lecture 2_irving fisher`s impatience theory of interest

8/18/2019 Lecture 2_Irving Fisher`s Impatience Theory of Interest

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Lecture 02: Irving Fisher’s Impatience Theory of

InterestThis lecture discusses in greater detail Fisher’s impatience theory of interest.Several concepts need to be introduced first. The financial economy is definedby lots of people in the economy and their utilities. Remember from last lecturethat we had two kinds of people A and B with utilities given by:

U A = 2

3 log(X 1) +

1

3 log X 2 (1)

U B = 12

log(X 1) + 12

log X 2 (2)

People also know today what their endowments are and they have some ideaof what they are going to be tomorrow. They possess labor today and they aregoing to be able to work again next year. The labor endowments were given by(eA1 , e

A2 ) = (1, 1) for A, and (e

B1 , e

B2 ) = (1, 0) for B . The stock endowments are:

θA

α = 1, θB

α = 0, θA

β = 1/2, θA

β = 1/2.Knowing that there are two stocks in the economy, they have to anticipate

what the dividends are going to be. As Fisher said, the main value of assets isthat they give you something, they produce a payoff. In this case they are goingto be dividends: β is producing dividends of 2, and α is producing a dividendof 1 next period (e.g., Dα2 = 1, D

β2 = 2). Thus the economy consists of:

(U A, U B , (e1A, e2

A), (e1B , e2

B), (D2

α, Dβ2 ), θA

α , θB

α , θA

β , θA

β ) (3)

This is the beginning of the economy and we want to define from this equilib-rium the following: the contemporaneous prices (q ) of the stocks, the ownershipof the portfolio of stocks, and the consumption levels. At first, this appears tobe a complicated problem. We can simplify it by looking at a general equilib-rium problem which is much shorter to describe. Finding a general equilibrium,requires finding a solution for:

(q 1, q 2, πα, πβ, (X A1 , X

A2 , θ

Aα , θ

Aβ ), (X

B1 , X

B2 , θ

Bα , θ

Bβ )) (4)

q 1X i1 + παθ

iα + πβθ

iβ ≤ q 1e

i1 + παθ

iα + πβθ

iβ (5)

X A1 + X B1 = e

A1 + e

B1 (6)

X A2 + X B2 = e

A2 + e

B2 + (θ

A

α + θB

α )Dα2 + (θβ

A

+ θβB

)D2β (7)

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We can simplify this problem by looking at this general equilibrium:

(U A, U B, ( eA1 , eA2 ), ( eB1 , eB2 )) (8)

where eA1 = eA1 = 1 and e

B1 = e

B1 = 1. We find that:

eA2 = eA2 + θ

A

αDα2 + θ

Aβ D

β2 = 1 + 1 ∗ 1 +

1

22 = 3 (9)

eB2 = eB2 + θ

B

α Dα2 + θ

Bβ D

β2 = 0 + 0 ∗ 1 +

1

22 = 1 (10)

Remember Fisher has no theory on contemporaneous prices, just relativeprices. Simplify the system by letting q 1 = 1, which means that p1 = 1. Further,we found that: p2 = 1/3, X

A1 = 4/3, X

A2 = 3 −

1

33 = 2, X B1 = 2/3, X

B2 = 1 +

1

33 = 2. So we have reduced the general equilibrium to a financial equilibrium.

What are Fisher’s insights?

0.1 No Arbitrage

Fisher said people look through the veil of things. They understand the worldand you can count on their understanding to guide your understanding of theeconomy. So if you know that πα = 1/3, Fisher says you don’t have to solve forthe whole equilibrium to figure out what πβ is. What would πβ be? Accordingto Fisher, stock β always pays off exactly what stock α pays off. So if thesepeople are rational they are not going to allow for an arbitrage. So arbitragemeans if there are two assets or two things that are identical, they have to sellfor the same price. If they sold for a different price there would be an arbitrage.You would sell the more expensive one and buy the cheaper one, and so you

wouldd have accomplished a perfect trade-off, but you would have gotten thedifference of money. So since πα = 1/3, it must be that πβ = 2/3.This is the first, most important principle of finance that Fisher introduced;

the idea of no arbitrage and making deductions for no arbitrage. A lot of whatwe do in Finance is actually being more and more clever about how to do noarbitrage.

