# Deriving factor allocation of production function

I am trying to solve an allocation problem for a nested CES production function with three factors.

The production function we posit is:

$$F(K, \mathbf L, \mathbf C) = [\alpha K^\rho + \sum_{i\not\in \text{AT}} \beta_i L_i ^\rho + \sum_{i\in \text{AT}} \beta_i (L_i + \eta_i C_i)^\rho]^{\frac{1}{\rho}}$$

where $$K$$ is capital, $$\mathbf L$$ is labour, $$\mathbf C$$ is compute, $$\rho$$ is the labour-capital substitution coefficient, $$\alpha + \sum_{0 \le i \le n} \beta_i = 1$$ are the task-specific share parameters, $$\text{AT}$$ is the set of automatable tasks and $$\eta_i < 1$$ are the task-specific substitution factors between labour and compute.

Assuming capital and labour are gross complements (ie $$\rho < 0$$) then the problem is equivalent to minimizing $$F'(\mathbf L, \mathbf C) = \sum_{i} \beta_i (L_i + \eta_i C_i )^\rho$$ subject to the budget constraints

$$\sum_i L_i = L$$ $$\sum_{i} C_i = C$$ $$C_i = 0, \text{ if } i\not\in \text{AT}$$

The point of minimization happens when the marginal returns to labour and compute on each task are equal.

This gives rise to the optimality conditions $$\frac{\partial F'}{\partial L_i} = \frac{\partial F'}{\partial L_j}$$ for all $$i,j$$; and $$\frac{\partial F'}{\partial C_i} = \frac{\partial F'}{\partial C_j}$$ for $$i,j \in \text{AT}$$.

$$\frac{\partial F'}{\partial L_i} = \beta_i \rho (L_i + \eta_i C_i)^{\rho - 1} = \beta_0 \rho (L_0 + \eta_0 C_0)^{\rho - 1} = \frac{\partial F'}{\partial L_0}$$

$$\frac{\partial F'}{\partial C_i} = \beta_i \rho (L_i + \eta_i C_i)^{\rho -1} \eta_i = \beta_0 \rho (L_0 + \eta_0 C_0)^{\rho -1} \eta_0 = \frac{\partial F'}{\partial C_0}$$

where we assumed that task 0 is automatable ie $$0\in \text{AT}$$.

After eliminating $$\rho$$ and resolving the exponent we have that:

$$\frac{1}{\beta_0^{\sigma}}L_0 - \frac{1}{\beta_i^{\sigma}}L_i + \frac{\eta_0}{\beta_0^{\sigma}}C_0 - \frac{\eta_i}{\beta_i^{\sigma}}C_i = 0$$

$$\frac{1}{\beta_0^{\sigma}\eta_0^{\sigma}}L_0 - \frac{1}{\beta_i^{\sigma}\eta_i^{\sigma}}L_i + \frac{\eta_0^{1 - \sigma}}{\beta_0^{\sigma}}C_0 - \frac{\eta_i^{1 - \sigma}}{\beta_i^{\sigma}}C_i = 0$$ where $$\sigma = \frac{1}{1 - \rho}$$ is the coefficient of elasticity of the CES production function.

Together with the budget conditions, these equations form a uniquely determined system of linear equations. Since $$F'$$ is convex, the solution is neccessarily the minimum.

So far so good, but I wanted to check that the solution works in an example.

I chose $$C = 1, L=1, \beta_1 = 1/3, \beta_2=2/3, \rho = -0.5, \eta_0 = 1, \text{AT} = \{0\}$$.

The solution that I get using a lineal algebra solver is $$L_0 = -0.22702358, L_1 = 1.22702358, C_0 = 1, C_1 = 0$$.

So clearly I have forgotten to account for the restriction that each individual factor must be non negative. The question is, how do I incorporate that in the solution?

You can use Karush-Kuhn-Tucker conditions. As you are considering a minmisation problem, this gives: $$\frac{\partial F'}{\partial L_i} \ge \lambda \text{ with equality if L_i > 0}\\ \frac{\partial F'}{\partial C_i} \ge \mu \text{ with equality if C_i > 0}$$ where $$\lambda$$ and $$\mu$$ are the Lagrange multipliers for the adding up constraintes. Wikipedia entry for KKT.
• I don't understand. How do I find $\lambda$ and $\mu$? Is this just saying that the derivatives need to be positive? How do I apply this in practice, eg in the example I gave? Jan 18 at 8:32
• @Jsevillamol Do you know Lagrangians and optimisation? In general there's no easy way to solve this except to consider all possible cases where $L_i > 0$ or $L_i = 0$ and see if you get a solution (or a contradiction). There are many resources online (e.g. Youtube) that show you how to use Kuhn-Tucker conditions.