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I'm trying to calculate the result of an simple example on page 326-327, in Harrison and Kreps(1978). It's pricing a piece of asset whose dividend stream is a simple Markovian process.

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Here's my attempt to replicate investor $1$'s evaluation of this asset.

$p^1(0) = 0 + \frac{3}{4} ( \frac{1}{2} p^{1}(0) + \frac{1}{2} p^{1}(1) )$

$p^1(1) = 1+ \frac{3}{4} ( \frac{2}{3} p^{1}(0) + \frac{1}{3} p^{1}(1) )$

But when I substitute the numerical value given in the bottom of second screenshot, they don't match. Do I miss something?

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Your equations are for cum-dividend prices, i.e. the price plus dividend today. The paper refers to ex-dividend prices. The correct two equations for investor group $a=1$ are \begin{align} p^1(0) =&\ \frac{3}{4} \left(\frac{1}{2}p^1(0) + \frac{1}{2}(1+p^1(1))\right) \\ p^1(1) =&\ \frac{3}{4} \left(\frac{2}{3}p^1(0) + \frac{1}{3}(1+p^1(1))\right) \end{align} When you solve these two equations for $p^1(0)$ and $p^1(1)$ you get the correct results.

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