# Cost Minimization and Profit Maximization in Economics

Classified in Mathematics

Written at on English with a size of 3.49 KB.

## Total Cost = w_{1}x_{1} + w_{2}x_{2} + … + w_{n}x_{n}

Isocost function:

x_{2} = -w_{1}x_{1}/w_{2} + tc/w_{2} where the slope is: -w_{1}/w_{2} ratio of relative factor prices with negative sign

Cost minimization mate: -w_{1}/w_{2} = -MP_{1}/MP_{2} (isoquant curve)

Cost minimization implies producing a certain amount of output (y > 0) with the lowest possible cost. As with the maximization of profits, the firm is getting the most out of the resources it is using.

What is the difference between cost minimization and profit maximization? Cost minimization is a necessary condition for the maximization of profits, but not the other way around. For example, if you bring costs down to zero then output and profits will be zero too, failing to maximize profits while minimizing costs. As before, when we represent this situation graphically, we see that anything to the right or left of the equilibrium point on the isoquant curve is non-optimal, as it will not be located at the lowest iso-cost function.

-----

Once we have found the unconditional demands for X_{1}* and X_{2}* that minimize the cost, we need to find the conditional demands for inputs X_{1}* and X_{2}*. We must plug the unconditional demands into the production function: y (X_{1}, X_{2}) = X_{1}^{4/3} · X_{2}^{1/3}.

Finally, we have found the conditional demands for both inputs, or the quantities of inputs needed to minimize costs given a certain output (y*):

## Suppose the own factor price (w) goes down, how would it affect X_{1}^{*}, and X_{2}^{*}? Now suppose output increases, how would it affect X_{1}^{*}, and X_{2}^{*}? Explain.

Notice that x_{1}* and x_{2}* depend on their own factor price (w_{1} and w_{2}) as well as on output (y*). It must be remarked that the own factor prices (w) inversely affect the demands for the inputs. This is consistent with economic theory, as lower factor prices will encourage producers to buy more inputs. Besides, output affects both demands for inputs positively, as higher output levels will require more inputs. So, we can say that:

- If w
_{1}or w_{2}goes down, then x_{1}or x_{2}will go up. (negative relationship) - If y goes up, then x
_{1}and x_{2}will go up. (positive relationship)

## Graphically derive the output expansion path for different hypothetical optimal demand bundles. What does this mean? (grafico de expansion path)

As output increases due to the natural trend in economics more inputs will be needed and, as a result, costs will be higher. The **output expansion path** is the line intersecting all the tangency points between the isocost and isoquant functions in the graph above (efficient points). The red dots indicate the optimal input bundles (x_{1}*, x_{2}*) that minimize costs for the different given production levels, indicated by the different isoquant curves.

So, the output expansion path is shown in a graph with x_{1} and x_{2} in the axis. While, the conditional demand for inputs x_{1} and x_{2} are shown in a graph with x_{1} / x_{2} and output in the axis.