Question 1 (22 marks). Adam consumes only beer (B) and whisky (W ). His utility function isU (B, W ) = √B + 5√W ,where B is the quantity of beer and W is the quantity of whisky. The price of beer is P B = 5, theprice of whisky is P W = 40, and Adam’s income is M = 100 dollars.(a) Obtain the equation of Adam’s indifference curve for the utility level U = 6, with the quantityof whisky as a function of the quantity of beer. (2 marks)(b) Obtain the amount of whisky required to attain the utility level U = 6 if the amount of beeris B = 1? Compute the marginal rate of substitution (MRS) at that bundle and explainhow you interpret that magnitude (of the MRS). (4 marks)(c) Obtain the equation of Adam’s budget line. What is the magnitude of the slope of thebudget line? What is the economic interpretation of this magnitude? (4 marks)(d) Adam’s cousin, John, has a utility function given byV (B, W ) = 3B 2 + W 2 .(d.1) Are John’s preferences monotone? Are they strictly monotone? Justify your answers.(3 marks)(d.2) Are John’s preferences convex? Are they strictly convex? Justify your answers. (3marks)(d.3) Do John’s preferences satisfy the diminishing marginal rate of substitution property?Explain. (2 marks)(d.4) Assume that the prices are as indicated above (P B = 5 and P W = 40), and John’sincome is I > 0. Obtain John’s optimal bundle. (4 marks)EXAMINATION CONTINUES ON NEXT PAGE1
Question
Question 1 (22 marks). Adam consumes only beer (B) and whisky (W ). His utility function isU (B, W ) = √B + 5√W ,where B is the quantity of beer and W is the quantity of whisky. The price of beer is P B = 5, theprice of whisky is P W = 40, and Adam’s income is M = 100 dollars.(a) Obtain the equation of Adam’s indifference curve for the utility level U = 6, with the quantityof whisky as a function of the quantity of beer. (2 marks)(b) Obtain the amount of whisky required to attain the utility level U = 6 if the amount of beeris B = 1? Compute the marginal rate of substitution (MRS) at that bundle and explainhow you interpret that magnitude (of the MRS). (4 marks)(c) Obtain the equation of Adam’s budget line. What is the magnitude of the slope of thebudget line? What is the economic interpretation of this magnitude? (4 marks)(d) Adam’s cousin, John, has a utility function given byV (B, W ) = 3B 2 + W 2 .(d.1) Are John’s preferences monotone? Are they strictly monotone? Justify your answers.(3 marks)(d.2) Are John’s preferences convex? Are they strictly convex? Justify your answers. (3marks)(d.3) Do John’s preferences satisfy the diminishing marginal rate of substitution property?Explain. (2 marks)(d.4) Assume that the prices are as indicated above (P B = 5 and P W = 40), and John’sincome is I > 0. Obtain John’s optimal bundle. (4 marks)EXAMINATION CONTINUES ON NEXT PAGE1
Solution
(a) To obtain the equation of Adam's indifference curve for the utility level U = 6, we need to set U(B, W) = 6 and solve for W in terms of B.
U(B, W) = √B + 5√W = 6
Squaring both sides to get rid of the square roots, we get:
B + 25W + 10√(BW) = 36
Rearranging to solve for W:
W = (36 - B - 10√(BW)) / 25
This is the equation of Adam's indifference curve for the utility level U = 6.
(b) To find the amount of whisky required to attain the utility level U = 6 if the amount of beer is B = 1, we substitute B = 1 into the equation from part (a):
W = (36 - 1 - 10√(1*W)) / 25
Solving this equation for W gives the amount of whisky required.
The marginal rate of substitution (MRS) is the rate at which Adam is willing to trade beer for whisky while maintaining the same level of utility. It is given by the negative of the ratio of the marginal utility of beer to the marginal utility of whisky. The marginal utility of beer is the derivative of the utility function with respect to B, and the marginal utility of whisky is the derivative of the utility function with respect to W.
(c) Adam's budget line is given by the equation M = PBB + PWW, where M is his income, PB is the price of beer, PW is the price of whisky, B is the quantity of beer, and W is the quantity of whisky. Substituting the given values, we get:
100 = 5B + 40W
Rearranging to solve for W:
W = (100 - 5B) / 40
The slope of the budget line is -PB/PW = -5/40 = -1/8. This represents the rate at which Adam can trade beer for whisky given his budget constraint.
(d.1) John's preferences are monotone if increasing the quantity of either good while keeping the quantity of the other good constant leads to an increase in utility. They are strictly monotone if a strict increase in the quantity of either good leads to a strict increase in utility. Looking at John's utility function, we can see that both the terms 3B^2 and W^2 are increasing in B and W respectively, so John's preferences are monotone. They are also strictly monotone because a strict increase in either B or W leads to a strict increase in utility.
(d.2) John's preferences are convex if a convex combination of two bundles yields a utility level that is at least as high as the utility level of the original bundles. They are strictly convex if the utility level of the convex combination is strictly higher. Looking at John's utility function, we can see that it is a sum of squares, which is a convex function, so John's preferences are convex. They are also strictly convex because the utility level of a convex combination of two bundles is strictly higher than the utility level of the original bundles.
(d.3) John's preferences satisfy the diminishing marginal rate of substitution property if the MRS decreases as the quantity of one good increases while the quantity of the other good is held constant. Looking at John's utility function, we can see that the marginal utility of beer is 6B and the marginal utility of whisky is 2W. The MRS is therefore 3B/W, which decreases as B increases and W is held constant, so John's preferences satisfy the diminishing marginal rate of substitution property.
(d.4) To find John's optimal bundle, we need to solve the following system of equations, which is obtained by setting the MRS equal to the price ratio and using the budget constraint:
3B/W = PB/PW
PBB + PWW = I
Solving this system of equations gives the optimal bundle.
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