can some one help to specify a continuous asymmetric function of univariate with one or two (or more) parameters that can generate a figure looks like:
The detail requirements for the function y=f(x;parameter) are:
1. y is continuous in x with one or two (or more) parameter;
2. y is asymmetric in the sense y is more responsive if x is positive than if x is negative. In math language, y'>0 (for continuity, it must be y'=0 at x=0? If it is, y'=0 at x=0 is ok), y''>0 if x>0 and y''<0 if x<0, and y'(x)>y'(-x) for x>0.
My point is that I need a elasticity, say e, that define the responsiveness of y with respect to x: e=d(ln(y))/d(ln(x)). e will be a function of x and parameters e(x, parameters) that characterize the features of the asymmetric continuous function as described in 1. and 2.
Put it into context, I am trying to define a supply function. This supply y responds to price change, say dp. I need y behaves in a way that if no price change dp=0, then y=0; if dp>0, y>0; if dp<0, y<0; but for the same magnitude of price change dp, supply is more responsive to positive changes y(dp)>y(-dp), and if dp1>dp2>0, y'(dp1)>y'(dp2)>0 where y' means derivative and if dp2<dp1<0, y'(dp2)>y'(dp1)>0.
Hopefully I make my question clear. Thanks a lot.