1. Intermediate Macroeconomics
Chapter 14
Investment

2. Model 1 – Net present value (NPV) Model 2 – Simple accelerator
Investment
Introduction
Definitions
Model 1 – Net present value (NPV)
Model 2 – Simple accelerator
Model 3 – Neoclassical
Model 4 – Tobin’s q
Complications
Policy Implications
Intermediate Macroeconomics

3. Introduction Change in investment vs change in GDP
Real GDP
Gross Investment
Intermediate Macroeconomics

4. Definitions Net investment
Net investment = increase in productive capital stock
Int = K t – K t-1
Int = Net investment during period t
K t = Capital stock at end of period t
Intermediate Macroeconomics

5. Definitions Replacement investment and depreciation
Replacement investment = spending necessary to maintain a constant productive capital stock
Irt = d K t-1
Irt = investment in replacement capital
in period t
d = rate of depreciation, percent/year
Intermediate Macroeconomics

6. 2. Definitions Gross investment
Gross investment = total spending on goods used to produce other goods and services
Igt = Int + Irt
Irt = gross investment in period t
Intermediate Macroeconomics

7. 2. Definitions Gross and net investment
Gross Investment
Net Investment
Intermediate Macroeconomics

8. 3. Model 1 - Net Present Value (NPV)
Present Value – present day value of a future revenue or expense
Present Value = cash flow year n
(1 + i)n-1
i = nominal interest rate
Net Present Value – total present day value of all current and expected future revenues and expenses
Present value simplifying assumption: Cash flow occurs at beginning of year n. Cash flow year 1 is today. Cash flow year 2 is 1 year from today. Avoids complication of compounded interest rates.
Net value = revenue - expense
Intermediate Macroeconomics

9. Model 1 - Net Present Value (NPV) Case 1
Intermediate Macroeconomics

10. Model 1 - Net Present Value (NPV) Case 2
Intermediate Macroeconomics

11. Model 1 - Net Present Value (NPV) Variables that affect investment
Demand (and income): increase in demand increases revenues and NPV of investment.
Nominal interest rate: increase in interest rate reduces the NPV of future cash flows. If future net cash flows are positive result is a lower NPV of investment.
Net Present Value analysis can be used by firms to evaluate potential investment projects. NPV analysis also provides some reassurance to macroeconomists that investment is positively related to income and negatively related to the interest rate. <i>But the NPV approach does not provide a theoretical foundation for determining how much investment will be undertaken.</i> How many potential investment projects are there? What is the pattern of their cash flows? On aggregate measures this firm-level NPV approach is silent.
Intermediate Macroeconomics

12. 4. Model 2 – Simple Accelerator
The desired level of capital stock is a fixed function of aggregate demand.
Kt-1 = ß Yt-1
Kt = ß Yt
Kt-1 = stock of capital at end of period t-1
Firms attempt to maintain a fixed ratio of their capital stock to expected sales. Net investment is a positive function of the change in expected output.
Intermediate Macroeconomics

13. Model 2 – Simple Accelerator Net investment
Net investment equals the change in the level of capital stock.
Int = Kt - Kt-1
Int = net investment in period t
Firms instantaneously adjust the level of capital to the observed level of demand.
Int = ß Yt - ß Yt-1
= ß (Yt - Yt-1)
Firms attempt to maintain a fixed ratio of their capital stock to expected sales. Net investment is a positive function of the change in expected output.
Intermediate Macroeconomics

14. Model 2 - Simple Accelerator Variables that affect investment
Demand (and income): increase in demand increases desired level of capital stock and investment.
As useful as this model is we must be aware of its limitations. For example:
The simple accelerator model assumes that the desired level of capital stock is a fixed ratio to output. But there are reasons to expect this ratio is not constant but varies depending upon the cost of capital and labor, interest rates and tax rates, etc.
The simple accelerator also assumes that firms can instantaneously adjust their capital stock to the observed level of demand. This is expecting too much. "Time-to-build" can be significant. Accelerating the installation of new factories and equipment may result in much higher costs.
We can consider these additional economic variables by constructing a more complex macroeconomic model of the desired level of the capital stock and investment -- the Neoclassical model with a flexible accelerator.
Intermediate Macroeconomics

15. Derive desired level of capital stock, K*
5. Model 3 - Neoclassical
Derive desired level of capital stock, K*
Calculate investment as a function of:
K* - Kt-1
where,
K* = desired level of capital
Kt-1 = stock of capital at start of the period
(end of preceding period)
Intermediate Macroeconomics

16. Model 3 – Neoclassical Desired stock of capital, K*
Profit Maximization
Marginal Product of Capital
Rental (User) Cost of Capital
Real Interest Rate
Expected Inflation Rate
Intermediate Macroeconomics

17. 5. Model 3 – Neoclassical Profit maximization
Profit = p • Y - c • K - w • L
p = average product price
Y = physical measure of output
= production function, Y = f(K,L)
c = rental (user) cost of capital
K = available capital stock
w = wage rate
L = quantity of labor input
dY/dk - marginal product of capital
p * dY/dK = value of marginal product of capital
If we had a specific function form for the production function Y = f(K,L), such as the Cobb-Douglas production function we could explicitly solve for the desired level of capital. But our objective here is to solve for general relationships between the desired level of capital and the cost of capital.
Intermediate Macroeconomics

