Minimizing a function with a minimum constraint

The Lagrange multiplier method involves setting the derivatives equal to 0 and solving for the variables, but in this case, we have a minimum function with a constraint, so I'm not sure how to apply it. In summary, the goal is to minimize the cost function rk + wL with respect to the constraint of producing q units of output. The Lagrange multiplier method can be used, but it is important to note that taking the derivative of a minimum function can be tricky. The optimal values for K* and L* will depend on whether rk is greater or less than wl.
  • #1
DannyCPA
7
0
Minimizing a function with a minimum constraint...

Homework Statement


A firm would like to produce q units of output at the lowest cost. It's cost structure is rk + wL. Minimize this function with respect to the constraint: min {sk, L/S} = q


K = represents capital
l = represents labor
q= represents output
Capital costs R per unit, Labor costs W per unit, s is just a constant

Minimize cost:

min rK + wL
(k,l)

subject to: min{sK, L/s} = q


Homework Equations



General minimizing with constraint problem.


The Attempt at a Solution



My biggest problem is taking the derivative of a minimum function. That is, dq/dk = S? dq/dL = 1/s?

Is that correct? If so, I got a conclusion that if rk > wl, the optimal value for L* = q/w. If rk < wl, the optimal value for K* = q/r.
 
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  • #2
The Lagrange equation would be: L(k,l) = rk + wL - λ(min{sK, L/s} - q)Differentiating with respect to k and l, we get: dL/dk = r - λs dL/dl = w - λ/s From here, I'm not sure how to proceed.
 

What is the concept of minimizing a function with a minimum constraint?

Minimizing a function with a minimum constraint involves finding the smallest possible value of a function while also satisfying a given minimum constraint. This means that the solution to the problem must be the smallest possible value that meets the minimum constraint.

What is the difference between minimizing a function and minimizing a function with a minimum constraint?

Minimizing a function without a minimum constraint involves finding the absolute smallest value of a function, while minimizing a function with a minimum constraint involves finding the smallest value that still satisfies a minimum constraint.

How is a function with a minimum constraint typically represented mathematically?

A function with a minimum constraint is typically represented as an optimization problem, with the function being the objective to be minimized and the minimum constraint being a constraint that must be satisfied by the solution.

What are some common techniques for minimizing a function with a minimum constraint?

Some common techniques for minimizing a function with a minimum constraint include using Lagrange multipliers, the method of Lagrange multipliers, and the Kuhn-Tucker conditions.

What are some real-world applications of minimizing a function with a minimum constraint?

This concept is commonly used in economics and finance, where there may be a minimum level of production or investment that must be met in order to be profitable. It is also used in engineering and optimization problems, where there may be physical or practical constraints that must be satisfied in order to find the most efficient solution.

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