251a
Fall Econ 2006
Computing Equilibrium in OLG Economies with Land 1 OLG Economies
In real life, the economy lasts much longer than the life of any individual. Indeed many economic institutions survive precisely because they are expected to continue forever. Social security is the chief example. If people understood that there was going to be a last generation, then they would anticipate that this generation would not contribute when it was young, hence that the previous generation would not contribute when it was young, hence that the third to last would not contribute, and so ultimately, that even todays young would not contribute. Maurice Allais and later (perhaps independently) Paul Samuelson invented the overlapping generations model (OLG) to capture this situation. Every period a new generation is born which lives for two periods. Thus at each time period there are two living generations, an old generation and a young generation. A period is thought of as approximately 30 years. Time goes on forever. Allais and Samuelson argued that the innity of both time periods and agents radically changed the nature of equilibrium. Samuelson suggested that equilibrium might not be Pareto ecient, and that the real rate of interest might be negative, even if the economy did not shrink over time. They also thought that a second, new kind of equilibrium would emerge in which the real rate of interest is divorced from any of the considerations like impatience that Irving Fisher had stressed. They thought that in this new kind of equilibrium the real rate of interest woud turn out to be equal to the rate of population growth, irespective of the impatience of the consumers or the distribution of their endowments. Furthermore, Samuelson argued that it might not be necessary for an asset to be valued according to the present value of its dividends, contradicting yet another one of Fishers central concepts. Samuelson suggested that a piece of green paper might be worth a lot, even though it pays no dividends, because the holder might think he could sell it to somebody later, who would buy it on the expectation that he could sell it to somebody else later, ad innitum. Later authors called this a rational bubble. It turns out that these views are incorrect if one includes in the model innitely lived assets like land, that do pay dividends in every period. We shall not discuss this new kind of equilibrium in this chapter, but instead go directly to the overlapping generations model with land. Overlapping generations economies give us the opportunity to study how longlived assets get priced in a world of certainty in a simple setting. With many periods, one would typically have to calculate equilibrium prices for every time period, which might get too complicated. But with OLG economies, we can assume a stationary stucture, with not only the forward rates, but also the asset prices, constant for all time. In a nite horizon model prices could never be constant over time, because as
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the end approaches the value of the asset would have to decline (becoming literally zero in the last period). When the population is stationary and identical across generations, asset prices will be constant. But if the population oscillates in size, as the American population has from one generation to the next over the last 100 years, asset prices will also oscillate. There will be periods of rapid stock price growth, but also periods when stock prices decline or stay at over decades. The dierence between a Samuelson OLG economy (without land) and a Fisherlike OLG economy (with land) has some practical implications for a trader. Most traders buy an asset with a view to its near term dividends, and the price they think they will be able to sell it for. They spend a lot of time thinking about what other people will be thinking about the asset when it comes time to sell it. That would be consistent with rational behavior in Samuelsons world. Fishers advice is, think about the dividends. Thats what the future buyers should be thinking about as well. To solve the OLG model for equilibrium we repeat the trick Fisher introduced earlier, reducing the nancial equilibrium to a timeless general equilibrium. We emphasize how quick this is by considering a model with 1000 dierent assets. Once we have the interest rates, we can price every asset by the present value of its dividends. So whether there are a thousand assets or a million, the calcuation of equilibrium is essentially the same.
