How to find the 'optimal' value (or threshold) of an independent variable

[Bill Austin]

Dear Stata friends,

I have data from a labour market experiment where employers could use money to make a wage offer to a worker, who in return could provide effort which generated earnings for the employer (it's a fairly common lab experiment).

In my dataset I now have the following variables: Employer's Earnings, Wage Offer, and a series of control variables. As I am comparing a control and treatment condition from the experiment I would like to answer the following question: "what is the optimal wage the employer can offer to maximise her earnings?".

I am thinking of a model like this: Employer's Earnings = Wage Offer + squared(Wage Offer) + Control

Estimating this with OLS, the sign of the squared term will tell me if the wage offer is increasing or if there's a threshold (i.e. negative sign). However, I still don't know what should be the STATA command or the econometric approach to answer my question and see what's the optimal wage offer. Is it going to be the coefficient of the Wage Offer variable?

Thanks for your help and time,

Bill

[Clyde Schechter]

So, assuming the quadratic model you propose, finding the value of wage offer that maximizes (or, if the quadratic term coefficient is positive, minimizes) the employer's earnings is standard algebra of parabolas. If the function is y = ax² + bx +c, the vertex is located at x = -b/2a. So, in Stata:

regress employer_earnings c.wage_offer##c.wage_offer // AND OTHER ADJUSTMENT VARIABLES

nlcom -_b[wage_offer]/(2*_b[c.wage_offer#c.wage_offer])

[Bill Austin]

Thanks! Very helpful. One more suggestion, if possible: how would you calculate the marginal increase in earnings from any additional unit of wage offer (e.g. I would expect that for an increase from 10 to 20 of wage offer the responses of worker's effort, and related employers' earnings, is higher than an increase 90 to 100)? Would you simply run an OLS and then estimate margins, or is there a slightly more sophisticated way to do this? Thanks again, Bill

[Clyde Schechter]

I would use -margins, dydx()- for this. May I ask what you find unsophisticated about -margins-?

[Bill Austin]

Nothing in particular, I guess another way to disentangle the relationship more in detail though would be using quantile or others (by sophisticated I guess I meant the type of 'story' one is able to tell from the results of the regression...)

[Clyde Schechter]

Got it! Thanks.

[Md Bayzidur Rahman]

Hi,
I have a similar situation but a bit more complicated. We are following up a group of patients after diagnosing a condition. During the follow-up they are given a certain medication daily. The dose may vary across days for some of the patients depending on their prognosis. The outcome is resolution of symptom. How can I determine the optimum dose for quickest resolution of the symptom?
Regards
Bayzid