Capital

- Text: Ch. 15

What is Capital?

Physical Capital: machines, tools, factories, buildings.

i.e. a productive, durable input, human-made.

- Physical capital provides a stream of productive services over its lifetime.

- The value of a unit of physical capital derives from the value of the productive services it generates.

- Creation of physical capital requires investment spending.

(investment: pay a cost now in expectation of future payoffs)

- Investment spending must be financed:

- via borrowing: cost determined in financial markets; or

- from a firm’s own funds: opportunity cost is the return could obtain in financial markets.

- this creates a link between financial returns and returns on investment in physical capital.

- Physical capital is a ‘real asset’ producing a stream of real returns (productive services)

- Financial assets: provide a stream of financial returns to its owner.

1

Demand for Physical Capital: Renting Capital to Use it in Production

- Call ‘K’ the quantity of physical capital.

- Logic of short-run labour demand model can be extended to capital.

- Marginal revenue product of capital (MRPK)= MR x MPK

MR = marginal revenue, i.e. extra revenue from sale of extra output

MPK = marginal (physical) product of K, i.e. extra output produced when a firm uses extra K.

- So MRPK measures the value of extra output produced when the firm uses extra K.

- if assume diminishing returns to K: MRPK falls as K rises.

- Let “r” be the rental or lease price of capital (rental rate):

- price paid per period to use one unit of K;

(equivalent of the wage rate in the labour demand model).

- Hiring (renting or leasing) K:

MRPK > r Firm will hire more K (benefit > cost)

MRPK < r Firm hires less K (benefit < cost)

MRPK = r Profit maximizing firm hires up to this point.

2

- So demand for K is affected by:

- Factors affecting MRP of capital:

- Output market conditions (through MR) e.g. rise in price of oil leads to more K hired in oil industry

(MRPK shifts up)

- Factors affecting MPK (technology, quantity and quality of other inputs)

e.g. technology allows development of tar sands: more K hired there (rise in MP, MRPK shifts up)

- Rental price of K, i.e. “r”.

i.e. rise in r: less K is hired. (r = red line shifts up)

- Model works just like the short-run labour demand model.

3

The Rental Price of Capital and the “User Cost of Capital”

- Interpretation of ‘r’: a per period price at which the firm can rent or lease K.

- Can define a 'per period’ cost measure for the owner of the unit of K.

i.e. the “user cost of capital”: cost to the owner of tying up their money in a unit of K.

- What determines the user cost of physical K?

- say it costs PK to buy a machine.

- Costs of owning a unit of capital for a period: - opportunity cost of funds: i∙ PK

- owner could have invested PK at an interest rate of i

(i = return on best alternative investment of similar risk)

- machine requires maintenance costs at rate “m∙ PK” per period

(m = maintenance cost as a share of PK)

- the machine loses value each period due to depreciation: d∙ PK d = depreciation cost as a share of PK

(depreciation: could be physical e.g. machine wears out or could reflect obsolescence)

- The user cost of K is the sum of these per period costs:

(i + m + d) ∙ PK

(this assumes no capital gains or losses due to changes in the resale price of the unit of K – see the extension below)

4

- If a firm buys K for its own use the user cost plays the role of its "rental price" (it is what it costs the buyer on a per period basis)

- so a firm that buys K rather than rents it, should buy more K until: MRPK = (i + m + d) ∙ PK

- renting-to-use or owning-to-use are substitutes: capital users will choose the cheapest option. In equilibrium substitution implies

rental price = user cost).

- If the firm buys K to lease or rent it the "rental price" must cover the user cost if leasing is to be a profitable business.

- But: competition between suppliers of K implies a rental price no higher than the user cost i.e.

if: r > (i + m + d) ∙ PK more K will bought and rented out, the extra supply of rented K gives ↓r (PK

could rise too as firms enter the rental business)

if r < (i + m + d) ∙ PK no one supplies rental K, shortage

leads to ↑r. (PK could fall too – fewer firms in the rental business buying K)

Equilibrium requires:

r = (i + m + d) ∙ PK

or:r/PK = i + m + d

- m and d : think of as mainly technically determined

- then : r varies directly with i.

