Question

In: Economics

Suppose a consumer seeks to maximize the utility function U (x1; x2) = (-1/x1)-(1/x2) ; subject...

Suppose a consumer seeks to maximize the utility function U (x1; x2) = (-1/x1)-(1/x2) ; subject to the budget constraint p1x1 + p2x2 = Y; where x1 and x2 represent the quantities of goods consumed, p1 and p2 are the prices of the two goods and Y represents the consumer's income. (a)What is the Lagrangian function for this problem? Find the consumer's demand functions, x1 and x2 . (b) Show the bordered Hessian matrix, H for this problem. What does the second order condition require for this problem? Show if it is satisfied. (c) Find the consumer's indirect utility function. (d) How does the consumer's optimal utility change when there is a small change in (i) p1 and in (ii) p2? (e)What is the interpretation of this value of the Lagrangian multiplier in this problem?

Expert Solution

(a) The utility function is:

subject to the budget constraint:

where x₁ and x₂ are the quantity of goods consumed and p₁ and p₂ are the prices. Y is the income of the consumer.

The Lagrangian function is:

The First-Order conditions are:

Hence, these are the amount of x₁ and x₂ consumed. The consumer's demand function is:

(b) The SOC boardered Hessian matrix is that it has to be greater than 0.

(c) Deriving the indirect utility function

Substituting the values of Marginal Utility of x₁ and x₂ we get,

putting the value of p₁ in the budget constarint we the indirect utility function.

(d) If there is a small change in the prices then the consumer's budget constraint will be significanlty changing. If the prices of both the goods that he consumes increases, then his demand for those goods will have to decrease to satisfy his budget constraint. If there is an increase or a decrease in the p₁ and p₂ respectively, then the consumer will spend less on the good x₁ and more on the good x₂. The consumer will buy two goods at that level so that his budget constraint is satisfied.

(e) The value of this Lagrangian multiplier is that it helps in finding the local maxima and minima of the given function. In utility maximisation problem, the lagrangemultiplier measures the marginal utility of income of the consumer which means how the utility is maximised as their income (Y) increases. gives the value at that point where the budget constraint is satisfied. It also represents the rate of change of the function as the inputs are increased.

Rahul Sunny answered 2 years ago

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