Lecture 13: A Real Intertemporal Model with Investment

• Construct a real intertemporal model that will serve as a basis for studying money and business cycles.

• Understand the investment decision of the firm. • Show how macroeconomic shocks affect the economy.

Real Intertemporal Model

• Current and future periods. • Representative Consumer – consumption/savings

decision • Representative Firm – hires labor and invests in current

period, hires labor in future • Government – spends and taxes in present and future,

and borrows on the credit market.

First, let’s consider the representative consumer

Consumer’s budget constraint

Consumer’s current-period budget constraint: Consumer’s future-period budget constraint:

Lifetime budget constraint

Solve for savings and substitute to obtain the consumer’s lifetime budget constraint:

Intratemporal Euler Equation

Consumer’s marginal condition: The Consumer optimizes by choosing leisure and

consumption so that the marginal rate of substitution (MRS) of leisure for consumption is equal to the real wage (MRT)

Intertemporal Euler Equation

Consumer’s intertemporal marginal condition: The consumer chooses consumption and savings such that

the marginal rate of substitution (MRS) of current for future consumption equals the relative price of current in terms of future consumption (MRT)

Consumer’s Current Labor Supply Behavior

• Current labor supplied increases with the real wage (substitution effects are assumed to dominate income effects).

• Labor supply increases with an increase in the real interest rate, through an intertemporal substitution effect. (higher r => higher return on savings => work harder today to save more)

• An increase in lifetime wealth (e.g. taxes fall) reduces labor supply.

The Consumer’s Current Labor Supply Curve

• The current labor supply curve slopes upward, under the assumption that the substitution effect of an increase in the real wage outweighs the income effect

Real Interest Rate and Current Labor Supply Curve

higher r higher return on savings work harder today to save more

Effects of an Increase in Lifetime Wealth

• More leisure is consumed in the present, due to an income effect • current labor supply shifts to the left

Now let’s consider the representative firm

Firm’s production function

Firm’s current-period production function: Firm’s future-period production function:

Law of Motion for Capital

Evolution of the firm’s capital stock:

Firm’s profits

Firm’s current-period profits: Firm’s future-period profits:

Firm’s Present Value of Profits

The firm maximizes the present value of profits,

The Firm’s Labor Demand

As in the one-period model, the firm’s labor demand schedule is the marginal product of labor for the firm, which is downward sloping.

Demand Curve for Current Labor and the Marginal Product of Labor Schedule

Marginal product of labor declines as labor input increases Downward sloping demand curve

Demand Curve for Current Labor and Changes in Current TFP and in Capital

• increase in z or K shifts the curve to the right

• reflecting the resulting increase in marginal product of labor

The Representative Firm’s Investment Decision

• The firm invests to the point where the marginal benefit from investment equals the marginal cost.

• The marginal cost of investment is 1, as the firm gives up one unit of current profits for each unit it invests, so:

Firm’s Optimal Investment

The marginal benefit of investment is the marginal product of future capital plus the quantity of capital that will be left in the future after depreciation, all discounted back to the present:

Firm’s Optimal Investment

The firm’s optimal investment rule, obtained by equating the marginal benefit and marginal cost of investment:

Optimal Investment Schedule for the Representative Firm

• Firm invests until MP’K - d = r • Future net marginal product schedule MP’K - d is the representative firm’s optimal investment schedule, • because this describes how much investment is required for the net marginal product of future capital to equal the real interest rate.

Decrease in Current Capital or an Expected Increase of Future TFP

• Either of these changes causes the future marginal product of capital to increase

Data for Christine’s Orchard

Discussion: Which level of investment is optimal?

We can use the firm’s problem to analyze financial crises in the presence of asymmetric information

• Suppose many firms in the economy – some good, some bad.

• Bad firms borrow in the credit market, consume the proceeds as executive compensation, then default.

• Lending rate of interest is r, loan rate is rl

• Asymmetric information: Lenders cannot distinguish good from bad firms.

Asymmetric Information and Investment

Asymmetric Information and Investment

• rl-r = x, where x is a default premium to compensate for borrowers who default.

• x is an interest rate spread that reflects the severity of the asymmetric information problem.

• For firms that borrow to invest,

rxdMPK =−−'

Asymmetric Information and Investment

• When the asymmetric information problem worsens, as in the financial crisis, x increases.

• Given the interest rate, this reduces the borrowing firm’s optimal quantity of investment.

The Effect of an Increased Default Premium

• An increase in the default premium shifts the optimal investment schedule down. • So firm invests less for any real interest rate

Investment and the Interest Rate Spread

• Spread tends to be high during recessions • Investment tends to be low, consistent with our theory

Scatter Plot: Investment vs. Interest Rate Spread

• The negative correlation is even easier to spot in the scatter plot

Finally, let’s consider the government

Government’s Budget Constraint

The government’s present-value budget constraint:

We have explored the agents in the economy: representative consumer, representative firm, and the government Before we analyze the competitive equilibrium of this economy, let’s first understand the goods and labor markets.

