Consider the problem of a social planner who wants to maximize utilitarian welfare in an exchange economy with two people. His objective is thus
maxu1 + u2 ,where u1 and u2 are the utilities of the two individuals. Both have the same utility overconsumption c, so ui = u(ci ) = ln ci , i=1,2 (ln stands for the natural logarithm). The total available quantity of the consumption good is X, so that c1 + c2 = X .
1. Show (preferably analytically) that the planner will allocate half of the available
consumption good to each individual.
2. Briefly explain why you generally would not expect this result if the utility functions of the two individuals were different.
What are the steps to solve this analytically?© BrainMass Inc. brainmass.com October 17, 2018, 3:34 am ad1c9bdddf
1. Show (preferably analytically) that the planner will allocate half of the available consumption good to each individual.
Let U = social utility, then the objective function becomes
max U = U1 + U2
U = ln(C1) + ln(C2)
Then the marginal utility of C1 and C2 would be:
MU1 = dU/dC1 = 1/C1
MU2 = dU/dC2 = 1/C2
According to the first order condition, U is maximized when ...
First order condition is clearly expressed in this solution. Natural logarithms are given.
System of non-homogenous first order differential equations
The volume of two tanks are V1 =100 gallons and V2 = 200 gallons .The inflow and outflow rates of the system are r = 10 gallons per minute. Suppose that the two tanks both contain fresh water initially, but the inflow to tank 1 is brine at 2 pounds per minute, so that 2 pounds of salt flow into tank 1 each minute. Write a matrix system of equations for the amounts and of salt in the two tanks after t minutes. Solve the system using matrix methods. Calculate how long it takes for tank 2 to have a salt concentration of 1 pound per gallon.
Please remember to calculate the eigenvalue using det |A-lambda*I|