<p>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. </p>

<p>Hence, from the perspective of the current period, your utility from accepting that offer is</p>

<p>$$ w + \beta w + \beta^2 w + \cdots =\frac{w}{1-\beta} $$</p>

<p>In case this equality isn't familiar, recall that the LHS is simply a <a href="https://en.wikipedia.org/wiki/Geometric_series" rel="nofollow noreferrer">geometric series</a>.</p>

CC BY-SA 3.0
2016-12-21T23:13:55.207Z
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
CC BY-SA 3.0
2016-12-21T08:53:51.113Z
2016-12-21T08:53:51.113Z
Theoretical Economist
<p>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.</p>

CC BY-SA 3.0
2017-01-21T19:42:35.517Z
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
<macroeconomics><search-and-matching><discount-rate>
McCall Search Model: Present value of lifetime income
user400860
# McCall Search Model: Present value of lifetime income

(Asked by: username_0 on 2016-12-21)

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.

**Tags:** macroeconomics, search-and-matching, discount-rate
> **Comments:**
> **username_1**: 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...  

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## 1 answers

### Answer by username_2 on 2016-12-21. Score: 2

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](https://en.wikipedia.org/wiki/Geometric_series).

> **Comments:**
> **username_0**: 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 ?  

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