Question

Sophie's utility function is given as U = 2(x1)^(1/2) + 8(x2) where x1 and x2 represent the two goods Sophie consumes. Sophie's income is $3400 and the prices are given as x1 = $2 and x2 = $160

a) derive & represent in two separate diagrams the demand for x1 and x2 ( for any income and prices)

b) If Sophie's income is increased to $6600, what is the income effect on x1? Is x2 a normal good? Justify your answer

Answer 1

Equation (3) represent demand curve for x1.

And equation (5) represents demand curve for x2.

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Figures (1) and (2) below represent the demand curve for good x1 and x2 respectively.

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Answer (b)

Since the given utility function is of the form,

U (x1, x2) = v (x1) + x2

This implies that the utility function represents ‘Quasilinear Preferences’.

As a result, increasing income of the consumer would not affect the demand for good x1 at all. This is also clear from the derived demand curve of good x1 as depicted by (3). As quantity x1 is dependent on the price p1 only and thus independent of income m.

Hence, it can be concluded that when the consumer’s income increases to $6600, then the income effect on x1 would be Zero.

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A good is called a normal good when its demand increases with the increase in income.

x2 is not a normal good. The justification is given as follows-