Equilibrium in the Labor Market Given the Real Interest Rate r

• In (a), the intersection of the current labor supply and demand curves determines the current real wage and current employment

• In (b), the production function determines the aggregate output

Construction of the Output Supply Curve

• In (a) a higher real interest rate implies higher labor supply • In (b), the production function determines the aggregate output • In (c), the relationship between the real interest rate and real output is graphed

An Increase in Current or Future Government Spending

• increase in gov’t spending • Increase the present value of taxes • current leisure falls • shifting labor supply curve to the right in (a) • resulting in higher production in (c)

An Increase in Current Total Factor Productivity

• increase in z • Increase marginal product of labor • shifting labor demand curve to the right in (a) • also shifting production function up in (b) • resulting in higher production in (c)

Demand for Current Goods

Construction of the Output Demand Curve

• higher interest rate implies lower consumption and investment in (a) • resulting in lower output demand in (b)

What Shifts the Output Demand Curve to the Right?

• A decrease in the present value of taxes (consumption ↑)

• An increase in future income (consumption ↑) • An increase in future total factor productivity

(consumption, investment ↑) • A decrease in the current capital stock

(investment ↑)

The Complete Real Intertemporal Model

• (a) The current real wage and current employment are determined by the intersection of the current labor supply and demand curves, given the real interest rate • (b) Current aggregate output and the real interest rate are determined by the intersection of the output supply and demand curves

Using the model, let’s look at the effects of • a temporary increase in government expenditure • a decrease in the capital stock (by natural disaster or

war) • a temporary increase in TFP • an expected increase in future TFP • credit frictions • sectoral shocks

A Temporary Increase in Government Purchases

• Output demand increases (G ↑ )

• Present value of taxes increase

• Labor supply and Output supply increase (income effect)

• From (b), real interest rate and output increase

• In (a), labor supply increases further

• As a result, real wage falls, consumption and investment decrease, employment rises.

• Government spending crowds out both consumption and investment.

Important Points

• The total multiplier – the ratio of the equilibrium increase in Y, to the increase in G – is less than 1.

• Consumption and Leisure decrease in response to an increase in G

• Compare with Keynesian style Government expansions during recessions (Japan’s Lost Decade)

What are the effects of a decrease in the current capital stock? (This could arise due to a war or natural disaster)

Equilibrium Effects of a Decrease in the Current Capital Stock

• Labor demand falls since (MPN ↓ )

• Output Supply falls, real interest rate increases

• Labor supply increases • Another effect:

investment increases since (MPK ↑ )

• This increases output demand

• As a result, output may rise or fall, depending on how large the output demand effect is relative to the output supply effect.

• The real interest rate rises, the real wage falls, employment may rise or fall. • Empirically, this seems to be consistent with episodes such as Katrina (2005)

What are the effects of an increase in current TFP?

• Labor demand and output supply increase

• Real interest falls and real wage increases

• Labor supply decreases in response to higher interest rate

• As a result, real interest rate falls, consumption and investment rise, employment rises, real wage rises

• Productivity shocks are a potential explanation for business cycles

Equilibrium Effects of an Increase in Current TFP

What if TFP is expected to increase in the future?

• Investment increases (MPK’ ↑)

• Output demand curve shifts right

• Interest rate increases • Labor supply increases • Real wage falls

• As a result, real interest rate rises, investment increases, consumption may rise or fall, employment rises, real wage falls, output rises.

• Important in explaining investment boom in the 1990s in the United States.

Equilibrium Effects of an Increase in Future TFP

Deviations from Trend in Investment and a Relative Stock Price Index

• Financial theory tells us that stock prices act to aggregate information

• Investment and the relative stock price index are positively correlated

• Consistent with the view that fluctuations in news about the future are an important source of business cycles

Credit Market Frictions and the Financial Crisis

• One feature of the financial crisis was more severe credit market frictions – asymmetric information and limited commitment.

• In our model, what happens when credit market frictions become more severe?

The Effect of More Severe Credit Market Frictions

• More severe credit market frictions reduce the demand for goods by consumers, so output demand curve shifts to the left

• Consumers increase labor supply (income effect), so output supply curve shifts to the right.

• As a result, real interest rate falls (which reduces labor supply) • Output decreases (if demand effect is stronger).

The Effect of More Severe Credit Market Frictions

• Spread: difference between interest rate on BAA-rated corporate debt (risky) and AAA-rated corporate debt (safe)

• Credit frictions (measured by spread) are negatively correlated with deviations from trend in real GDP

Sectoral Shocks and Labor Market Mismatch

• “Jobless recoveries” a feature of the last 3 recessions. • Could be due to sectoral reallocation – changes in labor

markets and the structure of production giving rise to movement of factors of production across sectors.

• For example, a negative shock to the manufacturing sector (auto and steel), and a positive shock to the services (health and education). Reallocation takes time.

• What does our model predict for sectoral shocks?

The Effects of a Sectoral Shock

• Sectoral shocks add frictions to the labor market

• More costly for firms to hire, labor demand shifts to the left

• Also more costly for workers to find employment, labor supply shifts to the left

• Output supply shifts to the left

• As a result, the shock reduces employment and output, and increases the interest rate

The Effects of a Sectoral Shock on Average Labor Productivity

• But notice that average labor productivity goes up!!

Real GDP in Two Recessions

• Large decline in real GDP

Employment in Two Recessions

• Larger decline in employment

• Prolonged, “Jobless Recovery”

Average Labor Productivity in Two Recessions

• Smaller decline and quick rebound of average labor productivity

• Consistent with model predictions