Example of a (not quasi-linear) production function whose inputs are not perfect substitutes but are not asymptotic at the axes

I'm looking for an example of a family of production functions indexed by, say, rho, where the inputs become closer and closer to perfect substitutes as rho approaches 1, and yet, the marginal product of, say, L, approaches a finite limit as L approaches 0. Of course one can construct quasi-linear production functions that exhibit this property for one input, but their behavior is really awful. One would think that CES production functions might exhibit this property, since their limit is an affine function, but they don't. Indeed, clearly, any production function in which L is raised to a power < 1 is going to give me the same problem. Any suggestion would be very much appreciated.

You could, for example take the function $$f: \mathbb{R}^2_+ \times[0,1] \to \mathbb{R}$$. $$f(L,K, \rho) = L + K + (1-\rho) L K.$$ For $$\rho = 1$$, we have $$f(L, K, 0) = L + K$$ which is a production function with perfect substitutes.
The marginal product (of say $$L$$) is given by: $$\frac{\partial f(L, K, \rho)}{\partial x} = 1 + (1-\rho)K,$$ which equals $$1$$ if $$\rho = 1$$.
• Thanks very much, @tdm, I ended up working with something that's effectively the same. $f(L,K) = L + K + \alpha ( log(L) + log(K) )$. May 13 '21 at 17:23