# McCall Search Model: Present value of lifetime income

In the Bellman for the search model, how is the present discounted value of future income equal to $\frac{w}{1-\beta}$? Sounds like a basic question but I just can't comprehend the math in my head, because for present value I tend to compare it with NPV and think $1-\beta$ should be raised to the power of something. If someone could show the math in the McCall model's bellman.

• Please make the question self-contained (without assuming complete knowledge of the model in question). Also, you could work on phrasing your question in general... – FooBar Jan 21 '17 at 19:42

## 1 Answer

In the version of the McCall search model I'm familiar with, once you accept an offer of wage $w$, you receive that wage $w$ in every period starting from the period of acceptance.

Hence, from the perspective of the current period, your utility from accepting that offer is

$$w + \beta w + \beta^2 w + \cdots =\frac{w}{1-\beta}$$

In case this equality isn't familiar, recall that the LHS is simply a geometric series.

• thanks! Its so simple.... I don't what I was thinking. Anyway what would be the present value of lifetime income when on accepting the offer you will receive w first period followed by w(t)=(phi)^t*w every consecutive period? will it also be a simple gp like the above ? – user400860 Dec 21 '16 at 23:13