I am working on a model of optimal payout percentages in the gambling industry.
Because the nominal price of a \$1 ticket is always \$1, we use an effective price strategy where Q = \$1 in won prizes. If a game pays out 50%, the effective price is \$2, since that is what would need to be spent to win an expected \$1 in prizes. Pretty simple, right?
Well, I ran into this footnote in some research, and can't figure out how they got to the First Order Condition for Profit Maximization from the first equation:
"Let $C(Q)$ represent operating costs as a function of quantity units, where one quantity unit is defined as one dollar in expected value of prizes.
The lottery agency's net profits are given by
$$N = PQ - Q - C(Q)$$
where $P$ is the price charged for a quantity unit.
The first-order condition for profit maximization can be written
$$-E_{PQ} = P(1 - C')/[P(1 -C')- 1] $$
If marginal operating costs are $6$ percent of sales and the payout rate is $50$ percent, we have $P = 2$ and $C' = .12$, implying that the price elasticity of demand at maximum profit is $-2.3$.
For an increase in the payout rate to increase profits, $E_{PQ}$ must exceed $2.3$ in absolute value."
-[Citation] Clotfelter, Charles T, and Philip J Cook. "On the Economics of State Lotteries." Journal of Economic Perspectives: 105-19.
In the FOC equation, $-E_{PQ}$ is the effective price elasticity of demand. That normally would be found by taking the derivative of $P$ with respect to $Q$ in the first equation.
How did they end up where they did? There has to be something I'm missing.
I am having trouble understanding how that particular First Order Condition was reached- whether it was a result of some some derivative process on the Net Revenue equation, or if it is simply an external condition being applied.
Thanks!