12)

zF(K ′ ,N

d′ ) + (1 −d)K′ = c′ + G′. (3.13)

From (3.1) to (3.13), we have 13 equations in 12 unknowns. Drop any of the market clearing conditions by Walras’ law, and we can solve the remaining 12 × 12 system of equations to find the competitive equilibrium.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

To proceed, let us drop Equation (3.13), and eliminate (Nd,Nd′,w,w′,r,π,π′, t′) by substitutions. It is easy to verify that we can reduce the 12 × 12 system into a 4 × 4 system in (c,c′,I,K′):

U1(c,c ′)

U2(c,c′) = z

′ F1(K

′ ,h ′ ) + (1 −d),

⇔ MRSc,c′ = MP ′K + (1 −d).; (3.14) zF(K,h) = c + I + G; (3.15)

z ′ F(K

′ ,h ′ ) + (1 −d)K′ = c′ + G′; (3.16)

K ′

= (1 −d)K + I. (3.17)

By solving Equation (3.14) to (3.17), we can find the equilibrium values of (c,c′,I,K′), and then recover the values of other variables.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

The Credit Market at the Equilibrium

Note that we did not mention anything about the market clearing in the credit market, where the household and the government borrow or lend to each other. It turns out that the credit market is also cleared at the competitive equilibrium. At the equilibrium, the supply of credit, i.e., savings by the household, is

s =y − t− c =wh + π − t− c =wh +

[ zF(K,N

d ) −wh− I

] − t− c

=zF(K,N d ) − I − c− t.

From the equilibrium condition (3.12), we have

G = zF(K,N d ) − I − c.

Therefore, s = G− t = B

where B is the government bonds issued by the government, i.e., the demand of credit. So the credit market automatically clears.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Outline

The Representative Household

The Representative Firm

Competitive Equilibrium

Applying the Theory: Ricardian Equivalence

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

The Ricardian Equivalence Theorem

We have built a two-period model of the market economy, and we can now introduce a key result in macroeconomics, called the Ricardian equivalence theorem.

To explain the idea of this theorem, let us continue our work on the competitive equilibrium in the two-period model by considering the effect of a decrease in the current lump-sum tax t while holding all the other exogenous variables constant.

Note that by definition, the amount of government debt B is G− t, so a decrease in the current tax t means the government must borrow more in the current period to finance its current expenditures. Let ∆t denote the change of current tax, then the change in government debt is ∆B = −∆t.

Based on our definition of the competitive equilibrium, we have 12 endogenous variables to pin down (c,c′,Nd,Nd′,I,K′,π,π′, t′,w,w′,r) .

1. Because t does not appear anywhere in Equation (3.14) to (3.17), we know the equilibrium values of (c,c′,I,K′) will remain the same after the decrease of t.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

2. Once we know (c,c′,I,K′), then from the other equilibrium conditions, we have

N d

= h,

N d′

= h ′ ,

w = zF2(K,h),

w ′

= zF2(K ′ ,h ′ ),

r = z ′ F1(K

′ ,h ′ ) −d,

π = zF(K,h) −zF2(K,h)h− I, π ′

= z ′ F(K

′ ,h ′ ) −z′F2(K′,h′)h′ + (1 −d)K′.

Again, we do not see t anywhere, so the equilibrium values of these endogenous variables are not affected by the decease of t, either.

3. Now the only endogenous variable left is the future tax t′. From the government budget constraint, we have

t ′

= (1 + r)(G− t) + G′.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Because we know r will not respond to a change in t from previous discussion, we have

∆t ′

= −(1 + r)∆t = (1 + r)∆B.

Intuitively, this is just saying that the extra government debt accumulated by the current tax cut must be paid by an increase of future tax.

Lessons: Assume that the government must honor its budget constraint: 1. A tax cut today has no effect on endogenous variables such as consumption,

investment, labor, wage and interest rate;

2. The tax cut (plus interest) must be paid back by an increase of tax in the future. (A tax cut is not a free lunch.)

The Ricardian Equivalence Theorem: A change in the timing of taxes by the government is neutral.

Remarks: Note that the Ricardian equivalence theorem states that changes in tax policy (hence the government deficits) do not matter, assuming that the government expenditures (G and G′ in the model) remain the same. It does not mean changes in government expenditures are neutral.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Ricardian Equivalence and Credit Market

When there is a tax cut today, the government has to borrow more, so why does the interest rate remain the same?

To understand what is happening in the credit market, let us consider the changes in the demand and supply of credit in response to the tax cut.

1. Change in credit demand: ∆B = −∆t; 2. Change in credit supply: The household savings at the competitive equilibrium is

s = y − t− c = zF(K,Nd) − I − c− t.

We have shown that Nd ,I and c will not respond to the tax cut, hence

∆s = −∆t.

So when the government offers a tax cut today, the demand of credit increases. However, because the household anticipates that the tax in the future must increase to pay for the tax cut today, it saves more, and the supply of credit also increases by the exact same amount. In the end, the effects on interest rate from the changes in demand and supply just cancel out with each other, and the equilibrium interest rates remains the same.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Why the Ricardian Equivalence May Not Hold?

The Ricardian equivalence is a very strong result, as it says that changes in tax policy (and changes in government deficits due to them) do not matter. However, this is also an important starting point for thinking about when these changes do matter.

Some complications are left out of our previous analysis, and they could potentially invalidate the Ricardian equivalence result.

1. Tax changes may not be the same among different households. Some may receive a larger tax cut than others.

2. The current households who receive a tax cut may be dead when the government debt due to this tax cut must be paid off in the future.

3. Taxes are distortionary, not lump sum, and they change the effective relative prices of goods in the market.

4. Some households may be borrowing-constrained, and they could be affected beneficially by a tax cut.

Lecture Notes: A Two-Period Model of the Macroeconomy Chunzan Wu

Readings

• Chapter 9 in Williamson. • Chapter 11 in Williamson: Only the part about the representative firm.

The Representative Household
The Representative Firm
Competitive Equilibrium
Applying the Theory: Ricardian Equivalence