0.2 Bond Pricing

Suppose we introduced a nominal bond with payoff $1 in period 2. Also supposefor now that q 1 = q 2 = 1. By definition, the price of this bond is

1

1+i where

i is the nominal interest rate. To figure out 1 + i we can use the no arbitrageprinciple. Thus, 1 dollar today can go into 3 units of stock α, which goes into 3

units of X 2 as dividends, which equals 3 dollars. So if you take 1 dollar todayby buying stock α you can get 3 units of it since its price is a 1/3, and sincestock α pays one unit of output next period you know that 1 dollar today givesyou 3 units of stock alpha. In turn, this gives you 3 units of good 2 as theoutput, which at a price 1 dollar tomorrow is worth 3 dollars. Therefore, bybuying stock α you can put in a dollar and get out 3 dollars. So it means that

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1 + i = 3, which means the interest rate is 200 percent. This a second resultthat can be deduced.

0.3 Real Interest Rate

Fisher defined the real interest rate as number of goods today goes into numberof good tomorrow. For instance, one good today or 1 unit of X 1 is 1 dollartoday. If you had one apple today you could sell it for q 1× 1 apple, which isq 1×1 = 1, or 1 dollar today. This one dollar today can get 3 shares or 3 units of stock α, which then gives you 3 units of X 2. Thus, 1 unit of X 1 today turns into3 units of X 2, so therefore 1 + r = 3 implies r = 200%. This is the real rate of interest. Fisher realized that people are going to look through all the gibberishof money and they are going to think about what apples are they giving uptoday and what apples are they getting tomorrow. They are not going to beconfused by all the holding of assets in between.

0.4 Inflation

Suppose we now started with q 1 = 1, q 2 = 2. Fisher said that there is always anormalization in equilibrium (in our case q 1 = 1). Walras originally introducedthe idea of normalization in a one period model in general equilibrium. Inmulti-period models there is a normalization every period. Every period there’sa choice of whether you are dealing with dollars, or francs, or centimes, etc.,and so there is a free normalization.

That means that inflation 1 + g = 2/1 = 2. This means inflation will be100%. In this case what is πα going to be? If we re-solved the equilibriumtaking q 1 = 1 and q 2 = 2 the equilibrium outcome will not be affected. Thus,πα = 1/3.

This is a big question in finance. People when they were buying and sellingstocks only traded for the output produced by that stock, in our case apples.One does not care about dollars or centimes or francs but cares about the goodshe is going to get. This is what looking through the veil means. The price of the stock is going to stay 1/3 because the number of apples it pays tomorrowhas not changed. It is still the same one apple. Since the stock is only payinga certain number of goods, the price of the stock today is going to equal thepresent value which is 1/(1 + r) times its dividend:

πα = p2 ∗ Dα2 (11)

In essence, we got the price of the stock by saying the stock pays off onegood tomorrow, but one good tomorrow is only worth a third of one good today,

so therefore the value of the stock is only equal to a third times 1 or 1/3. Soassuming p1 is 1 you figure out how many units of today’s goods is it worth.

Now, if p1 weren’t 1 then what do we do? Then we would have to write:

p1 ∗ πα = p2 ∗ Dα2 (12)

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Therefore the following equality must hold:

πα = 1 p1/p2

Dα2 = 11 + rDα2 (13)

This is Fisher’s famous equation. Fisher said the way to figure out the valueof a stock is to look at its dividends and discount them by the real rate of interest (i.e., 1 unit of output tomorrow, since the value of an apple tomorrow isonly a third of the value of an output today). Remember that the real interestrate 1 + r, is equal to the ratio of the two goods. Thus, p1/ p2 is just 1 + r.Another way of saying that is that the real interest rate is the tradeoff betweenapples tomorrow and apples today which is p1/p2. The apple tomorrow is worth

p2 times the dividend.Therefore, the value of a stock is the real dividends it is paying in the future

discounted by the real rate of interest. You are turning tomorrow’s next year’s

goods, finding the equivalent in terms of this year’s goods, and the ratio of thosetwo prices is the real rate of interest.Another way of saying the same thing is you could turn cash next year into

cash this year. Assuming q 1 = 1, another way of saying that is:

