Answer to Question #108423 in Macroeconomics for David Allison

Question

Assuming a consumer consumes commodity X1 to the power of one and half quantities and 1 quantity commodity X2 after spending $3 on X1 and $4 X2.
i.Compute the optimum quantities of X1 and X2 that gives the consumer highest level of satisfaction.

ii.Algebraicaly verify if the suficient condition is met using relevant Hessian Bordered Matrix.

Answer 1

X1 * 1,5 = 3$

X2 * 1 = 4$

3+4=7$

Recall that the consumer's objective is assumed to be to maximize his or her total satisfaction. Given the amount of available income and many possible goods, the problem is to distribute the spending in such a way as to achieve this objective. The key to the problem is provided by the principle of diminishing marginal utility and the consumer's ability to substitute one good for another. As more of a good is consumed each additional unit of the good adds less to total utility and consequently its marginal utility falls. As less of a good is consumed, the marginal utility of the good increases. The consumer can control the marginal utility of goods by shifting spending from one good to another. This should be done until the marginal utility of another dollar spent on one good is exactly the same as the marginal utility from another dollar spent on any other good. When this point is reached and the consumer has exhausted the available income, the total utility from the combination of goods chosen is a maximum. No further shifting of expenditure among or between the goods will increase the total -- in fact it will cause it to be less.

Bordered Hessian is a matrix method to optimize an objective function f(x,y) where there are two factors ( x1 and x2 ), the word optimization is used here because in real life there are always limitations ( constraints ) which we have to consider and we have to maximize.

for calculations it is necessary to know the total and marginal utility of the goods x1 and x2.

https://homes.chass.utoronto.ca/~reak/eco100/maxingq.htm