The problem is as follows:

A firm uses $k$ units of capital and $l$ units of labor to produce $(k^{\alpha}l^{1-\alpha})^{1/\beta}$ units of output, where $\alpha$ and $\beta$ satisfying $0 < \alpha < 1$ and $\beta > 1$, are parameters.

If the firm obtains these inputs at the competitive rental rate $r$ for capital and the competitive wage rate $w$ for labor, and seeks to minimize the cost of producing at least $y$ units of output, it solves the problem

$$\min rk + wl \text{ subject to } (k^{\alpha}l^{1-\alpha})^{1/\beta}\geq y$$

where, in this cost-minimization problem, the output requirement y is another parameter.

(1) Using Kuhn-Tucker theorem, show that the minimum cost function $c(r,w,y)$ is

$$c(r,w,y) = \frac{1}{\alpha^{\alpha}\times(1-\alpha)^{(1-\alpha)}} \times r^{\alpha}w^{(1-\alpha)}y^\beta $$

(2) Using the Envelop theorem, and the equation (1) derive the optimal solutions $k*(r,w,y)$ and $l*(r,w,y)$.

This is my homework problem. Humble request to anyone for the solution. Thanks.

  • $\begingroup$ so is your problem with applying K-T, or with using the K-T conditions to get the cost function? $\endgroup$
    – Trurl
    May 12, 2013 at 17:24
  • $\begingroup$ Also, is there a typo? for example, for non-zero values of $\alpha$, either capital or labor decreases output. the usual convention would be $\alpha$ and $1-\alpha$. $\endgroup$
    – Trurl
    May 12, 2013 at 18:01
  • $\begingroup$ Thanks Trurl. I have corrected the typo. $\endgroup$
    – Sohel
    May 13, 2013 at 1:21

1 Answer 1


Set up the Lagrangian function:

$$\min_{k,l} rk+wl-\lambda\left((k^{\alpha}l^{1-\alpha})^{1/\beta}-y\right)$$

Note that the function $(k^{\alpha}l^{1-\alpha})^{1/\beta}$ is quasi-concave and homogeneous and of degree less than 1 (since $\beta>1$). A quasi-concave function of degree between 0 and 1 is concave. The negative of a concave function is convex ($\lambda$ is positive). Further, the two linear functions $rk$ and $wl$ are convex. The sum of convex functions is convex. Thus, the entire Lagrangian function is convex. The first order conditions will be sufficient for a minimum. It is also noted that the production function has infinite marginal product for either capital or labor when one is set to $0$. Thus, we will not have to worry about corner solutions where only one of the two inputs is used. Further, prices are strictly positive and so no more than y will be produced. The first order conditions are:




Solving these gives:



Gives one key relationship:


Now, we use this along with the production constraint to solve for the inputs:





Plugging these back into the cost function gives the desired result:





By the envelope theorem, the derivative of an optimized function with respect to one of its parameters can be taken treating the choice variables as fixed. At the optimum, a very small change in these choice variables has no effect on the optimized value. This means, when we take the derivative of the optimized cost function with respect to $r$ and $w$ we will get back $k^*$ and $l^*$ respectively. I will leave this to you to check.


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