Boris has the utility function U(X,Y) = X + 3Y and an income of 20. The price of Y is 4. The price of X falls from 4 to 1. What are the income and substitution effects of this pric... Boris has the utility function U(X,Y) = X + 3Y and an income of 20. The price of Y is 4. The price of X falls from 4 to 1. What are the income and substitution effects of this price change on the demand for X?
Understand the Problem
The question is asking to analyze the impact of a price change on the demand for good X, using the utility function provided. It requires calculating both the income and substitution effects resulting from the change in price of good X from 4 to 1.
Answer
Income effect: 12, Substitution effect: 3.
Answer for screen readers
The income effect on the demand for X is 12, and the substitution effect is 3.
Steps to Solve
- Identifying Initial Conditions
Define the initial conditions with the given information. The utility function is ( U(X, Y) = X + 3Y ). The initial price of good X is ( P_X = 4 ), price of good Y is ( P_Y = 4 ), and income is ( I = 20 ).
- Calculate Initial Demand for X and Y
Using the budget constraint ( I = P_X \cdot X + P_Y \cdot Y ).
Substituting initial values:
$$ 20 = 4X + 4Y $$
- Determine Initial Consumption
Let's set up the system. Rearranging yields:
$$ X + Y = 5 $$
From the utility function, we maximize utility by fully spending the budget. If ( Y = 0 ):
$$ X = 5 $$
If ( X = 0 ):
$$ Y = 5 $$
Since the utility function is linear, Boris will choose the maximum combination of ( X ) within budget.
- Calculate New Demand after Price Change
After the price of X drops to ( P_X = 1 ), we need to recalculate the budget constraint:
$$ 20 = 1X + 4Y $$
- Determine New Consumption
Rearranging gives:
$$ X + 4Y = 20 $$
Assuming ( Y = 0 ):
$$ X = 20 $$
If ( X = 0 ):
$$ 4Y = 20 \quad \Rightarrow \quad Y = 5 $$
- Calculate Substitution Effect
Use the intermediate price to find optimal consumption without making any utility changes. Set the intermediate price at ( P_X = 2.5 ) (the average):
$$ 20 = 2.5X + 4Y $$
Rearranging gives:
$$ X + 1.6Y = 8 $$
Assuming ( Y = 0 ):
$$ X = 8 $$ If ( X = 0 ):
$$ Y = 4.2 $$
- Aggregate Demand Effects
Final comparison:
- Original ( X = 5 ) (at ( P_X = 4 ))
- New ( X = 20 )
Calculate effects:
- Total Change: ( 20 - 5 = 15 )
- Identify Income and Substitution Effects
The income effect is the difference between the total change and substitution effect.
Income Effect:
$$ IE = 20 - 8 = 12 $$
Substitution Effect:
$$ SE = 15 - 12 = 3 $$
The income effect on the demand for X is 12, and the substitution effect is 3.
More Information
In this problem, we explored how changes in price affect demand using a utility function and budget constraints. The linear nature of the utility function simplifies the analysis and allows clear insights into consumer choices.
Tips
- Forgetting to correctly derive the budget constraint from the utility function.
- Not adequately separating the income and substitution effects.
- Assuming linear utility implies equal relative preference without checking constraints.
