Altruistic Preferences: Finding the Optimal Distribution

4.4.1 of The Economy 1.0 & The Economy #2

How can Anil’s problem, when it comes to dividing his lottery winnings with Bala, be described by a constrained optimisation problem? Assume that:

x = Anil’s money

y = Bala’s money

Choose x and y to maximise U(x,y) subject to the constraint x+y=10,000

-The constraint x+y=10,000 is the feasible frontier.

How can we find the optimal allocation for this constrained optimisation problem, where the utility function is such that

U(x,y)=x^αy^β, such that α and β are positive constants and the production function is such that:

x + y = 10,000

-Using the MRT = MRS, we use the first order condition to find the optimal point.

-The MRS will be found via the absolute value of the marginal utility of x divided by the marginal utility of y. Thus, using the utility function, we find that:

MRS = αy/βx

-The MRT is simply the absolute value of the slope of the feasible frontier. Thus, using the production function, we find that:

MRT = 1

-Therefore, we see that the first-order condition is such that:

αy = βx

-Making the optimal points:

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