# Is this a case for Lambert?

## Main Question or Discussion Point

Dear Forum,

I am a researcher in the field of microeconomics and I came across this equation which I would like to solve for $k$. It looks a little bit like the Lambert function. But I am stuck here.
$\Omega = \rho^k (1-k\cdot \ln \rho)$

Do you have an idea how I could proceed?

Kind regards,
Samuel

Yes, it is !

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Dear Forum,

I am a researcher in the field of microeconomics and I came across this equation which I would like to solve for $k$. It looks a little bit like the Lambert function. But I am stuck here.
$\Omega = \rho^k (1-k\cdot \ln \rho)$

Do you have an idea how I could proceed?

Kind regards,
Samuel
Jacquelin gave the answer... but I'm interested, how did this come up?

Thank you Jacqueline!!

This is from a model where the demand of a consumer accumulates if he does not make a purchase in one period. This accumulated deteriorates with a factor ρ (e.g.0.9). After k periods without purchase, the demand is $\rho + \rho^2 + ... + \rho^k = \frac{1-\rho^k}{1-\rho}$.

The term shown in my problem is from a firm's FOC who chooses a set of prices for high-valuation consumers (who purchase in each period) and low-valuation purchases (whose demand accumulates).

Best,
Samuel