Question

the text discussed the explanation path as a graph that shows the cost-minimizing input quantities as output changes, holding fixed the prices of inputs.what the text didn't say is that there is a different expansion path for each pair of input prices the firm might face. in other words, how the inputs vary with output depends, in part, on the input prices. Consider now, the expansion paths associated with two distinct pairs of input prices, (w1,r1) and (w2,r2). assume that at both pairs of input prices, we have an interior solution to the cost-minimization problem for any positive level of output. also assume that the firms isoquants have no kinks in them and that they exhibit diminishing marginal rate of technical substitution. could these expansion paths ever cross each other at a point other than the origin (L=0, K=0)?

Answer 1

Consider the given problem here the “marginal rate of technical substitution (MRTS)” measure the rate at which inputs itself are substituted to each other to produce the same level of output or we can say it is the slope of isoquant. Now, the expansion path shows all the efficient inputs as output changes given the input price ratio. So, here the mathematical equation of expansion path is given by.

=> MRTS = w1/w2, where “w1=price of good1” and “w2=price of good2”.

Now, as “input price ratio” changes the expansion path will change, => we have different expansion path for different input price ratio. Now, let’s we have two different input price ratio “p1” and “p2” and “p1 < p2”. So, the equations of two different expansion paths are given by.

=> “MRTS=p1” and “MRTS = p2”, where “MRTS” measure the slope of the isoquants.

Now, if expansion path cut to each other, => at that particular point the MRTS” is equal to “p1” and “p2”, => MRTS = p1 = p2. But we know that “p1 < p2”, => that condition can’t hold, => expansion path can’t intersect to each other.

So, two expansion path cross only at “L=K=0” and will never cross at any point.