Ambrose's utility function is . The price of x is $1 and the price of y is $2. If his income rises from $100 to $150, his consumption of y increases by more than 10% but less than 50%.

Question

Ambrose's utility function is  . The price of x is $1 and the price of y is $2. If his income rises from $100 to $150, his consumption of y increases by more than 10% but less than 50%.

Quizplus · Accepted Answer

To solve this problem, we first need to determine Ambrose's optimal bundle and consumption of x and y at his initial income of $100, given his utility function and the prices of x and y. 
We can use the marginal rate of substitution (MRS) to find the optimal bundle. The MRS is the ratio of the marginal utility of x to the marginal utility of y, and it represents the rate at which Ambrose is willing to trade one good for the other while remaining at the same level of utility. At the optimal bundle, the MRS must be equal to the price ratio of the two goods (MRS = Px/Py). 
Differentiating Ambrose's utility function with respect to x and y, we get: 
MUx = 6e8b4d92_aad2_5e73_81a8_ef047a7e2c2d_TB6162_11 = 22x^2y 
MUy = 6e8b4d92_aad2_5e73_81a8_ef047a7e2c2d_TB6162_11 = 11x^3 
Setting the MRS equal to the price ratio, we have: 
MRS = Px/Py = MUx/MUy 
22x^2y / 11x^3 = 1/2 
y/x = 1/4 
So Ambrose's optimal bundle is to consume y at a rate of 1 unit for every 4 units of x. 
At an income of $100, Ambrose can afford to purchase any bundle of x and y that lies on his budget line, which is given by: 
Px*x + Py*y = I 
1*x + 2*y = 100 
So Ambrose can purchase 100 units of x and no y, or 50 units of x and 25 units of y (which lies on his optimal bundle), or 25 units of x and 50 units of y. We know that Ambrose chooses the bundle that gives him the highest utility, which is the bundle that lies on his budget line and has an MRS equal to the price ratio. 
At an income of $150, Ambrose's budget line shifts outward, and his optimal bundle and consumption of x and y will change. We can find the new optimal bundle and consumption of y by following the same steps as above. However, we can make some general observations based on the given information. 
Since Ambrose's income has increased, he can afford to purchase more of both x and y. However, we are told that his consumption of y increases by more than 10% but less than 50%. This means that his consumption of x must also increase, but not by as much as his consumption of y. If his consumption of y increased by less than 10%, then his increase in income would not be sufficient to change his optimal bundle, as his original bundle was already feasible (and he would not be able to increase his consumption of y by more than 50% without exceeding his budget). If his consumption of y increased by more than 50%, then his optimal bundle would change drastically, as he would be substituting x for y at a much higher rate than before. 
Therefore, we can conclude that Ambrose's optimal bundle at an income of $150 will have a higher consumption of both x and y compared to his optimal bundle at an income of $100, with a larger increase in y than in x, but not so large as to change his preferences drastically. Hence the answer is A.

Ambrose's Utility Function Is