Question

Tom Hanks was stuck in an island alone. Until somebody comes to rescue him, he has to live alone and fight for survival in the island. Tom considers two things essential to his survival, rest (R) and food. The only food Tom can eat in the island is Fish (denoted by F).

Tom’s utility function is: U(R,F)=R.F

In order to get fish, Tom has to spend time fishing. The following production function gives the relationship between time spent by Tom in fishing and the number of fishes Tom can collect. Production function: F(T)=T ^(1/2)

Tom has a total of 24 hours in a day, so R+T=24.

How much time will Tom spend on fishing and how much fish will Tom consume?

Answer 1

Consider the given problem here the utility function of Tom is, “U = R*F”, where “R=time devoted to rest” and “F=the amount of fish consumed”.   

The collection of fish can be represented by the production function, => F = T^0.5. If we put the value into the utility function then then the new utility function is given by.

=> U = R*T^0.5, the time constraint is given by, “R+T=24”. Given the utility function the marginal utility function are given by.

=> MUR = T^0.5 and MUT = 0.5*T^(-0.5)*R, => MRS between R and T are.

=> MRS = MUR/MUT = [T^0.5] / [0.5*T^(-0.5)*R] = [T]/[0.5*R] = 2T/R. the absolute slope of the constraint is “1”. So, at the equilibrium MRS must be equal to “1”.

=> MRS = 1, => 2T/R = 1, => R = 2T. Substituting the condition into the budget constraint we have.

=> R+T=24, => 2T+T=24, => T=24/3 = 8 hours, => R=24-T=16hours.

So, Tom will devote 8 hours for fishing and will consume “F=T^0.5 = 8^0.5 = 2.83 units” of fish.