Question

Consider a two person firm where each individual has total time = 24 hours per day and a utility function U=5ln(x+1)+y. The team production function is such that each dollar of income generated by the firm requires two hour of effort per day as input. a) (8.5 marks) Derive the efficient level of effort for each partner, and the corresponding level of utility. b) (8.5 marks) Derive the equilibrium level of effort when each partner acts independently, taking as given the effort level of the other partner. c) Are the effort levels you derived in (b) efficient? Briefly explain why or why not.

Answer 1

There is a little ambiguity in this question i.e., what is x and what is y are not specified. so, I will solve both the cases.

CASE 1:

Income is denoted as X and leisure is denoted as Y. We already know that because they are a single firm, at the end of the day, both of them will share the income equally irrespective of their effort

(a) Here, we are working on efficiency. So, both of them must be working collectively. So, they must be dedicating one hour each to earn a dollar in which they get half dollar each.

So, price of half dollar is one hour.So, one dollar costs two hours of time.(I was talking about them individually.)

Now, U = 5ln(X+1)+Y

Equilibrium occurs when ratio of marginal utilities = ratio of prices. Now, let's derive marginal utilities.

MU_X =5/(X+1) [Differentiate 5ln(X+1)]

MU_Y = 1

Price of 1 dollar for a person is 2 hours. So, effectively P_X/P_Y = 2

Equilibrium is 5/(X+1) :1 = 2 : 1

That gives X(Income) = 1.5

So, income of each individual is 1.5. That means they worked three hours each. So, totally they worked 6 hours.

(b) In this problem, they are working independently which implies they need to share money even if others don't work. So, if a person works for two hours then firm gets 1 dollar, out of which only half dollar goes to person who worked. (Here, they lack team work. So, if we really about a particular person then he gets just half dollar for two hours of work.)

That means for 1 dollar he needs to work for 4 hours which implies P_X/P_Y = 4 for a particular person.

Now, equlibrium is 5/(X+1) = 4

which gives X = 0.25

Both of them earned 0.25 each. So, total income is 0.5 and effort is 1 hour.

(c) Effort in (b) was not efficient because they didn't work collectively which raised price of their income in terms of their effort atleast in their selfish outlook. This also arrived because income of firm was shared equally irrespective of their effort. Even, (b) part would have resulted in efficient outcome had there been a clause that states income will be shared in proportions to their efforts.

CASE 2:

Leisure is denoted as X and Income is denoted as Y.

(a) Equilibrium is 5/(X+1) = 1/2 [LOOK AT CASE 1 CAREFULLY. WE JUST REVERSED THE VARIABLES]

Which gives X = 9 that means Effort = 24 - Leisure = 24 - X = 15.

Which means effort of each individual is 15 hours.

(b) Equilibrium is 5/(X+1) = 1/4

Which gives X = 19

That means Effort of each individual is 24 - 19 = 5 Hours.

(c) Effort in (b) was inefficient [REFER TO CASE 1: PART(c) FOR COMPLETE EXPLANATION]

NOTE: Case 2 is 100% meaningful. I answered in two cases just for your convenience.