Here is a demonstration without using code. If the probability density function (PDF) of the available expenditure for a given cost-effectiveness follows a Pareto distribution (power law), it is f(x)=αxmαx−(α+1), where x is the cost-effectiveness, xm>0 is the minimum cost-effectiveness, and α is the tail index. The total expenditure required for the marginal cost-effectiveness to drop to a given value x is E(x)=∫+∞xf(t)dt=αxmα∫+∞xt−(α+1)dt=−xmα[t−α]+∞x=−xmα(0−x−α)=(xm/x)α. So the marginal cost-effectiveness is x=xm(E(x))−1/α, which is an isoelastic function.
If the total utility U(x) gained until the marginal cost-effectiveness drops to a given value x is an isoelastic function of the aforementioned total expenditure, with elasticity −η, x=U′(x)=E(x)−η. Comparing this with the last expression above for xm=1, η=1/α.
What you wrote looks clean and correct and, indeed, i used the Pareto distribution α parameter incorrectly and will change that line of the post. Thank you!
Great post!
Here is a demonstration without using code. If the probability density function (PDF) of the available expenditure for a given cost-effectiveness follows a Pareto distribution (power law), it is f(x)=αxmαx−(α+1), where x is the cost-effectiveness, xm>0 is the minimum cost-effectiveness, and α is the tail index. The total expenditure required for the marginal cost-effectiveness to drop to a given value x is E(x)=∫+∞xf(t)dt=αxmα∫+∞xt−(α+1)dt=−xmα[t−α]+∞x=−xmα(0−x−α)=(xm/x)α. So the marginal cost-effectiveness is x=xm(E(x))−1/α, which is an isoelastic function.
If the total utility U(x) gained until the marginal cost-effectiveness drops to a given value x is an isoelastic function of the aforementioned total expenditure, with elasticity −η, x=U′(x)=E(x)−η. Comparing this with the last expression above for xm=1, η=1/α.
I think you mean η=1/α.
What you wrote looks clean and correct and, indeed, i used the Pareto distribution α parameter incorrectly and will change that line of the post. Thank you!