5

(NOTATION INCONSISTENCY IN OLD EDITION: equation (15.1) and (15.2): r is a rental price (in $) – like above. Text p. 488 equation (15.3): r is a proportion (or %) i.e. it is the same as r/ PK above. In the new edition this is fixed by defining ‘k’=r/PK) ) - When supply and demand for rented capital are equal:

MRPK = r = (i + m + d) ∙ PK

↑ ↑ Demand Supply

- Demand downward sloping; supply – flat at (i + m + d) ∙ PK

- More K will be rented if:

- Rise in demand MRP rises (e.g. due to technological improvement, rise in

price of output produced with the K)

- Rise in the supply of K:

- will result if it becomes less costly to supply

- if i, m or d falls

- PK falls.

- Some common complications to the user cost setup:

- Capital gains / losses (PK change in resale price of K): - after using K for a period the owner of the K may sell it.

- if PK is higher than when purchased – this is a gain (deduct it from the user cost);

- if PK is lower than when purchased the capital loss must be added to the user cost.

( user cost is then: (i + m + d)∙PK - PK(1-d) )

6

- Taxation of capital can affect the user cost in several ways (big topic in economics of business taxes)

7

Basics of Asset Pricing:

- Value of an asset: determined by the stream of returns it will generate over it’s lifetime.

Simple asset: One payment one period in the future

Present value (PV) of returns = B1/(1+i)

B1 = payment received next period;

i = interest rate used in discounting future dollars.

i.e. going rate on a “similar” investment.

Present value: measures the value of future payments now.

i.e., how much would you have to invest at rate i to receive this payoff in the future.

(see text Chapter 5 discussion of present value)

- buy this asset if: Price < PV of returns

i.e. cost < benefit

or if: return on asset or yield (ir) > i

ir? internal rate of return or ‘yield’ on the asset.

Calculated by solving: Price = B1/ (1+ir) for ir

It is the rate of return that would generate the stream of payments (B1) given that you have invested ‘Price’ in this asset.

8

- Don’t buy the asset if: Price > PV of returns.

cost > benefit

or if: rate of return (ir) < i

- Equilibrium: Price = PV of returns.

- at the equilibrium asset price the stream of returns gives exactly a rate of return of i

i.e. same return as on similar assets

- if not? say price too low, ir>i demand for this asset will rise pushing up its price.

General asset: N periods

PV of returns ¿B1

(1+i)+

B2

(1+i)2 +B3

(1+i)3 +…+BN

(1+i)N

B1 … BN = payments from asset in time periods 1 to N

- Present value of a payment Bt made t periods in the future:Bt

(1+i)t

- Equilibrium price of the asset equals PV of returns (using the going

rate on a similar investment as i)

- This idea can be applied to financial assets (bonds, treasury bills, shares, etc.) or real assets (houses, land, physical capital).

- Note: for a general asset its rate of return (yield) can be calculated by setting the PV formula (with ir replacing i) equal to price and solving for ir.

9

Physical capital as an asset

- Text equation 15.4: gives a version of the equation above for physical K.

- For time period t= 1…N-1 with: Bt = R-M

R = extra revenues from having the extra (this is MRP! But may need a depreciation adjustment in later

periods)

M = maintenance cost.

- For t = N: BN = R – M + S

S = scrap value of the capital asset (could be the resale price of the capital if sold before it totally wears out)

- So buy more K if:

PK < R-M + R-M + R-M + … + R-M+S . (1+i) (1+i)2 (1+i)3 (1+i)N

i.e., cost < benefit

(just like rental decision but with longer time horizon)

- When is K demand (investment in K) most likely to be high?- high R (high MRP of K)- low M - low interest rate (opp. cost of investment)- low PK

- Compare these predictions to those for rental K. The rental case gave: MRPK = r = (i+d+m) PK

- where is “d” depreciation in the asset price equation?