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A Simple OLG Economy with Land
Imagine an OLG economy in which every period one agent is born who lives for two periods. The trader has utility for consumption when young y and old z , and also an endowment of goods when young and old. Consider the following example: U (y, z ) = log y + log z (ey , ez ) = (3, 1) If we want to remind ourselves that there is a new trader born every period, we should index each trader by the date t of his birth, and as usual, index commodities by the date of their consumption, so that consumption of the apple at time period t is denoted xt . U t (xt , xt+1 ) = log xt + log xt+1 (et , et+1 ) = (3, 1) tt Let us also suppose there is one acre of land in the economy that produces a dividend Dt = 1 apple every period forever. The economy begins in period 1, with an old agent who owns the land and has an endowment of one apple, and a newly born agent as above. The apple from the land at time 1 is owned by the old agent at time 1 (who bought the land a time 0 and hence has the claim on the apple). Recall our convention that buying an asset at time t gives ownership of all dividends from time t + 1 up to and including the dividends in the period in which the asset is sold. 2
2.1
Financial Equilibrium
At every period t we need to nd the price qt of the commodity and the price t of the asset. Every agent in the economy must decide how much to consume when young, and what assets to hold when young, and how much to consume when old. The decision in old age is trivial, since the agent cannot do better than selling every asset he has and using the proceeds to buy consumption goods. Thus for every t 1 we can describe the decision problem of generation t by max U t (y, z ) = log y + log z
y,z,
such that qt y + t = qt et = qt 3 t qt+1 z = qt+1 et+1 + Dt+1 + t+1 = qt+1 1 + 1 + t+1 t For the original old generation, he optimizes simply by setting x0 = e0 + D1 + 1 = 1 + 1 = 2 + 1 1 1 Denote the optimal choice of agents t 1 by (y, z, ) = (xt , xt+1 , t ) tt
Market clearing requires for each t 2 that consumption of the old plus consumption of the young is equal to total output of goods, and also that demand equals the supply of land xt1 + xt = et1 + et + Dt = 1 + 3 + 1 = 5 t t t t t = 1 In period t = 1 we must have x0 + x1 = e0 + e1 + D1 = 1 + 3 + 1 = 5 1 1 1 1 Equilibrium is thus a vector (x0 , (qt , t , (xt , xt+1 , t )) ) satisfying the above conditt 1 t=1 tions on agent maximization and market clearing.
2.2
Stationarity
In all the equilibrium models we will consider in this course, money will not be in the model. So there is no reason for ination. So we can look for an equilibrium in which the contemporaneous price qt = 1 every period.1 The stationarity of the economy also suggests that there may be a stationary equilibrium in which the asset price also does not change through time. With the single kind of land, there is indeed an equilibrium in which the asset price is constant through time.
In other contexts we will consider models in which the prices have already been specied in advance, so there we may have ination. These models do not explain why there is ination,however. In fact they are not full equilibirum models, since in an equilbrium, every price is explained by supply and demand.
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2.3
How Fisher would Compute Equilibrium
Fishers recipe for computing equilibrium with assets is to put the assets into the endowments of their owners, and then nd the usual general equilibrium. He would have done the same thing in this innite horizon economy. He would simply look for the present value (as of period 1) prices (p1 = 1, p2 , p3,.. ). These present value prices implicitly dene real interest rates 1 + rt = which in turn satisfy for t 2, pt = Each agent t 1 should solve max U t (y, z ) = log y + log z
y,z,
pt pt+1
1 1 1 ... 1 + r1 1 + r2 1 + rt1
such that pt y + pt+1 z = pt et + pt+1 et+1 = 3pt + 1pt+1 t t or equivalently max U t (y, z ) = log y + log z
y,z,
such that 1 1 1 z = et + et+1 = 3 + 1 y+ t t 1 + rt 1 + rt 1 + rt Notice that Fisher has reduced two budget constraints to one. The original old agent 0 would have a more complicated calculation (though no choice). His endowment consists of an innity of goods, namely the apple x1 in his original endowment, the apple in period 1 paid as a dividend from the land on account of his prior ownership of the land, and one apple for every period t 2 which Fisher would put in his endowment in lieu of the asset. He would choose x0 = 1p1 + 1p1 + 1p2 + 1p3 + 1p4 + ... 1 or equivalently, x0 = 1 + 1 + 1 1 1 1 1 1 1 1+ 1+ 1 + ... 1 + r1 1 + r1 1 + r2 1 + r1 1 + r2 1 + r3