πα = 1

1 + iDα2 q 2 (14)

From (13) and the relation above we can write that: 11+r

= q21+i

.One takesthe nominal rate of interest times the money that is being produced, becausethe nominal rate of interest gives the trade-off of a dollar today for a dollar inthe future. A dollar in the future is not worth, usually, as much as a dollar todayso you have to discount it. Hence, a certain number of dollars in the future areworth less dollars today. If you take the payoff of dollars in the future discounted

by the nominal rate of interest you get today’s price, whic would also result fromtaking the real dividends in the future discounted by the real rate of interest.Both rules are an application of the principle of no arbitrage, looking throughthe veil.

So what would the nominal interest rate be in this case? Because the realinterest rate hasn’t changed, it is still 200%. Since Dα2 = 1, q 2 = 2, and πα = 1/3then:

1 + i = 1

παDα2 q 2 (15)

which means that 1 + i = 6 or that i = 500%. In words, you take 1 dollartoday at price q 1 = 1 that buys 3 units of alpha still, and that tells you that youget 3 units of X 2, the dividend. The 3 units of α pay 1 apple each, or 3 apples,

which in period 2 are worth 3×2 or 6 dollars tomorrow. Basically, we have turned1 dollar today into 6 dollars in the future. The relationship between the nominaland real rate of interest and inflation is the so-called Fisher Equation. Thesetwo famous equations are called the Fundamental Theorem of Asset Pricing.

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1 + r = 1 + i

1 + g (16)

So why is this theorem true? The real rate of interest trades off applestoday for apples tomorrow, and as we had, 1 apple today is trading for 3 applestomorrow. That is why r was 200%. Inflation is 100% and since 1 apple todayat a price of $1 is worth 3 apples or 6 dollars in the future the nominal rate of interest has to equal 500%. We can see the principle of no arbitrage at work.Any banker can take a dollar, buy a stock, turn it into 3 units of dividends andthen sell it for 2 dollars apiece and get 6 dollars. And so a banker can take adollar and turn it to 6, so competition will force the bankers to give you 6 dollarsfor every 1 dollar you give. So the interest rate has to be 1 + i = 2 × 3 = 6 andthe real rate of interest is the nominal rate of interest divided by inflation.

0.5 An ExampleLet’s go back to the equilibrium with q 1 = 1 and q 2 = 1. Suppose China offeredto lend money to the U.S. at a 0% interest. Would that be a great deal? Wouldpeople rush to do that? Obviously, since borrowing in the U.S economy fromanother American would require paying a 200% interest, everyone would rushto take advantage of the low interest rate.

Let’s try another question. Suppose you invented a new technology thatturns 1 unit, 1 apple today, into 2 apples tomorrow. Is this something peoplewould rush to do or not? Could this new technology be used to make a Paretoimprovement and make everybody better off? The answer is no, because thereal prices, Fisher would say, are 1 and 1/3 and no matter how you look at itthe interest rate is 200%% – you would be losing money, because you are givingsomething up that is worth 1 and getting something that’s only worth 2/3.

The proof is that if it did offer a Pareto improvement, and if in the end itled to an allocation ( X A1 , X

A2 , X

B1 , X

B1 ) that made everyone better off. Then,

it would mean that p1 X A1 + p2 X

A2 > p1e

A1 + p2e

A2 . Or similarly it would imply

that, p1 X B1 + p2 X

B2 > p1e

B1 + p2e

B2 .

This would be the case because in this Fisher economy, if this allocationreally made A better off than what he’s gotten (4/3, 2), he would have chosenit. And B would have chosen the new allocation if it was better than (2/3,2). Soclearly they must have been too expensive for A and B to choose because theywere rationally choosing the right thing given what they could afford. You wouldfind that total consumption value was bigger than total endowment value whichis impossible. This is a contradiction because in the end the total consumptionof the people has to be the total of what there is available and what is produced

in the economy.So that’s how we know that no new technology could possibly make every-

body better off, and we know trivially it makes everyone better off if and onlyif it makes a profit. So if and only if it makes a profit can it be used to makeeverybody better off, and amazingly, in a free market economy, people are goingto use it if and only if it makes a profit. So they’re going to use it if and only if

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Nominal rate

Inflation

Real Rate


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it’s a good thing for the economy. So that’s the basic laissez-faire argument–thatthere are new discoveries all the time. Every other day somebody’s thinking of

something new. Are we going to use it? Should we use it? Is it something weneed to read about in the papers and use? Well, there are a whole bunch of peo-ple, the discoverers themselves they are going to talk to their business friends,and they’re going to say, ”Do you want to lend me the money to get this thinggoing,” and all of them are going to do this profit calculation. If they decide itloses money they’re not going to do it. This is the main lesson of laissez-faire.

