Asked by Randall Armstrong on Jul 09, 2024
Verified
Nick insists on consuming 3 times as much of y as he consumes of x (so he always has y 3x) .He will consume these goods in no other ratio.The price of x is 2 times the price of y.Nick has an endowment of 20 x's and 75 y's which he can trade at the going prices.He has no other source of income.What is Nick's gross demand for x?
A) 21
B) 115
C) 23
D) 95
E) We can't determine the answer without knowing the price of x.
Gross Demand
The total amount of a product or service that consumers and organizations wish to buy at any given price level before taking into account external factors.
Endowment
An endowment refers to the total assets, both financial and non-financial, owned by an individual or institution, often used to generate income or donations.
Price Ratio
The proportional relationship between two or more prices, indicating the relative cost of goods or services.
- Inspect the influence of price changes on the consumer demand for different products.
- Grasp the relationship between consumption ratios and price changes.
Verified Answer
SM
Sahil MeangiJul 16, 2024
Final Answer :
C
Explanation :
Since Nick always consumes 3 times as much y as he consumes x, we can set up the following equation for his utility function: U(x,y) = x * (y^3). This ensures he only consumes them in a 3:1 ratio.
Now, since the price of x is twice the price of y, we can set up the following budget constraint:
2x * 20 + y * 75 = M, where M is Nick's income.
If we solve for y in terms of x, we get: y = (M - 40x)/75.
Substituting this into Nick's utility function, we get: U(x) = x * ((M-40x)/75)^3.
To find the amount of x that maximizes this function, we take the derivative with respect to x, set it equal to zero, and solve for x:
dU/dx = (M/25) * x * ((M-40x)/75)^2 - 8/5 * x^2 * (M-40x)^2/5625 = 0.
Solving for x, we get: x = 23.
Therefore, Nick's gross demand for x is 23.
Now, since the price of x is twice the price of y, we can set up the following budget constraint:
2x * 20 + y * 75 = M, where M is Nick's income.
If we solve for y in terms of x, we get: y = (M - 40x)/75.
Substituting this into Nick's utility function, we get: U(x) = x * ((M-40x)/75)^3.
To find the amount of x that maximizes this function, we take the derivative with respect to x, set it equal to zero, and solve for x:
dU/dx = (M/25) * x * ((M-40x)/75)^2 - 8/5 * x^2 * (M-40x)^2/5625 = 0.
Solving for x, we get: x = 23.
Therefore, Nick's gross demand for x is 23.
Learning Objectives
- Inspect the influence of price changes on the consumer demand for different products.
- Grasp the relationship between consumption ratios and price changes.