Fisher would also require market clearing in all the commodity markets (but of course not in the asset markets, since Fisher ignores those until after equilibrium has already been found). Market clearing requires for each t 2 that consumption of the old plus consumption of the young is equal to total output of goods xt1 + xt = et1 + et + Dt = 1 + 3 + 1 = 5 t t t t 4
In period t = 1 we must have x0 + x1 = e0 + e1 + D1 = 1 + 3 + 1 = 5 1 1 1 1 Equilibrium is thus a vector ((pt ) , x0 , (xt , xt+1 ) ) = ((rt ) , x0 , (xt , xt+1 ) ) tt tt t=1 1 t=1 t=1 1 t=1 satisfying the above conditions on agent maximization and market clearing. Note how much simpler equilibrium has become. Again we have dropped all considerations of asset market clearing, and dropped all asset prices. The only tricky equation seems to be the period 1 market clearing equation, since it involves an innite sum. Since the interest rates will turn out to be constant, the innite sum is easy to compute. But in fact it need not be computed at all. By Walras Law, if all the other markets t 2 clear, the t = 1 market must clear as well! From stationarity we might expect that the equilibrium real interest rate rt would be constant, or equivalntly pt = pp...p = pt1 . Thus we need only nd a single variable p = 1/(1 + r). Using the fact that Cobb-Douglas agents spend a constant proportion of their wealth on each good, no matter what the prices, we can reduce the Fisher equilibrium to one equation in one unknown. 2.3.1 Computing the Fisher Equilibrium
For each t 2, we must nd the interest rate r such that the demand of the old at t plus the demand of the young at t sums to the endowment of the od at t plus the endowment of the young at t plus the output of land. Using the single symbol p in place of 1/(1 + r) we must solve 1 [3 + p1] 1 + [3 + p1] = 1 + 3 + 1 2 p 2 which gives a quadratic equation p2 6p + 3 = 0 whichis solved by 6 (36 12).5 2 One of these roots is greater than one, and could not be right, because it would give a real interest rate less than zero, which would make the present value of land innite. Thus the right answer is p= p = (6 24.5 )/2 = .55051
1 + r = 181.7% Hence consumption when young and old is
(y, ) z = (1.775, 3.225) Clearly these values clear the consumption market for all t 2. We know by Walras Law that if all markets but one clears, then the last will as well, so we dont really have to check the period 1 market. But we will check it anyway. 5
The interest rate in turn gives a present value of land P V Land = p1 + p2 1 + ... = p/(1 p) 1 1 1+ 1 + ... = 1.225. = 1+r (1 + r)2 In period 1 the old will consume their endowment of 1 plus the dividend of 1 plus the value of the land they will sell, which gives exactly 3.225. So the time 1 market clears as well. Thus Fisher has solved the problem. We can now translate this general equilibrium back into a nancial equilibrium by taking (x0 , (qt , t , (xt , xt+1 , t )) ) = tt 1 t=1 (3.225, (1, 1.225, (1.775, 3.225, 1)) ). t=1 Note that for the problem with one type of land, we did not even have to bother to compute the present value of land. Once we knew that consumption when young was 1.775, we could have deduced that the young must be putting the rest of their income when young into land, giving the price of land 1.225 = 3. 1.775. Annualized rate of interest We have interpreted the period as 30 years. Therefore to gure the annualized rate of interest we should take 1 + rannual = 1.811/30 = 1.020
2.4
Comparative Statics
Despite what Allais and Samuelson said, the rate of interest does respond in exactly the way Fisher argued. Consider the same model as before, but make all the consumers more impatient U (y, z ) = log y + .5 log z Then our master equation would become 1 [3 + p1] 2 + [3 + p1] = 1 + 3 + 1 3 p 3 giving p = .419 1 + r = 239% 1 + rannual = 1.029 P V Land = .721 The rate of interest does indeed increase, and the price of land decreases. Incidentally, the increased impatience leads to a radical increase in young consumption and a drop in old consumption to y = 2.28 z = 2.72 6
2.5
Social Security
Imagine that in the original economy (with utilities log y + log z ) the government, seeing the endowments of 3 when young and 1 when old, decides to institute a social security system in which every period the young give the old 1 unit. Since this is a major reallocation of goods, it will have a nontrivial eect on equilibrium prices. Computing the equilibrium for the OLG economy in which endowments are (2, 2) for every generation t 1, and assuming the old generation 0 has an endowment of 2 apples at time 1 plus the land, which pays 1 apple every period, we get p = .38 1 + r = 261% 1 + rannual = 1.03 P V Land = .62 This conrms Fishers contention that decreasing early endowments and increasing later endowments should raise the rate of interest and lower land values.
