Question

Question 2: Bobby inherited a baseball card collection so large that he decided to quit his job to sell the cards on eBay. A major part of the “production” process involves shipping the cards to collectors across the U.S. The production function corresponding to shipping q baseball cards is given by q = 1/2 U + F where U indicates the number of cards shipped by USPS and F indicates the number of cards shipped by Fedex. (USPS loses the shipment half the time whereas Fedex never does).

(a) Suppose Bobby needs to ship 10 baseball cards. With U on the horizontal axis and F on the vertical axis, draw an isoquant corresponding to all the shipping combinations Bobby could use to have 10 baseball cards delivered to customers.

(b) If pU = pF=1, what is the least cost way to ship those 10 baseball cards?

(c) On your diagram from (a), draw the isocost curve corresponding to the least cost combination of inputs you found in (b) and indicate the amount of that cost.

(d) How high would the price of Fedex shipping have to rise to for Bobby to start shipping through USPS?

Answer 1

Solution:

Production function: q = (1/2)*U + F

(a) q = 10

Notice that the production function shows that U and F are perfect substitutes. So, we will have straight lines as the isoquant curves. Further slope of isoquant curve = marginal rate of technical substitution, MRTS = $\frac{(\partial q/\partial U)}{(\partial q/\partial F)}$

$\partial q/\partial U$ = 1/2

And, $\partial q/\partial F$ = 1

So, the slope of isoquant = (1/2)/1 = 1/2

For q = 10, the isoquant carrying all combinations of U and F can be found as follows:

(1/2)*U + F = 10

U/20 + F/10 = 1

So, the horizontal intercept is 20 and vertical intercept is 10. We have the following figure:

Cards shipped 20 by Fedex, 1 18 16 14 10 6 4 2 0 2 4 6 8 10 12 14 16 18 20 Cards shipped by USPS, u

(b) With pU = pF = 1; so the price ratio = pU/pF = 1/1 = 1

The cost-minimizing way for perfect substitutes occur in following way:

If MRTSuf > price ratio, produce entirely through U (one of the extremes)

If MRTSuf = price ratio, produce on any point of the isoquant (including the extremes)

If MRTSuf < price ratio, produce entirely through F (one of the extremes)

In our example, MRTS = 0.5 (as already found in part (a)) and 0.5 < 1 = price ratio

Since MRTS < price ratio, least cost will be incurred is all baseball cards are shipped through Fedex.So, to ship 10 baseball cards, cost is minimized when U = 0 and F = 10 (one of the extremes, vertical intercept).

(c) Total cost = pU*U + pF*F

Iso-cost curve has slope of price ratio = 1. Corresponding to the least cost combination: (U*, F*) = (0, 10), isocost can be graphed as follows:

Cards shipped 20 by Fedex, 18 16 14 Optimal Point (0,10) 6 correspohding to corresponding to cpmbnation for q-10 0 2 4 6 81O 12 14 16 18 20 Cards shipped by USPS, u

Minimum cost of shipping 10 baseball cards = 1*0 + 1*10 = $10

(d) We can answer this using the explanation in part (b). For Bobby to not ship only through Fedex, that is to shift a bit (not all) of production from F to U (to be producing or shipping on the isocost line, and not the extreme). For this we know, MRTS = price ratio.

That is, 1/2 = pU/pF

Since, we are interested in seeing the change in pF only, pU = $1

So, 1/2 = 1/pF

So, pF = 2

Thus, we can say that Fedex shipping price would have to rise to $2, for Bobby to start shipping using USPS as well.