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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?

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  • $\begingroup$ 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. $\endgroup$ – emeryville Apr 27 '20 at 17:02
  • $\begingroup$ 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. $\endgroup$ – london Apr 27 '20 at 18:02
  • $\begingroup$ I assume the products to be somewhat sustitutes with $\sigma>1$. $\endgroup$ – emeryville Apr 27 '20 at 18:54
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    $\begingroup$ Please could you define the terms of the function? $\endgroup$ – emeryville Apr 27 '20 at 19:47
  • $\begingroup$ @emeryville, yes $\sigma>1$ implies gross substitutability between the factors. However, my empirical estimates give $\sigma<0$ which is theoretically implausible. $\endgroup$ – london Apr 28 '20 at 9:57

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