2.6
Growth
It is often said that if only every generation had more children, social security would work better, since the young would be able to share the burden of helping the old. The trouble with that reasoning is that if ignores the fact that higher population and output growth would mean higher real interest rates, making the social security rate of return as bad as before relative to market interest rates. Suppose now that every agent has two children instead of 1, so that generation t has endowment 2t1 (3, 1) and land produces dividends of Dt = 2t1 . In Fishers general equilibrium we must have that 2t1 1 [3 + p1] 1 + 2t [3 + p1] = 2t1 1 + 2t 3 + 2t 1 2 p 2 1 1 [3 + p1] + 2 [3 + p1] = 1 + 2 3 + 2 1 = 9 2 p 2 2 2p 11p + 3 = 0 11 121 24 = .2878 p= 4 1 + r = 347%
so interest rates have gone up by 166%. It would now be possible for each old agent to give 1/2 an apple when young, and still get 1 apple when old. But the present value loss of this would be as bad as in the economy when the young had to give up a whole apple to get one back when old.
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3
Demography in OLG
In America over the last 100 years, the generations have alternated in size between big and small. Everybody knows about the baby boom and echo baby boom, but the same thing happened before. What happens to the value of land and the real interest rate if we alternate generation sizes? Suppose the small generation is exactly as before, but now we alternate that small generation with a large generation that is identical in every respect, except that it is twice as big U b (y, z ) = log y + log z (eb , eb ) = (6, 2) yz As before suppose that land produces 1 unit of output each period. Begin at time 1 with a small generation of young. We would expect the price of land and the real interest rate to alternate between periods. Let rb be the interest rate that prevails when the big generation b is young, and ra prevail when the small generation a is young. Equilibrium can be reduced to two equations. The rst describes market clearing for goods in odd periods when the small generation is young and the big generation is old, and the second equation describes market clearing in even periods, when the big generation is young and the small generation is old. As before, we let pa = 1/(1 + ra ) and pb = 1/(1 + rb ). Then 1 [6 + pb 2] 1 + [3 + pa 1] = 2 + 3 + 1 2 pb 2 1 [3 + pa 1] 1 + [6 + pb 2] = 1 + 6 + 1 2 pa 2 Using solver these can be simultaneously solved to get pa = .418 1 + ra = 239% 1 + rannuala = 1.029 pb = .912 1 + rb = 109.6% 1 + rannualb = 1.003 Also ya = 1.71 za = 4.09 yb = 3.91 zb = 4.29 8
Now for the present value of land we must compute P VLanda = 1pa + 1pa pb + 1pa pb pa + ... P VLandb = 1pb + 1pb pa + 1pb pa pb + ... It is evident that the price of land is higher in the periods when b is young, since the interest rate is lower. To compute these innite sums looks a little dicult, but is quite easy, as we shall see. However, again, since there is only one type of land, we can take a shortcut and simply observe that the price of land will necessarily be equal to the money which the young are spending to buy it. This alternates between periods and is P VLanda = 3 1.71 = 1.29 P VLandb = 6 3.91 = 2.09 However, the right way to deduce the price of land (since it workds for multiple kinds of land) is to nd the present values. And indeed we can use the following derivation to rewrite the innite sums as: P Va = pa + pa P Vb P Vb = pb + pb P Va Solving these two simulateneous equations in two unkowns we get P Va = P Vb = pa + pa pb 1 pa pb pb + pa pb 1 pa pb
Plugging in the values for pa and pb gives the same answers. Observe nally that the young generation did much better than the old on a per capita basis. (Dividing b consumption by 2, to take into account that there are really two agents in each b generations, it is clear that the a generation comes out much better).
