Econ 313.1 - Wissink - Fall 2005
PS#4 – XtraQ - ANSWERS
1. DISK, Inc., x = 10K1/2L1/2, and FLOPPY, Inc., x = 10K6/10L4/10.
a. Let x1 be the output of DISK, Inc., x2 be the output of FLOPPY, Inc., and Z be equal amounts of capital
and labor for the two firms. Then, x1 = 10(Z).5(Z).5 = 10Z and x2 = 10(Z).6(Z).4 = 10Z. So both
companies generate the same amount of output in this setting since x1 = x2.
b. With K = 9, x1 = 10(9).5(L).5 = 30L.5 and x2 = 10(9).6(L).4 = 37.37 (L).4
So: mpL[making x1] = 15(L)-0.5 and mpL[making x2] = 14.95(L)-0.6 . The ranking of marginal products
will depend on the value of labor. The two marginal products will equal each other very close to L=1.
If L>1 then, the marginal product of labor is greater in DISK, Inc than in FLOPPY.
c. For DISK: x=10K.5L.5 and now you want to minimize long run costs = rK*(x) + wL*(x)
1st: Set MRTS = ERTS (putting L on the horizontal and K on the vertical)
We know that ERTS = w/r and that MRTS = mpL/mpK = K/L
So: K/L = w/r implies that K = (wL)/(r)
2nd: Now sub into the production function: x = (10)[(wL/r)].5L.5 = (10)(w).5(.r)-.5(L)
Now solve for L*(x) and get L*(x) = (1/10)(w)-.5(r).5(x) and then K* = (1/10)(w).5(r)-.5(x)
3rd: Plug the L*(x) and K*(x) functions back into the lrtc definition equation and simplify to get: lrtc(x)
= w[(1/10)(w)-.5(r).5(x)] + r[(1/10)(w).5(r)-.5(x)] = (x)[(1/5)(r).5(w).5]
d. For FLOPPY: x=10K.6L.4 and now minimize long run costs = rK*(x) + wL*(x)
1st: Set MRTS = ERTS
We know that ERTS = w/r and that MRTS = mpL/mpK = (2/3)[K/L]
So: (2/3)[K/L] = w/r implies that K = (3/2)(wL)/(r) = (3/2)(w/r)L
2nd: Sub into production function: x = (10)[(3/2)(w/r)L].6L.4 = (10)(3/2).6(w).6(.r)-.6(L)
Now solve for L*(x) and get L*(x) = (1/10)(2/3).6(w)-.6(r).6(x)
while K* = (3/20)(2/3).6(w).4(r)-.4(x)
3rd: Plug the L*(x) and K*(x) functions back into the lrtc definition equation and simplify to get lrtc(x)
= w[(1/10)(2/3).6(w)-.6(r).6(x)] + r[(3/20)(2/3).6(w).4(r)-.4(x)] = (x)[(5/20)(2/3).6(r).6(w).4]
e. If K=9 then for DISC you get x = [(10)(3)](L).5 so it's straightforward to get that L*(x) = x2/900 so then
srtc = 9r + (w/900)x2
2. For the long run cost minimization problem we know that K=L (since we need to be at the kink in the
isoquant/isocost diagram). So we get that K*(x)=x and L*(x)=x. Note that lrtc = wL*+rK* so that plugging
in you get: lrtc(x) = wx + rx = (w+r)x.
In the short run, suppose K is fixed at Ko. Then fc=rKo. You are now capacity constrained to x≤Ko. So,
L*(x) = x for x≤Ko. So you have srvc = wx for x≤Ko. And srtc = rKo + wx for x≤Ko. At x=Ko it is
typical to then draw the costs vertical to demonstrate you are at a capacity constraint and can’t go beyond,
no matter what the cost.
3. Because you are now measuring units of output in a meaningful measurable way where the number actually
means something. We can count widgets!