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### CHAPTER 4

Consumption, Saving and Investment

The amount of capital that allows the firm to earn the largest expected profit.

To invest, benefits must > costs of using additional capital.

Desired capital stock is when:

MPKf = uc expected futuremarginal product of capital = expected user cost of capital

uc = rpK + dpK = (r+d)pK

expected real interest rate

depreciation rate

real price of capital goods

pK = cost of capital goods = real price of capital goods

i.e. buying a new oven at $100 measured in real dollars, pK=100

d = cost of depreciation = depreciation rate

i.e. oven produces less 10% cookies after one year, d=0.1

r = cost of interest earnings sacrificed or interest payments paid

= expected real interest rate

i.e. government bond pays 8% interest rate = r=0.08

uc = interest cost + depreciation cost

= rpK + dpK = 0.08(100) + 0.1(100) = 18

MPKf vs uc

MPKf

At K = 4000, MPKf > uc. Increase K.

uc

At K = 5000, MPKf = uc. K*.

uc

At K = 4000, MPKf < uc. Decrease K.

MPKf

Desired capital stock is given by the intersection between MPKf and uc, where MPKf = uc.

4000

5000

6000

K

MPKf

uc decreased from uc1 to uc2 due to a decrease in real interest rate. (note: uc = rpK + dpK)

At K = 5000, MPKf < uc. Increase K.

At K = 6000, MPKf = uc. K*.

↓ r results in ↑ K

uc

uc1

uc2

MPKf

5000

6000

K

MPKf

MPKf increased from MPKf1 to MPKf2 due to an increase in technology.

At K = 5000, MPKf < uc. Increase K.

At K = 6000, MPKf = uc. K*.

↑ tech results in ↑ K

uc

uc

MPKf2

MPKf1

5000

6000

K

At MPKf = 20, every additional K will increase revenue by $20.

If tax = 20%,

then firm pays extra $4 of tax from each additional $20 revenue, which gives firm after-tax additional revenue (or after-tax MPKf) equals to $16.

Before tax = MPKf = 20. After tax (1-t)MPKf = (1-0.2)(20) = 16.

To find K*, we must now consider MPKf=16 (after-tax).

By formula … MPKf = uc (without tax)

(1-t)MPKf = uc (with tax)

(1-t) MPKf = uc ↑t will ↓MPK thus ↓K

An increase in tax rate, t, reduces the after-tax future marginal product of capital, reduces desired capital stock.

MPKf

uc

uc

MPKf1

MPKf2 (with tax)

K2

K1

K

MPKf = uc / (1-t) ↑t will ↑uc thus ↓K

An increase in tax rate, t, increases the tax-adjusted user cost of capital, reduces desired capital stock.

MPKf

uc

uc1

Tax-adjusted user cost of capital = uc / (1-t) … shows how large before-tax MPKf must be if add one unit K.

uc2

MPKf

K2

K1

K

Gross investment (It) = total purchase or construction of new capital goods.

Depreciation (dKt) = the amount of capital that wears out.

Net investment (Kt+1 – Kt) = Gross investment (It) – Depreciation (dKt)

Kt+1 – Kt = I - dKt

I = Kt+1 – Kt + dKt

I = K* – Kt + dKt

Gross investment (I)

Desired increase in capital (K* - K)

Investment needed to replace depreciated K

=

+

It = K* – Kt + dKt

Any increase in K* will increase It

Case 1: Real interest rate increases from 5% to 10%.

Effect: uc = rpK + dpK increases, K decreases. r↑ K*↓ It↓

Case 2: Effective tax rate increases from 5% to 10%.

Effect: tax-adjusted uc = uc/(1-t) increases, K decreases.t↑ K*↓ It↓

Case 3: Expected MPKf from 16 to 20.

Effect: Intersection MPKf=uc moves upward. K increases. MPKf↑ K*↓ It↓

- Desired Capital Stock
- User Cost of Capital
- MPKf versus uc
- Changes in uc
- Changes in MPKf
- Tax Impact on MPKf
- Tax Impact on uc
- Net Investment
- Desired Investment

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