Let a utility function be given by some arbitrary function that is strictly conc
ID: 2428999 • Letter: L
Question
Let a utility function be given by some arbitrary function that is strictly concave and increasing in both x1 and x2 (these conditions ensure that the consumer’s utility maximization problem has a unique solution and the re- sulting value function is differentiable).
How would homgenity / marginal utlity of income / marginal utility of the prices differ from utility given by a standard cobb douglas utility function?
More info:
Question (C) given the answer to (1)A - K below.
(https://www.chegg.com/homework-help/questions-and-answers/suppose-anna-s-utility-function-given-u-xi-t2-inxit-1-x2-answers-problem-1-change-nuary-3--q26186512)
Explanation / Answer
A strictly concave function means utility increases with both the products but with a diminishing marginal rate. Any cobb douglas utility function (standard or non-standard) is a subset of strictly concave utility functions.
Any utility function would be homogenous with degree 0 for U (m,p1,p2). The rationale behind this is utility functions are concerned with real income and relative prices of the goods. If all goods are doubled in price and also the consumer's income is doubled, it does not affect the consumer's purchasing power and hence does not affect utility. Similarly quantity of each good consumed would be same in this scenario. This part is same for every utility funstion including standard cobb douglas.
By similar argument, for any concave utility function, the marginal utility of income would be positive as it increases purchasing power and marginal utility of prices would be negative as it reduces purchasing power. But in case of a standard cobb douglas function with degree 1, marginal utility of income would be a constant and that would not necessarily be true for any other concave utility function.
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