Chapter1~13

Chapter13 investment demand

● the present value criterion for investment

PV criterion: ranking investment programs by the present discounted value of the projects’ income streams

()

1....11t t t n R R PV C R r r +=?++++++t n +) (1) Rationale:

Firm owned by persons who maximize their utility:

(2)

(01,,...,T U U c c c =s.t income possibility curve:

increases future income by investing the firm’s retained earnings.

()010,,...,T y y y φ= (3)

Two-period case

Figure13-1 show the income possibility curve

Marginal rate of transformation (MRT, differentiating equation (3) )

010

1//y dy dy y φφ??=??? Choosing an income stream 01,y y along the income possibility curve that gets the owners to the highest possible budget line.MRT=-(1+r)

Figure13-2 show the investment and consumption decisions

Marginal rate of substitution (MRS, differentiating equation 2)

010

1//U c dc dc U c ??=??? The owners reach their highest possible indifference curve by finding an indifference curve tangent to the highest possible budget line. MRT=-(1+r)=MRS

And the intercept of the budget line on the horizontal period 0-axis is given by the present value of the 01,y y income steam, 1001y PV y r

=++ The manager does not need any information on the shape of the owner’s utility function and indifference curves: the production and consumption decisions are completely separate.

Thus, the firm that is trying to maximize its owner’s welfare will follow the rule: maximize PV .

Fig 13-3 ranking investment projects in order of their PV’s

Present value of the marginal project is zero

An increase in the market interest rate shift the curve down

Expected returns increase shift the curve up

● the marginal efficiency of investment

(1) the marginal efficiency of an investment project, m , is defined as the rate of interest that

will discount the PV of the project to zero:

()

10....11t t t n R R C R m m ++=?++++++n Project can be ranked by m. as the size of the total investment program is increased, we go to projects with lower and lower R streams, so that as I rises, m falls. Figure 13-4

(2) the supply of funds

the shape: constant and then rises as investment increasing(investing with retained profit →external fund resource: bonds or stocks)

rinsing profit of firm could shift the r curve right. Figure 13-4.

(3) Present value and marginal efficiency

PV rankings can be different from m rankings

Eg:

Project 1, cost 1, return (0,0,4)

Project 2, cost 1, return(0,2,1)

Calculate the marginal efficiency of the above two projects

Calculate the PV of the above two projects when r=0.1

Investment demand and output growth

The relationship between the growth rate of output and the level of net investment is called the accelerator principle since it suggests that an increase in the growth rate of output is needed to increase the level of investment.

● Equilibrium capital stock

The determinants of the desired capital stock can be derived from the present value maximization of a firm that looks to its expected future profit stream when it makes current investment decisions.

Production function:

(),,,t t t y y y y N K N K

??=>??0 Capital stock evolves according to:

()11t t t t t K K i K K δδ+=+?=?+t i (10)

Present value of its future profit stream:

()()001,1I t

t t t t t t t PV P y N K W N P i r ∞??=???+∑? (12) Problem:

Max (12) s.t.(10)

Solution: ()11I I I I t t t t K t

P rP P P y P δ??+??= Equilibrium capital stock as s function of output, the user cost, and the price of output:

E K (18)

(,,E E K K y C P =)0t /0,/0,/E E E K y K P K C ??>??>??<

A cobb-douglas example

● The investment demand function

Components of gross investment:

1g t t t i K K K δ+=?+

Net investment: , replacement investment 1n t t i K K K +=?=ΔE r t i K δ=.

Net investment depends on changes in the equilibrium level of the capital stock, whereas replacement investment depends on the level of the capital stock.

Cobb-douglas case:

n E Py P i K C C αα????=Δ=Δ=Δ????????

y This relation between the change in output and the level of net investment is the accelerator principle.

Using saving function we get the growth rate of y

/y s y P C

αΔ= The case of total investment

(),,g E i K

Y C P K δ=Δ+ And g i y c K αδ=

Δ+

● The accelerator and stabilization policy

The accelerator effect will produce a temporarily rapid growth of demand which will subside

● Capacity utilization and the accelerator

The accelerator effect might appear with a delay as an upsurge in replacement investment

● The q-theory of investment

Another version of that condition:

()()111I I t K t t P y P r P δ?+??+=0

Or

()()1

1/11I t K t I t P y P r P δ???+?+??= ()()1

1/1arg I t K t I t P y P r m inalq P δ???+?+??= The denominator is the cost of acquiring a small increment to the capital stock in period t-1

The numerator is the increment from that acquisition to the value of the firm in period t, discounted back to t-1

So the tobin’s marginal q is the ratio of the change in the value of the firm to the added capital cost for a small increment to the capital stock

I PV averageQ P K

=

● The user cost and liquidity effects

Construct a measurement of the user cost

1. the interest cost of the capital

2. the depreciates

3. capital gains or losses

I I I I I dP C rP P P r P dt δδ???=+?=+?????

I r P ?

?is the real interest rate

① eliminating the capital gains or losses ();0I I C P r P δ???=+=???? ② the lag of capital price change

③ weighted average of the interest rates

int l I D E ernalfunds bondissue equityissue r r r r tota investment totalinvestment totalinvestment ??????=++????????????

● Ages in investment demand

Time lags involved in changing the level of capital stock

(Advanced macroeconomics)

Summarize the theory of investment demand

● Investment in the static model

(),;0,0,r y i i r y i i ′′=<>,

in the static model, the replacement investment is a function of the level of output and the interest rate.

IS curve be flatter, even positive slope