So let me just put this in perspective a little bit. In the old Russian economyof the 1930s and ’40s there was no profit system, so the central planner had tofigure out, should a new invention be used or not. So every time there was anew invention a committee had to get together, of central planners and decidewhether to use it or not. There was a famous guy named Kantorovich who wasin charge of a lot of that. He won the Nobel Prize in economics. He sharedit with a Yale economist named Koopmans and so Kantorovich told this veryamusing story.

He said that there were two central planning bureaus. One was in chargeof allocations and one was in charge of prices. One had to set the prices. Theother had to set the allocations. And of course the whole message here is thatyou have to combine these. You don’t know whether it’s worthwhile to changethe allocation until you know whether the new technology is going to make aprofit or not, and here they had the two things separated. They were tellingpeople what to do before knowing whether they made a profit or not becausethey didn’t have prices because there weren’t free markets.

So the bottom line of the Fisher story is that you take a complicated financialeconomy, you reduce it to something very simple, solve for the equilibirum, andyou can understand a lot about this economy. That is something that most

people did not realize at the time and still don’t realize it now. If you ask atypical person if there is inflation, that means the dividends next year are goingto be higher, is that going to raise the value of the stock today? One might say,”Yes of course because it makes the price of the dividends higher tomorrow.”Fisher would say no, it doesn’t change anything real in the economy. If thereis more inflation there will be a higher nominal interest rate, so discounted bythe higher interest rate payoffs of the stock will give you the same stock priceas before.

To summarize, how do you know that a final allocation that emerges as acompetitive equilibrium is Pareto efficient? And the argument was if you cando better–that means, make everybody better off–then each person, if you lookat the value of what they’re getting under the new regime it must be morethan the value of their endowments. Otherwise they would have chosen the new

regime. That means everybody would have had to pay more for this new regimeallocation than the value of their endowments. So this is more than person Aand person B can afford. Their consumption would be more than the value of their endowments but people can only eat what is being produced. Everythingthat is being produced is part of somebody’s endowment. So whatever the newallocation is it has to add up to the new endowment.

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Because the new technology loses money the value of the new endowmentwill be lower than the value of the old endowment. The contradiction is the

value of the new endowment after the technology is used, at the old equilibriumprices, is lower than the value of the old endowment at the old equilibriumprices. Hence, you cannot make everybody better off.

The simple argument, that Ken Arrow and Gerard Debreu gave is the sim-plest and most important argument in all of economics. So we get as a conclusionthat, putting it another way, that owners of firms should maximize the value of their firms, the stock market value of their firms, and they do so because if theyfind some new way of producing that is going to lose money it is going to makethe stock market value go down. Remember the stock market value is just thesame calculation, the value of all the output they are producing. If they findsome way of losing money and they try to use it it willl make their stock marketvalue go down.

0.6 Impatience Theory of Interest

So far we haven’t introduced risk. When that happens things are going toget more complicated. Fisher couldn’t deal with risk. So without risk, whereeverybody is anticipating the dividends in the future, that means that you canalways reduce a financial economy to a general equilibrium.

The solution to that problem with marginal utility and Pareto efficiencytells us an enormous amount about how the stock market and everything works.It tells us that the value of every stock is just the discounted real dividends,discounted at the real rate of interest, or the discounted nominal payoffs, cashflows, discounted at the nominal rate of interest.

And it tells us that the real rate of interest is the nominal rate divided bythe rate of inflation. And it tells us that it’s a good thing all these owners of companies are maximizing profits or share value, which is the same thing, andthat’s helping society.