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A variant with 1000 assets
To see the power of Fishers transformation of nancial equilibrium into general equilibrium, it is useful to keep in mind a variation of our example with 1000 assets. Suppose there are actually 1000 dierent kinds of land, each with 1/1000th of an acre. Let the rst kind of land pay 1 apple in periods 1, 1001, 2001, etc, i.e, in periods 1(mod1000). Let the second kind of asset pay 1 apple in periods 2, 1002, 2002, i.e. in periods 2(mod1000). Let the other kinds of land be similar, so the 1000th kind of land pays 1 apple in periods 1000, 2000, 3000, etc, i.e, in periods 1000(mod1000) = 0(mod1000). Each of these assets will typically sell for a dierent price, since their dividends occur in dierent periods. In period 1, for example, the rst kind of land 9
will sell for the lowest price, since it will not pay again for another 1000 periods. Worse still, the asset prices will not be constant over time, since for example asset 2 will drop suddenly in price just after paying its dividend at time 2. So the variant economy seems much more dicult to analyze. 4.0.1 Financial Equilibrium in the variant example
Similarly, in the model with 1000 types of land, we can describe the decision problem of generation t by
y,z,1 ,..., 1000
max
U t (y, z ) = log y + log z
such that qt y + 1t 1 + ... + 1000t 1000 = qt et = qt 3 t qt+1 z = qt+1 et+1 + 1 D1t+1 + ... + 1000 D1000t+1 + 1000t+1 1 + ... + 1000t+1 1000 t For the original old generation, he optimizes simply by setting x0 = e0 + 1 1 1 1 1 [D11 +...+D1000,1 ]+ [11 +...+1000,1 ] = 1+1+ [11 +...+1000,1 ] 1000 1000 1000
Denote the optimal choice of agents t 1 by (y, z, 1 , ..., 1000 ) = (xt , xt+1 , t t , ..., t t ) tt 1 1000 Market clearing requires for each t 2 that consumption of the old plus consumption of the young is equal to total output of goods, and also that demand equals the supply of land xt1 + xt = et1 + et + t t t t t t 1 1 [D1t + ... + D1000t ] = 1 + 3 + 1 = 5 1000 = 1/1000, ..., t t = 1/1000 1000
In period t = 1 we must have x0 + x1 = e0 + e1 + 1 1 1 1 1 [D1t + ... + D1000t ] = 1 + 3 + 1 = 5 1000
Equilibrium for the variant economy is thus a vector (x0 , (qt , (1t , , , , , 1000t ), (xt , xt+1 , t t , ..., t t )) ) tt 1 1000 t=1 1 satisfying the above conditions on agent maximization and market clearing. With the 1000 kinds of land, there is no stationary equilibrium, for the simple reason that the assets must go down in value immediately after they pay o (and up in value as they get closer to their dividend dates). So the 1000 asset model seems much more complicated, both because there are 1000 prices each period, and because each of these 1000 prices will change over the course of 1000 years. Note however that all these assets in the variant economy pay o in total exactly 1 apple every period. Hence for Fisher there is really no dierence between the single asset economy and the 1000 asset economy. The total dividends are the same in each 10
economy. One only needs to nd the interest rates, which will be the same in both economies, and then trivially compute present values to get the land prices at each point in time. In a world of certainty, such as we have, every asset must give the same rate of return. Nobody would bother to buy an asset with a lower rate of return. The real rate of return 1 + rt on an asset at time t is the number of apples that can be obtained in period t + 1 from an investment of one apple in the asset at time t. In the single asset economy, that is 1 + rt = Dt + t+1 t
In the multiple asset economy, each asset will have its own rate of return 1 + rjt = Djt + jt+1 , j = 1, ..., 1000 jt
In equilibrium these must all be the same. The reason is that an agent who bought an asset with lower rate of return would be getting less money in old age then he could have had, and thus could not be optimizing. Equilibrium requires every agent to be optimizing. (Note that since supply of each asset is 1/1000 > 0, market clearing requires each asset to be held in positive quanity by somebody, so no asset can earn a lower return than any other asset, without contradicting the denition of equilibrium.) Thus Fisher would say just look for the single rate of return rt each period, and forget about all the dierent asset prices. The variant economy The equilibrium interest rates and consumptions of the variant economy are identical to the one-land economy. The only dierence is that in the last step we need to do a dierent present value computation for each of the 1000 dierent kinds of land. In the multiple-land economy there is no shortcut to present value, since it does not help to deduce that the young will put 1.225 into land without knowing how that will be split up between the dierent kinds of land.
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