Cost Minimization and Profit Maximization in Economics

Classified in Mathematics

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Total Cost = w1x1 + w2x2 + … + wnxn

Isocost function:

x2 = -w1x1/w2 + tc/w2 where the slope is: -w1/w2 ratio of relative factor prices with negative sign

Cost minimization mate: -w1/w2 = -MP1/MP2 (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 X1* and X2* that minimize the cost, we need to find the conditional demands for inputs X1* and X2*. We must plug the unconditional demands into the production function: y (X1, X2) = X14/3 · X21/3.

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 X1*, and X2*? Now suppose output increases, how would it affect X1*, and X2*? Explain.

Notice that x1* and x2* depend on their own factor price (w1 and w2) 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 w1 or w2 goes down, then x1 or x2 will go up. (negative relationship)
  • If y goes up, then x1 and x2 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 (x1*, x2*) 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 x1 and x2 in the axis. While, the conditional demand for inputs x1 and x2 are shown in a graph with x1 / x2 and output in the axis.

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