All three major religions thought interest was a terrible thing. They allthought that the nominal rate of interest should be 0. But what Fisher says isthe nominal rate of interest is irrelevant. Nobody cares about the nominal rateof interest. They look at apples today and apples next year. The money andstuff just gets in the way. It is the real rate of interest that you care about, andthe real rate of interest doesn’t have to be positive.

The real rate of interest is obtained by solving for p1 and p2 in the generalequilibrium model above. So what would change the real rate of interest when allyou have are the utilities and the endowments? The first thing Fisher though of

was impatience. So in fact one of his most famous articles is called an Impatience Theory of Interest , so let’s call it that, Impatience Theory of Interest.

Fisher said that in his view people are impatient. Why? That means anapple today they thought was more valuable that an apple next year. Why?Because of the poor imagination, it was easy to think about eating the appletoday. You can just hold it in your hand and it’s so close, but to think about

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eating it in a year requires some imagination. They had poor imagination, andsecondly, the second main reason is mortality. They might die between today

and next year.So those are the two main reasons. So what does it mean? An apple next

year is not a sure thing. There is the Impatience Theory of Interest . So he saidthat is why it makes sense to have this guy A as impatient because he valuesthe apple today more than a value tomorrow. He has a discount rate of δ = 1/2say (i.e., δ = 1

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= 1/2). However, B is not impatient because the discountfactor is 1.

Samuelson was the one who introduced the discount factor to capture Fisher’sidea that the good next year, the same apple next year is not worth as much toA as an apple this year. So suppose I change 1/2 to a 1/3. What will happento the real rate of interest? That makes people more impatient. This makesthem more impatient, because now they care even less about the good nextyear. In the Reagan years, everybody talked about the now generation. Peoplewere getting more impatient. So what happens to the real rate of interest whenpeople get more impatient? In order to get anybody to save, because they wantthe stuff now, you are going to have to give them a higher real rate of interest.

We can write:

X A1 = 1

1 + δ ( p1e

A1 + p2e

A2 ) (17)

where δ is the discount. If p2 = 1 then:

X A1 = p11 + δ

(eA1 + 1/p1eA2 ) (18)

So obviously as p1 goes down, your demand goes up or if p1 goes up thedemand goes down. So if you make δ smaller, that is going to raise demandfor A at the old prices. Why? If δ goes down it implies X A1 goes up. Sothis persion is demanding more now, but if he’s demanding more at the oldequilibrium prices the only way to clear the market is to raise p1. Thus, p1must go up to clear the market.

In other words, if you care less about the future (i.e., low δ means you careless about the future) to get anybody to save you’re going to have to raise theinterest rate. To say it formally if we solve for equilibrium with a lower delta atthe old equilibrium prices, A would now shift and try to demand more of good1. But if he demanded more of good 1 that would mean too much demand forgood 1, and the only way to clear the price of good 1 is to raise the price P1.But if you raise p1 holding p2 fixed (i.e., p1/p2), the interest rate has to go up.That s Fisher’s Impatience Theory. That is the main determinant of interest

according to Fisher.What is the second one? He says suppose people are more optimistic aboutei2. Everybody thinks the world’s going to be much better next year. We’regoing to have more endowments. What do you think is going to happen tothe interest rate, the real interest rate? If people thought they were going tobe richer at the old prices what would they do today for X 1, demand more or

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less today? The point is there’s going to be so much stuff around for peopleto eat tomorrow, you’ve got to get them to want to eat all that extra stuff

tomorrow. So you have to give them an incentive to want to eat all that extrastuff tomorrow, so you have to raise the interest rate, not lower it.

Now, how can you actually give a formal proof of that so you know you’renot confused? Again, at the old prices what’s going to happen to the demandfor X 1? At the old prices, since you’re going to be so rich in the future, youthink you’re just incredibly rich now, so of course you’re going to consume moretoday. So there’s going to be more demand today and the endowment todayhasn’t changed. So there’s going to be more demand today with the sameendowment today, so therefore in order to clear the market today you’re goingto have to raise p1 relative to p2 so the interest rate has got to go up.