18. 5. Model 3 – Neoclassical Profit maximization
Profit = p ∙ Y - c ∙ K - w ∙ L
∂ profit = p  ∂ Y - c = 0
∂ K ∂ K
c = p  ∂ Y
∂ K
K* = f(p, c, w)
K* is derived from first order conditions with respect to dK and dL
A change in p represents a change in product price when c and w are unchanged. This does not characterize a change in the average level of prices, inflation, where p, w, and c all change by the same rate.
A change in p would represent a change in demand for the product.
Intermediate Macroeconomics

19. 5. Model 3 – Neoclassical Rental cost of capital
c* = r + d
c* = rental cost of capital
r = real interest rate
d = depreciation rate
c* not the same as c
c* is unitless, or % of $1 worth of capital
c = p dY/dK depends on a specific type of capital
Since depreciation rate is assumed not to change, focus of interest is on changes in the real interest rate
Intermediate Macroeconomics

20. 5. Model 3 – Neoclassical Real interest rate
r = i - E()
r = real interest rate
i = nominal interest rate
E() = expected inflation rate
Intermediate Macroeconomics

21. 5. Model 3 – Neoclassical Nominal interest and inflation rates
Nominal Interest Rate
Inflation Rate
Source: Nominal interest rate based on U.S. bank prime rate (
Inflation rate based on CPI measure of inflation (
Intermediate Macroeconomics

22. 5. Model 3 – Neoclassical Nominal and real interest rates
Nominal Interest Rate
Real Interest Rate
Source: Nominal interest rate based on U.S. bank prime rate (
Real interest rate based on prime rate - CPI measure of inflation (
Intermediate Macroeconomics

23. 5. Model 3 – Neoclassical Expected inflation rate
Naive expectations:
E(t) = t-1
Adaptive expectations:
E(t) = E(t-1) + a  [t-1 - E(t-1)]
Rational expectations:
E(t) = t + random error
Intermediate Macroeconomics

24. 5. Model 3 – Neoclassical Desired level of capital, K*
Positive function of:
Product price, p
Product demand, Y
Labor wage rate, w
Negative function of
Rental Cost of Capital, c
nominal interest rate (+), i
expected inflation rate (-), E
expected depreciation rate (+), d
Some macroeconomic models assume that all changes in the inflation rate are reflected in changes in the nominal interest rate. Changes in the inflation rate do not have real effects.
If your macro model has lags between changes in the inflation rate and changes in the nominal interest rate then changes in inflation can have real effects
Intermediate Macroeconomics

25. 5. Model 3 – Neoclassical Flexible accelerator
How do you go from desired level of capital to investment?
Flexible Accelerator Model - firms close a portion of the gap, a, between the desired and the current levels of capital
It = a  (K* - Kt-1)
Intermediate Macroeconomics

26. 5. Model 3 – Neoclassical Adjustment of the capital stock
It = 0.4  (K* - Kt-1) K*
Intermediate Macroeconomics

27. Model 3 – Neoclassical Variables that affect investment
Demand (and income) (+)
Product price (+)
Wage rate (+)
Nominal interest rate (-)
Expected inflation rate (+)
As useful as this model is we must be aware of its limitations. For example:
The simple accelerator model assumes that the desired level of capital stock is a fixed ratio to output. But there are reasons to expect this ratio is not constant but varies depending upon the cost of capital and labor, interest rates and tax rates, etc.
The simple accelerator also assumes that firms can instantaneously adjust their capital stock to the observed level of demand. This is expecting too much. "Time-to-build" can be significant. Accelerating the installation of new factories and equipment may result in much higher costs.
We can consider these additional economic variables by constructing a more complex macroeconomic model of the desired level of the capital stock and investment -- the Neoclassical model with a flexible accelerator.
Intermediate Macroeconomics

28. q = company’s market value replacement cost of capital
6. Tobin’s q
q = company’s market value
replacement cost of capital
As the value of the stock market increases relative to the total stock of real capital then the rate of investment should increase.
Intermediate Macroeconomics

29. Credit Rationing Capacity Utilization 7. Complications
Credit Rationing. Firms unable to borrow money for investment even though willing to pay going interest rate. If credit rationing increases during recessions then the rate of investment may decline even if there is no change in the interest rate.
Intermediate Macroeconomics

30. Policy Implications Government spending and investment
Intermediate Macroeconomics

31. Policy Implications Investment tax Credits
Temporary tax credit
small impact on desired capital stock
large impact on current period investment spending
Permanent tax credit
larger impact on desired capital stock
smaller impact on current period investment spending
Investment Tax Credits - Temporary or permanent?
Temporary investment tax credit:
Temporarily lowers the cost of capital
Small change in desired level of capital stock because of temporary lower cost of capital
Rate of investment accelerated to take advantage of temporary tax credit
Intermediate Macroeconomics

32. Policy Implications Corporate Income Tax
Is investment financed from borrowed funds or equity funds (e.g., stock sale)?
Borrowed Funds - interest payments on borrowed funds deducted from firm’s income before income tax calculated.
Equity Funds - interest payments (e.g. dividends) on funds are not deducted from firm’s income before tax is calculated.
Firm’s Balance Sheet. Condition of the firm's balance sheet (cash in the bank, ability to borrow money or sell new stock), not just cost of capital determined by interest rate affect investment. Another reason investment is lower during recessions.
Borrowed Funds Equity Funds
Product sales Product sales
- Raw material and labor costs Raw material and labor costs
- Depreciation Depreciation
- Interest payments on borrowed funds
Gross profit
- Corporate income tax Corporate income tax
Net profit Net profit
(i.e., dividend payments on equity funds)
Intermediate Macroeconomics