Does the Lambert Function Apply to Solving This Economic Model?

[Sammuueel]

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

 

[JJacquelin]

Yes, it is !

 

[pierce15]

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?

 

[Sammuueel]

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).


Samuel