In other words, if you increase the endowments tomorrow the supply today of goods hasn’t changed, but people are richer tomorrow. So clearly they’re goingto consume a higher fraction of their wealth. You tell anybody, ”You’re goingto be rich next year. You’re going to be worth a fortune,” the normal person,Cobb-Douglas person, is going to consume more stuff today anticipating thathe is going to be so rich tomorrow. He is going to borrow against tomorrow’swealth. And so therefore, in order to clear today’s market where the supplyhasn’t changed, with all these people trying eat more today you have to raisetoday’s price relative to tomorrow.

The third example is Fisher’s most famous one. Suppose you transfer money,transfer wealth, from poor to rich. What would happen? We have to make anextra assumption here. Fisher felt that the people who were rich were richbecause they were patient. They could charge interest and get lots of money.So if you change wealth you take away some money from the poor. That’swhat’s happened in the American economy over the last 15 or 20 years. The

rich have gotten richer and the poor are pretty much back where they werebefore. So suppose the rich get rich at the expense of the poor. What’s thatgoing to do to the real rate of interest?

That would lower the interest rate. There’s an intuitive way of saying itmeaning that the rich, because they are patient, are probably the lenders. Nowthey are even more willing to lend and so the interest rate has to go down to getthese other people to borrow. A formal way of saying it is that if you transfermoney from the poor to the rich that means the rich guys always consume ahigher proportion in the future because they are more patient. So a more patientguy will consume more in the future. If you take away wealth from an impatientguy and give it to a patient guy you’re going to increase the size of the economy.The economy is going to be more in the hands of the patient people, and so thepatient/impatient mix is going to change. People on average are more patient

than they were before so on average in the economy they are going to consumeless than they were of today’s good.

Because you’ve made people, a lot of them impatient, a lot more patient,you’ve increased the patient ones and decreased the impatient ones, so in balanceyou’re going to decrease demand today because it was the impatient ones whowanted to eat today and the other guys were willing to wait. Now the guys

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who aren’t willing to wait don’t have any money. They’re the ones doing all theconsuming today and now they can’t afford to do much consuming, so you’re

going to reduce consumption today. So to get the market to clear again youhave to lower the interest rate this time.

Those are the three famous conclusions of Fisher, more impatient people,higher interest rate, more optimistic about the future, higher interest rate, trans-fers from the poor to the rich lower interest rate. So what happens to the stockmarket in this case? Suppose people are more impatient. Does the stock marketgo up or down? The answer is down because the stock market price is just the1/(1 + r) times the dividends. So I haven’t told you the dividends changed, soif the dividends are the same and the real interest rate has gone up the stockmarket has gone down.

Now, suppose you transfer wealth from the poor to the rich, what is going tohappen to the stock market? It’s going to go up. So what happened in the last20 years? The rich got richer, the poor got poorer, the interest rates got lowerand lower and the stock market got higher and higher just as Fisher would havesaid.

To summarize, Fisher’s theory of interest made sense of thousands of yearsof confusion, and the main idea is that you shouldn’t think of the nominalinterest. People look through all that. They look at the real rate of interest andthe real rate of interest is just the ratio of two prices just like everything elsein equilibrium, so therefore there is no such thing as a just price. The price, infact, that equilibrium finds is the best price is because that is the price that isgoing to lead new firms and inventors to use technologies that help the economyas opposed to hurting the economy and wasting resources. So the price that themarket finds is the just price and the real rate of interest is the right real rateof interest provided that people are rational and see through this veil.

So, why is it that the real rate of interest is typically positive? Well, it’sbecause people are impatient. Now Fisher said one other reason that screws upthe real rate of interest is people sometimes get confused by inflation.

As an ”amusing” fact, he said that all contracts should be inflation indexed,and he forced his Yale secretary and his secretaries at his company to changetheir contracts and accept deals where their wage was indexed to inflation. Andof course the Great Depression happened and all of the prices collapsed, and soall his secretaries got less money out of the deal so he wasn’t too popular withthem either.

He said impatience is a fundamental attribute of human nature. As longas people like things today rather than tomorrow there is going to be interest.So interest is, as it were, impatience crystallized into a market rate, and thereasons for impatience are the lack of foresight, possibility of dying and self

control etcetera.In conclusion, those patient accumulate wealth and by waiting and lend-

ing they make production possible because the people with the good ideas aregetting the money to produce from the patient people who are willing to wait.

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lecture 2_irving fisher`s impatience theory of interest

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