If a consumer has a utility function u(x1, x2)=x1 x2^4, what fraction of her income will she spe

step 1 In this question, we have been asked about the fraction of the income that will be spent on good

If a consumer has a utility function u(x1, x2)=x1 x2^4, what...

Key Concepts

Cobb-Douglas Utility Function

A Cobb-Douglas utility function has the form u(x?, x?) = x?? x??, where the exponents a and b are positive constants. One of its key properties is that the consumer spends a fixed fraction of their income on each good, regardless of income level or changes in market prices. This characteristic simplifies the determination of optimal consumption bundles since the expenditure shares are constant and directly related to the exponents in the utility function.

Expenditure Shares

In the context of consumer choice, expenditure shares refer to the proportion of the consumer's total income allocated to purchasing each good. For a Cobb-Douglas utility function, the fraction of income spent on a particular good is the exponent on that good divided by the sum of the exponents of all goods. This makes it straightforward to calculate how income is allocated without needing to solve a system of equations from the budget constraint and the utility maximization problem.

Budget Constraint

The budget constraint represents all combinations of goods that a consumer can afford given their income and the prices of the goods. It is fundamental in consumer choice theory, as the optimal choice is determined where the highest attainable indifference curve is tangent to the budget line. In the case of Cobb-Douglas preferences, the budget constraint reinforces the result that expenditure shares remain constant, as the tangency condition leads directly to the ratio of spending being equal to the ratio of the exponents.

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