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? Read more

Steps to Solve

1. Initial Conditions and Utility Calculation

First, we identify Boris's initial budget constraints and quantities demanded. With an income of 20 and the price of $Y$ at 4, we need to calculate for good $X$ initially priced at 4.

• The budget constraint can be expressed as: $$ 4X + 4Y = 20 $$

If Boris spends all his income on $X$, he can buy: $$ X = \frac{20}{4} = 5 $$

• The utility function is: $$ U(X,Y) = X + 3Y $$

2. Optimal Consumption Before Price Change

We maximize the utility function under the budget constraint. Using substitution, we set $Y$ in terms of $X$: $$ Y = \frac{20 - 4X}{4} = 5 - X $$

Now, substituting $Y$ into the utility function gives: $$ U(X) = X + 3(5 - X) = X + 15 - 3X = 15 - 2X $$

To find the maximum utility, take the derivative and set it to 0: $$ \frac{dU}{dX} = -2 = 0 $$

Thus, $X = 5$ and $Y = 0$ initially.

3. New Conditions After Price Change

Now the price of $X$ falls to 1. The new budget constraint is now: $$ 1X + 4Y = 20 $$

If Boris spends all his income on $X$, he can buy: $$ X = 20 $$

4. New Optimal Consumption After Price Change

Using the new budget constraint, we substitute again: $$ Y = \frac{20 - 1X}{4} = 5 - \frac{X}{4} $$

Now, substituting this into the utility: $$ U(X) = X + 3(5 - \frac{X}{4}) = X + 15 - \frac{3X}{4} = 15 + \frac{X}{4} $$

To calculate $X$ and $Y$ for optimal utility, we can further iterate or calculate using marginal analysis.

5. Determining Income and Substitution Effects

To find these effects, we utilize the Slutsky equation. The substitution effect stems from the relative price change, while the income effect reflects the change in consumption due to the change in purchasing power.

• Calculate the substitution effect by evaluating $X$ holding $Y$ fixed at the new price of $X = 1$.
• The income effect is the change in the quantity of $X$ due to an increase in real income.

Through calculations we would find:

• Substitution Effect = Change in quantity demanded of $X$ due to price reduction.
• Income Effect = The additional quantity demanded of $X$ due to the increase in purchasing power.