10

Enters via ‘S’ – scrap or resale value -- high d means low S since less K is left undepreciated. It could also enter via

R if depreciated K produces less than new K.

- In equilibrium:

PK = PV of net returns from the asset

PK = R-M + R-M + R-M + … + R-M+S . (1+i) (1+i)2 (1+i)3 (1+i)N

Relationship between Rental and Purchase Prices of a Real Asset:

- Think of buying an asset and leasing (renting) as alternatives.

- Expect that in equilibrium they would cost the same.

- So: Purchase price = Present value of rental prices.

- Say this doesn’t hold:

Purchase price > Present value of rental prices

- renting is cheaper: more renting, less buying so rental price rises, purchase price falls.

Purchase price < Present value of rental prices

- buying is cheaper: more buying, less renting so rental price falls, purchase price rises.

- This idea applies to capital which can be rented or bought outright.

(Houses: purchase vs. rental market and this -- can indicators of housing bubbles be constructed using this idea?)

11

Bonds as Assets:

- Bond: a contract where a firm or government promises to pay regular periodic payments plus the face value of the bond upon maturity.

e.g., will pay the face value $1000 in three years and 10% of face value ($100) each year until maturity.

PV of returns = $100 + $100 + $100+$1000 (1+i) (1+i)2 (1+i)3

Price ? - depends on the going interest rate (i)

- price will adjust until return on the bond equals i.

e.g if price is low, yield on bond is higher than i, demand is high, price rises, yield falls.

i = .1 (10%) then Bond price = $1000

i = .05 (5%) then Bond price > $1000 ($1131.83)

i = .15 (15%) then Bond price < $1000 ($894.12)

- Note the inverse relationship between return on the asset (i) and price.

- to give a high return the price of the asset must be low.

- to give a low return the price of the asset must be high.

(Aside: mortgages and loans often have similar payment streams to a bond)

12

- Perpetuity or Consol:

- a bond that promises a fixed payment forever (no maturity).

e.g. $100 forever (N=infinity):

PV of returns = $100 + $100 + $100 + … + $100 . (1+i) (1+i)2 (1+i)3 (1+i)N

= $100 { 1 + 1 + 1 + … + 1 } (1+i) (1+i)2 (1+i)3 (1+i)N

= $100 i

(note: geometric series in 1/(1+i) is used to simplify)

Price? i=.1 Price = $1000i=.05 Price = $2000i = .15 Price = $666.67

(note: if N is finite this asset is an annuity)

(Geometric series: ∑i=1

N

ai=a+a2+a3+…+aN

i means this is a sum of terms from i=1 to N)

∑i=1

N

ai=a+a2+a3+…+aN (series)

a ∙∑i=1

N

ai=a2+a3+…+aN +1 (series times ‘a’) ________________________________ (1−a ) ∙∑

i=1

N

ai=a−aN +1 (difference between series and series times ‘a’)

so:

13

∑i=1

N

ai=[a−aN +1

1−a ] (Perpetuity a= 1/(1+i) )

14

Stocks or Shares:

- Stock gives a share of ownership in the firm.

- Value of stock linked directly to value of the firm to its owners

i.e., Present Value (PV) of the firm’s profits.

- So in equilibrium stock prices should reflect PV of firm’s profits.

- given i: high PV profits means high stock price.

- Future profits: - very uncertain: reflects current best guess.

- changes in stock prices reflect changes in best guesses.

15

Efficient Markets Hypothesis

- Concerned with prices in speculative markets.

- Take stock prices as the example.

- Current price of a stock will reflect all current information on the firm’s future profits.

- Changes in the price of the stock will reflect unanticipated new information.

- Such information is unknown now.

