Question

Hal is stranded on an island. He only has 24 labor hours in a day, but must sleep for 8 of these hours. The remaining time can be divided between two tasks: fishing (F) and harvesting coconuts (C) . Let Lf and Lc represent time spent allocated to fishing and coconuts respectively. The production of fish and coconuts correspond to the functions C = sqrt(Lc) and F = sqrt(Lf)

What is the marginal rate of (commodity) transformation at the allocation (C, F) = (0, 4)?

(treat fish like the y-axis commodity)

Answer 1

Total time = 24 hours

sleeping time = 8 hours

working time = 24 - 8 = 16 hours

Therefore LC  + LF = 16 (i)  

C = LC1/2

C2 = LC  

F = LF1/2

F2 = LF  

Substitute LC = C2 and LF = F2 in equation (i)  

LC  + LF = 16

C2 +   F2 = 16

Thus equation of PPC or PPF curve is

C2 +   F2 = 16

MRT = dF/dC = - fc/ff  

fc = 2Cat

ff = 2F  

MRT = - 2C/2F

= - C/F

we can also find MRT by differentiating equation of PPF curve with respect to C which is given below

C2 +   F2 = 16

2C + 2FdF/dC = 0  

2FdF/dC = - 2c

dF/dC = - 2C/2F

= - C/F

Hence MRT = - dF/dC = - fc/ff    = - C/F

at (C,F) = (0,4)  

MRT = - 0/4  

= 0