Jack consumes only Bread (good x) and Cheese (good y). His utility function is given byU (x, y) = ln(x) + ln(y)(a) Find Jack’s marginal rate of substitution (MRS) at bundle (x, y).(b) The price of good x is $6 and the price of good y is $3. Jack has an income of $120. Findhis utility-maximising consumption bundle.(c) Calculate Jack’s utility from the bundle you have found in part (b). (You can leave youranswer expressed in logarithm.).(d) In a graph with the quantity of x on the x-axis and the quantity of y on the y-axis, drawJack’s budget line, clearly marking the intercepts and the utility maximising bundle youhave solved in part (b) (call it bundle A). Also sketch an indifference curve passing throughbundle A.(e) Now the price of good y increases to $6 while the price of good x remains at $6. Jack’sincome remains at $120. Calculate his new utility-maximising consumption bundle.(f) On the diagram you have drawn for part (d), draw Jack’s new budget line, clearly markingthe intercepts and the new optimal bundle you have solved in part (e) (call it bundle B).Also sketch an indifference curve passing through bundle B.(g) Find the minimal expenditure required for Jack to achieve his original utility level (i.e.,your answer to part (c)) under the new prices. As you are finding this minimal expenditure,solve for the expenditure-minimising bundle that achieves the original utility level underthe new prices.(h) On the diagram you have drawn for parts (d) and (f), draw the budget line associatedwith the minimal expenditure you have solved in part (g), clearly marking the interceptsand the expenditure-minimising bundle you have solved in part (e) (call this bundle H).
Question
Jack consumes only Bread (good x) and Cheese (good y). His utility function is given byU (x, y) = ln(x) + ln(y)(a) Find Jack’s marginal rate of substitution (MRS) at bundle (x, y).(b) The price of good x is 3. Jack has an income of 6 while the price of good x remains at 120. Calculate his new utility-maximising consumption bundle.(f) On the diagram you have drawn for part (d), draw Jack’s new budget line, clearly markingthe intercepts and the new optimal bundle you have solved in part (e) (call it bundle B).Also sketch an indifference curve passing through bundle B.(g) Find the minimal expenditure required for Jack to achieve his original utility level (i.e.,your answer to part (c)) under the new prices. As you are finding this minimal expenditure,solve for the expenditure-minimising bundle that achieves the original utility level underthe new prices.(h) On the diagram you have drawn for parts (d) and (f), draw the budget line associatedwith the minimal expenditure you have solved in part (g), clearly marking the interceptsand the expenditure-minimising bundle you have solved in part (e) (call this bundle H).
Solution
(a) The marginal rate of substitution (MRS) is the rate at which a consumer is willing to trade off one good for another while maintaining the same level of utility. It is given by the negative ratio of the marginal utility of good x to the marginal utility of good y.
The marginal utility of good x is the derivative of the utility function with respect to x, which is 1/x. Similarly, the marginal utility of good y is 1/y. Therefore, the MRS is - (1/x) / (1/y) = -y/x.
(b) To find the utility-maximizing consumption bundle, we need to solve the consumer's budget constraint, which is given by the equation pxx + pyy = I, where px is the price of good x, py is the price of good y, and I is the consumer's income.
Substituting the given values, we get 6x + 3y = 120. Simplifying, we get 2x + y = 40.
Since the utility function is Cobb-Douglas, the utility-maximizing bundle is found when the consumer spends all his income and the ratio of the quantities of the two goods is equal to the ratio of their prices. Therefore, x = 40/3 = 13.33 and y = 40 - 2*13.33 = 13.33.
(c) Substituting these values into the utility function, we get U(13.33, 13.33) = ln(13.33) + ln(13.33) = 2.59 + 2.59 = 5.18.
(d) In the graph, the budget line is a straight line with a slope of -px/py = -6/3 = -2 and intercepts at x = 120/6 = 20 and y = 120/3 = 40. The utility-maximizing bundle A is at (13.33, 13.33). The indifference curve passing through A is a downward-sloping curve that is tangent to the budget line at A.
(e) With the new prices, the budget constraint is 6x + 6y = 120, or x + y = 20. The utility-maximizing bundle is now at x = 20/2 = 10 and y = 20 - 10 = 10.
(f) The new budget line has a slope of -1 and intercepts at x = 120/6 = 20 and y = 120/6 = 20. The new optimal bundle B is at (10, 10). The indifference curve passing through B is a downward-sloping curve that is tangent to the new budget line at B.
(g) To find the minimal expenditure required to achieve the original utility level, we need to solve the equation U(x, y) = 5.18 for the new prices. This gives us the equation ln(x) + ln(y) = 5.18, or x*y = e^5.18. Substituting the new prices, we get 6x + 6y = e^5.18, or x + y = e^5.18/6. The expenditure-minimizing bundle is at x = e^5.18/12 = 9.93 and y = e^5.18/12 = 9.93.
(h) The budget line associated with the minimal expenditure has a slope of -1 and intercepts at x = e^5.18/6 = 19.86 and y = e^5.18/6 = 19.86. The expenditure-minimizing bundle H is at (9.93, 9.93).
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