U(x, y) = 2xy.(a) Obtain the marginal rate of substitution. Obtain the equation of the indifference curves forU = 16 and U = 36 (solving for y as a function of x). Draw these indifference curves. Dothese indifference curves intersect the y-axis and/or the x-axis? Explain in your own wordswhy this is the case. (4 marks)(b) Are Julian’s preferences homothetic? Justify mathematically your answer. (2 marks)(c) At the current prices (which Julian knows, but you don’t), initially, Julian spends 30 dollarsin x and 20 dollars in y. Suppose that Julian’s income increases by 5 dollars (and pricesremain the same). Compute Julian’s expenditure in x and Julian’s expenditure in y afterthis increase in income. (2 marks)(d) There is a decrease in the price of x. Is the (total) price effect on Julian’s demand for xlarger, the same, or smaller than the substitution effect? Explain. (1 mark)
Question
U(x, y) = 2xy.(a) Obtain the marginal rate of substitution. Obtain the equation of the indifference curves forU = 16 and U = 36 (solving for y as a function of x). Draw these indifference curves. Dothese indifference curves intersect the y-axis and/or the x-axis? Explain in your own wordswhy this is the case. (4 marks)(b) Are Julian’s preferences homothetic? Justify mathematically your answer. (2 marks)(c) At the current prices (which Julian knows, but you don’t), initially, Julian spends 30 dollarsin x and 20 dollars in y. Suppose that Julian’s income increases by 5 dollars (and pricesremain the same). Compute Julian’s expenditure in x and Julian’s expenditure in y afterthis increase in income. (2 marks)(d) There is a decrease in the price of x. Is the (total) price effect on Julian’s demand for xlarger, the same, or smaller than the substitution effect? Explain. (1 mark)
Solution
(a) The marginal rate of substitution (MRS) is the rate at which a consumer is willing to substitute one good for another while maintaining the same level of utility. It is obtained by taking the negative ratio of the marginal utility of x to the marginal utility of y. For the utility function U(x, y) = 2xy, the marginal utility of x (MUx) is 2y and the marginal utility of y (MUy) is 2x. Therefore, MRS = -MUx/MUy = -2y/2x = -y/x.
The indifference curves are obtained by setting U(x, y) equal to a constant and solving for y. For U = 16, we have 16 = 2xy, or y = 8/x. For U = 36, we have 36 = 2xy, or y = 18/x. These curves do not intersect the x-axis because y cannot be zero for any positive x (as this would imply zero utility, which contradicts the given utility levels). They do intersect the y-axis at the points where x = 0, but these points are not part of the indifference curves because x cannot be zero (for the same reason).
(b) Preferences are homothetic if the marginal rate of substitution is the same at all points along an indifference curve. In this case, the MRS is not constant but depends on the ratio of y to x. Therefore, Julian's preferences are not homothetic.
(c) If Julian initially spends 30 dollars on x and 20 dollars on y, his total expenditure is 50 dollars. If his income increases by 5 dollars, his total expenditure becomes 55 dollars. If prices remain the same, Julian will distribute this additional income between x and y in the same proportions as before. Therefore, his expenditure on x will be 30/50 * 55 = 33 dollars, and his expenditure on y will be 20/50 * 55 = 22 dollars.
(d) The total price effect on Julian's demand for x consists of the substitution effect and the income effect. The substitution effect is the change in demand for x due to the change in its relative price, holding utility constant. The income effect is the change in demand for x due to the change in real income, holding relative prices constant. If the price of x decreases, the substitution effect will increase the demand for x (as x becomes relatively cheaper). The income effect will also increase the demand for x (as the decrease in the price of x effectively increases Julian's real income). Therefore, the total price effect on Julian's demand for x is larger than the substitution effect.
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