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?

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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

  1. 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 ).

  1. Calculate Initial Demand for X and Y

Using the budget constraint ( I = P_X \cdot X + P_Y \cdot Y ).

Substituting initial values:

$$ 20 = 4X + 4Y $$

  1. 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.

  1. Calculate New Demand after Price Change

After the price of X drops to ( P_X = 1 ), we need to recalculate the budget constraint:

$$ 20 = 1X + 4Y $$

  1. Determine New Consumption

Rearranging gives:

$$ X + 4Y = 20 $$

Assuming ( Y = 0 ):

$$ X = 20 $$

If ( X = 0 ):

$$ 4Y = 20 \quad \Rightarrow \quad Y = 5 $$

  1. 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 ):

$$ Y = 4.2 $$

  1. Aggregate Demand Effects

Final comparison:

  • Original ( X = 5 ) (at ( P_X = 4 ))
  • New ( X = 20 )

Calculate effects:

  • Total Change: ( 20 - 5 = 15 )
  1. 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.
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