# 6-ECON251 Irving Fisher's Impatience Theory of Interest. Yale - Financial Theory : Lecture 6 Professor John Geanakoplos

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Financial Theory: Lecture 6

Professor John Geanakoplos: All right, so we spent a long time reviewing general equilibrium and we've now switched to finance, and you're hopefully going to see that the principles of finance emerge very quickly from the principles of general equilibrium. So that although it seems it was a long interlude we've actually learned a lot about the financial economy. So I'm going to continue with the example that we started with the last time. So we have a financial economy. So in a financial economy — what is a financial economy?

On this top board the financial economy is defined by lots of people in the economy and their utilities. So here we have for simplicity two kinds of people A and B with utilities given by the log X1 + 1 half log X2 etcetera. It's also people know today what their endowments are and they have some idea of what they're going to be tomorrow. They're labor powered today and they're going to be able to work again next year. So the labor endowments are given by (1, 1) for A, and (1, 0) for B.

And then they also know that there are two stocks in the economy and they have to anticipate what the dividends are going to be. And as Fisher said, the main value of assets is that they give you something, they produce something. In this case they're going to be dividends and beta's producing dividends of 2, and alpha is producing a dividend of 1 next period, and then the ownership of shares.

So that's the beginning of the economy and we want to define from that equilibrium which involves: what are the contemporaneous prices going to be, that's Q for contemporaneous, what are the prices of the stocks going to be, and who's going to hold which portfolio of assets of stocks, and who's going to consume what. And so Fisher said that's a very complicated problem. You can simplify it by looking at a general equilibrium problem which is much shorter to describe. And so the general equilibrium economy is going to be a much simpler one.

It's going to consist of UA and UB the same as before, and E-hatA1, the endowments, E-hatA2 and (E-hatB1 E-hatB2). So we've left out half the variables up there and we define E-hatA1 = EA1 = 1 and E-hatB1 = EB1 = 1, but E-hatA2 (this is the Fisher insight) = EA2 + what A owns of the payoffs of the future dividends, [theta-barAalpha times Dalpha2 plus theta-barAbeta times] Dbeta2. Since A owns half of the alpha stock, sorry, all of the alpha stock and half of the beta stock, his endowment is 1, his original thing, plus what the stock is going to produce, and after all he's the owner. So he's going to get all of 1 + a half of 2 which is = 3. I took more space than I thought. And so similarly E-hatB2 is going to be 1 + a half of 2 which = 2. So here endowments are this and also let's just write it here, E-hatA2 = 3, so this.

So Fisher said we start with a financial equilibrium, we can switch to the economic equilibrium and solve this problem, and having solved that one go back and figure out how to solve this one. And you remember what the prices were. They turned out to be q1 — I might as well write it up there what the prices we had, we solved. We said first of all Fisher has no theory for the contemporaneous prices. It's all relative prices. I'm going to write that. Relative prices, is all we can ever figure out. Someone might always come along and change dollars to cents.

Professor John Geanakoplos: All right, so we spent a long time reviewing general equilibrium and we've now switched to finance, and you're hopefully going to see that the principles of finance emerge very quickly from the principles of general equilibrium. So that although it seems it was a long interlude we've actually learned a lot about the financial economy. So I'm going to continue with the example that we started with the last time. So we have a financial economy. So in a financial economy — what is a financial economy?

On this top board the financial economy is defined by lots of people in the economy and their utilities. So here we have for simplicity two kinds of people A and B with utilities given by the log X1 + 1 half log X2 etcetera. It's also people know today what their endowments are and they have some idea of what they're going to be tomorrow. They're labor powered today and they're going to be able to work again next year. So the labor endowments are given by (1, 1) for A, and (1, 0) for B.

And then they also know that there are two stocks in the economy and they have to anticipate what the dividends are going to be. And as Fisher said, the main value of assets is that they give you something, they produce something. In this case they're going to be dividends and beta's producing dividends of 2, and alpha is producing a dividend of 1 next period, and then the ownership of shares.

So that's the beginning of the economy and we want to define from that equilibrium which involves: what are the contemporaneous prices going to be, that's Q for contemporaneous, what are the prices of the stocks going to be, and who's going to hold which portfolio of assets of stocks, and who's going to consume what. And so Fisher said that's a very complicated problem. You can simplify it by looking at a general equilibrium problem which is much shorter to describe. And so the general equilibrium economy is going to be a much simpler one.

It's going to consist of UA and UB the same as before, and E-hatA1, the endowments, E-hatA2 and (E-hatB1 E-hatB2). So we've left out half the variables up there and we define E-hatA1 = EA1 = 1 and E-hatB1 = EB1 = 1, but E-hatA2 (this is the Fisher insight) = EA2 + what A owns of the payoffs of the future dividends, [theta-barAalpha times Dalpha2 plus theta-barAbeta times] Dbeta2. Since A owns half of the alpha stock, sorry, all of the alpha stock and half of the beta stock, his endowment is 1, his original thing, plus what the stock is going to produce, and after all he's the owner. So he's going to get all of 1 + a half of 2 which is = 3. I took more space than I thought. And so similarly E-hatB2 is going to be 1 + a half of 2 which = 2. So here endowments are this and also let's just write it here, E-hatA2 = 3, so this.

So Fisher said we start with a financial equilibrium, we can switch to the economic equilibrium and solve this problem, and having solved that one go back and figure out how to solve this one. And you remember what the prices were. They turned out to be q1 — I might as well write it up there what the prices we had, we solved. We said first of all Fisher has no theory for the contemporaneous prices. It's all relative prices. I'm going to write that. Relative prices, is all we can ever figure out. Someone might always come along and change dollars to cents.

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