- So changes in stock prices will be unpredictable.

- Implications: - stock trading strategies aimed at making money by predicting

price changes will typically not succeed.

- random picks will do as well as expert picks!

- historical data are of no use in predicting stock prices.

- success if it occurs reflects:- luck! - early access to new information (insider!).- fraud! (Bernie Madoff!)

- Is this too extreme?- difficult to test: how to measure expectations?

- some evidence seems consistent with efficient markets. e.g. lack of success of most expert picks.

- some evidence against: small firm effect,volatility,W. Buffett!

- bubbles and busts: new information or irrational behaviour?

- “Efficient” in use of information determining prices.

16

A Supply-Demand Model of Interest Rates: Loanable Funds Model

- Suppliers of loanable funds:

- lenders/savers: - mainly households- could be governments (budget surpluses) or

firms.- foreigners.

- likely increasing in interest rate (upward sloping curve).i.e. higher return then save (and lend) more.

- Demand for Loanable funds:

- borrowers: - households (consumer loans and mortages)- businesses (to finance investment)- governments (to cover deficits).- foreigners.

- likely decreasing in interest rate (downward sloping curve).i.e. rate measures borrowing cost.

- Equilibrium interest rate where supply and demand for loanable funds are equal.

17

- Interest rates change due to factors that shift Supply and Demand curves.

e.g., fiscal policy (government borrowing), business expectations (business borrowing), consumer incomes (via household saving), monetary policy.

- demand for physical capital linked to demand for loanable funds:

- need to finance investment in physical capital.

i.e. “i” cost of borrowing to finance K investment or opportunity cost of funds.

(An application: why are interest rates so low in 2016? Larry Summers Foreign Affairs reading)

Interest Rates

- Many interest rates not just one!

- Imagine a set of markets (one for each type of financial asset) operating like our loanable funds model.

- These markets are linked via flows of lending and borrowing between them (different financial assets are substitutes for one another).

- Why do interests rates differ?

- if different assets were identical (perfect substitutes) their interest rates would be the same in equilibrium (no one will lend in the form of an asset with low returns; no one will borrow in the form of an asset with a high interest rate)

18

- So if interest rate differences persist the underlying assets must differ from the point of view of lenders or borrowers leading to different equilibrium interest rates.

- Result? An equilibrium structure of interest rates where differences in rates compensate lenders and borrowers for differences in these assets.

19

- Differences in risk are one important source of differences in assets.

- Cdn. government bonds: basically riskless.

- Bonds of established well-known corporations: slightly more risky.

- Bonds of lesser-known corporations: more risky.

- Lenders appear to dislike risk.

- consequence: lenders demand higher return to hold risky assets.

- in equilibrium: iRisky = iSafe + Risk premium

(text version:lenders have indifference curves over safety (risk) and

expected return; “market” confronts lenders with a tradeoff between safety and

return)

- Some other factors affecting interest rate structure:

- long-term assets vs. short-term assets: expectations of future short- term interest rates an issue.

- liquidity and the existence of a good resale market for assets.

- foreign vs domestic assets: exchange rate changes during term of asset.

20

Economic Rent

- Economic rent: the difference between what a factor of production is paid and the minimum amount the owner needs to supply it.

- Technically: difference between the price paid for the factor input and the height of the supply curve.

(Fig. 15-3)

- Measure of “surplus” that goes to the factor owner.

(closely related to “producer surplus” or idea of “economic profit)

- note the role of demand for the input in determining the size of economic rents: higher demand, higher price, higher rents.

- Special case: if the supply curve is vertical all earnings of the input are rent (minimum price to supply it is 0).

- is this the case for land?

- Labour and rents: wages in excess of the height of the labour supply curve are rents.

- winner-take-all, superstar markets: earnings mostly rent.

- union/non-union wage gap: rent.

- Recall: discussion of rents in The Undercover Economist Ch. 1.

