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TRANSCRIPT
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.
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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.
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- 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.
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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)
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- 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.
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(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) )
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- Taxation of capital can affect the user cost in several ways (big topic in economics of business taxes)
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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.
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- 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.
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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?
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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?)
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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)
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- 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:
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∑i=1
N
ai=[a−aN +1
1−a ] (Perpetuity a= 1/(1+i) )
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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.
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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.
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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.
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- 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)
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- 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.
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- 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.
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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.
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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?
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- 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.
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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.
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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.
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- 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!
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- 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).
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