# Negative elasticity of substitution in a CES production function

I have empirically estimated the elasticity of substitution parameter in the following model: $$Y_t=[(A_1L_tK_{t})^{\rho} +(A_2M_{t})^{\rho}]^\frac{1}{\rho}$$

here, $$Y_t$$ is output, $$A_i$$ is a factor-augmenting technology index, $$K, L$$ and $$M$$ are factors of production and the elasticity of substitution is $$\sigma = 1/(1 - \rho)$$. I estimated $$\rho$$ combining the logorithmised form of the above function and its FOC conditions in a system of nonlinear equations.

I am getting $$\rho>1$$ which gives a negative elasticity of subsitution, $$\sigma<0$$. I am now struggling to put a plausible interpretation on this result; theoretically, the lower bound for $$\sigma$$ is 0.

I am guessing that I could still interpret the negative $$\sigma$$ as an evdience that factors are strong complements. Would this be a correct interpretation?

• In my models, I assume $0<\rho<1$ so I'm shocked :) I guess you double-checked your estimations and calculations. In this paper, link.springer.com/article/10.1007/s13209-019-00205-0 they interpret $σ<0$ as strong complementarity. – emeryville Apr 27 '20 at 17:02
• I am afraid, I cannot assume it away. I have to estimate it :(. Why do you ussume $0<\rho<1$. It can be negative and still be theoretically plausible. – london Apr 27 '20 at 18:02
• I assume the products to be somewhat sustitutes with $\sigma>1$. – emeryville Apr 27 '20 at 18:54
• Please could you define the terms of the function? – emeryville Apr 27 '20 at 19:47
• @emeryville, yes $\sigma>1$ implies gross substitutability between the factors. However, my empirical estimates give $\sigma<0$ which is theoretically implausible. – london Apr 28 '20 at 9:57