Answer

Solution

1. 1

   Present value of income stream:

2. 2
   $$PV_y = \sum_{t=0}^{\infty}\beta^t y_t = y_h + \beta y_l + \beta^2 y_h + \beta^3 y_l + \cdots = (y_h + \beta y_l)\sum_{k=0}^{\infty}\beta^{2k} = \frac{y_h + \beta y_l}{1 - \beta^2}.$$
3. 3

   Present value of constant stream $\bar c$:

4. 4
   $$PV_c = \bar c \sum_{t=0}^{\infty}\beta^t = \frac{\bar c}{1-\beta}.$$
5. 5

   Equating $PV_y = PV_c$:

6. 6
   $$\frac{\bar c}{1-\beta} = \frac{y_h + \beta y_l}{(1-\beta)(1+\beta)} \;\Longrightarrow\; \bar c = \frac{y_h + \beta y_l}{1+\beta}.$$

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