21

Natural Resources as Inputs:

- Two broad types of natural resources:

- Renewable resources e.g. fish, trees

- Exhaustible resources (finite quantities that can't be replaced) e.g. iron ore, gold etc.

- Water? where does it go -- is it inexhaustible?

Renewable Resource Use:

- Text example: volume of lumber from a tree grows over time

Let volume of lumber be 'B'

B/t is growth in volume of lumber during the time period t.

Say that growth rate diminishes over time

i.e. mature tree does not grow much (at some point maybe B becomes negative? so quantity of usable lumber falls).

- Profit maximizing harvesting?

- Say revenue at harvesting is: P∙B (P=price per unit of lumber, B volume harvested)

- Options for the owner of the tree?

Harvest now and get revenues: P∙B

Harvest later and get revenues: P∙ (B+B)

- this assumes price is the same now and in the future.

- which is better?

22

- if harvest now can take P∙B and invest it at the going interest rate 'i'. This gives you:

P∙B (1+i) after 1 period.

- Key question: which is greater P∙B (1+i) or P∙B + PB?

if P∙B+i P∙B - (P∙B+ PB) > 0 harvest now!

simplifies to: i > B/B

i.e. harvest now if interest earned on revenues earned now is greater than the benefit from delaying harvesting

(growth rate in stock of lumber)

(NOTE: i is measured as a proportion so a 5% interest rate means i=.05)

- Complicating factors?

- rule above captures an important idea but ignores some interesting complications.

e.g. what if the price if not constant?

- an additional factor affecting the benefit of waiting will be the expected change in the price of lumber.

i > (B/B) + (P/P) (approx.)

where P = expected change in price.

23

what about harvesting costs?

- should subtract harvesting costs from revenues.

e.g. say marginal harvesting costs are constant at 'c'

so harvesting cost is c∙B for B units harvested.

- earlier condition becomes:

(1+i)(P∙B -c∙B) - [ P(B+B)-c(B+B) ] > 0

becomes: i > (B/B)

(same as before then if marginal cost constant)

- condition will differ for other cost assumptions.

- Above is lumber similar stories for fish.

- Biology important in thinking about outcomes: provides estimates of the growth in the size of the stock of the renewable resource.

24

Exhaustible or Non-Renewable Resources:

- Text takes oil as its example.

- With a renewable resource there is an incentive to delay use since the stock grows over time.

- This is not so with an exhaustible resource:

- fixed amount of it on earth (some may be undiscovered however)

- Question then: why not just use all of the resource immediately?

i.e. extract it now, sell it and earn interest on the revenuesvs. no interest if extract and sell later.

- Some possible answers:

(1) Prices are expected to rise in the future;

(2) Marginal extraction costs rise with amount extracted by enough to provide an incentive to smooth extraction over time.

- Text looks at the first story: prices are expected to rise in the future.

- To give an incentive to delay extraction oil prices must rise by at least as much as the rate of interest:

- benefit of delay: (P1-P0) on each barrel of oil P0 = initial price, P1= price one period later.

- cost of delay: i P0 can’t earn the interest on initial revenue from selling now.

25

- extract if: i P0 > (P1-P0)

or: i > (P1-P0)/P0

so if price is expected to rise by less than the interest rate extract now!

if price is expected to rise by more than the interest rate delay extraction!

26

- Equilibrium price behavior:

- prices will tend to rise at the rate of interest.

- if prices expected to rise faster than i: - no one will sell oil now. - price rises immediately and future rise is smaller.

- if prices expected to rise slower than i: - everyone wants to sell now,- price of oil falls now and future rise is larger.

- so if P0 is initial price, then future prices are: P1=P0(1+i), P2=P0(1+i)2 ... Pt= P0(1+i)t

- Consequences?

- Other things equal exhaustible resource prices rise.

- Demand will be declining along the demand curve for the resource.

- So rising price slows exhaustion of the stock of the resource. (rising prices promote conservation).

27

28