# Minimizing the cost of production by choosing between two options with different resource demands

Suppose that the ﬁrm has two possible activities to produce output. Activity $A$ uses $a_1$ units of good $1$ and $a_2$ units of good $2$ to produce $1$ unit of output. Activity $B$ uses $b_1$ units of good $1$ and $b_2$ units of good $2$ to produce $1$ unit of output. Factors can only be used in these ﬁxed proportions. If the factor prices are $(w_1,w_2)$, what are the demands for the two factors?

Please help me with this question as I don't know what to do when there are two simultaneous activities involved.thank you

The cost function for one unit of output is $c(a,b)=w_1\cdot x_1+w_2\cdot x_2$

You can sketch the points $A(a_1/a_2)$,$B(b_1/b_2)$ in a diagram. The line, which goes through the points A and B have to have a negative slope. Otherwise activity A or B is not efficient. Calcutlate the slope.

Then set $c(a,b)$ equal to 0 and solve $c(a,b)$ for $x_2$. Draw the line through the point $C(0/0)$ with the slope $-\frac{w_1}{w_2}$ Then push the line parallel upward till the line touches first point A or point B. The Point, which is touched first by the line, is the activity, which you have to apply, if you minimize the costs.

You can insert some values for $w_1,w_2,a_1,a_2,b_1$and $b_2$. You will recognize the following rules for minimizing the costs:

1. Use activity A, if $\left|\frac{a_2-b_2}{a_1-b_1} \right| < \left| \frac{w_1}{w_2}\right|$

2. Use activity A or B, if $\left|\frac{a_2-b_2}{a_1-b_1} \right| = \left| \frac{w_1}{w_2}\right|$

3. Use activity B, if $\left|\frac{a_2-b_2}{a_1-b_1} \right| > \left| \frac{w_1}{w_2}\right|$

• Tip: @Noel.campbell04091992 if you think this answer is correct, please accept it as such by clicking the checkmark just below the up and downarrows for voting for the answer quality. – PortMeadow Aug 6 '14 at 15:02