Question

Suppose Frank has an income of \(£ 50\), the unit price of \(X\) is \(P_{X}=£ 2\) and the unit price of \(Y\) is \(P_{Y}=£ 1\). Write down the budget constraint for Frank. Knowing that the marginal rate of substitution (in absolute value) between \(X\) and \(Y\) is \(M R S=\) \(X / Y\), find the optimal bundle that Frank should consume. (Hint: at the optimal bundle, the absolute value of the \(M R S\) must be equal to the absolute value of the slope of the budget constraint. Moreover, the budget constraint must be satisfied. You need to solve a system of two equations in two variables, \(X\) and \(Y\).)

Short answer

Frank should buy 20 units of X and 10 units of Y.

Step-by-step solution

Define the Budget Constraint

The budget constraint represents the combinations of goods Frank can buy with his income. The formula is \( P_X \cdot X + P_Y \cdot Y = I \), where \( I = £50 \), \( P_X = £2 \), and \( P_Y = £1 \). Thus the budget constraint equation becomes \( 2X + Y = 50 \).

Find the Slope of the Budget Line

The slope of the budget line is given by \(-\frac{P_X}{P_Y}\). With \(P_X = 2\) and \(P_Y = 1\), the slope is \(-\frac{2}{1}\) or simply \(-2\).

Use Marginal Rate of Substitution

The problem states that the marginal rate of substitution (MRS) is given by \(\frac{X}{Y}\). For the optimal bundle, \(|MRS| = |\text{slope of the budget line}|\). So, \(\frac{X}{Y} = 2\).

Set up the System of Equations

We now have two equations: \( 2X + Y = 50 \) from the budget constraint, and \( \frac{X}{Y} = 2 \) from the MRS condition. These are the system of equations to solve for \( X \) and \( Y \).

Solve for Y in Terms of X

From \( \frac{X}{Y} = 2 \), rearrange to find \( Y = \frac{X}{2} \).

Substitute and Solve for X

Substitute \( Y = \frac{X}{2} \) into the budget constraint: \( 2X + \frac{X}{2} = 50 \). Multiply through by 2 to clear the fraction: \( 4X + X = 100 \). This simplifies to \( 5X = 100 \), giving \( X = 20 \).

Solve for Y

Substitute \( X = 20 \) back into \( Y = \frac{X}{2} \) to find \( Y = \frac{20}{2} = 10 \).

Verify the Solution

Check that these values satisfy both conditions. The budget constraint \( 2 \cdot 20 + 10 = 50 \) holds true. The MRS, \( \frac{X}{Y} = \frac{20}{10} = 2 \), matches the slope. Thus, \(X = 20\) and \(Y = 10\) is the optimal bundle.

Key concepts

Budget Constraint

In Consumer Choice Theory, a budget constraint represents the various combinations of goods or services that a consumer can purchase given their income and the prices of these goods. It's a visual depiction of all possible spending options within a set budget. For instance, if Frank has £50, with good X costing £2 per unit and good Y costing £1 per unit, his budget constraint is expressed by the equation: \[ 2X + Y = 50 \]This equation indicates all the combinations of X and Y that Frank can afford without exceeding his budget. Key components:
This budget constraint helps us understand the trade-offs Frank must make—if he wants more of good X, he must sacrifice some of good Y, or vice versa.- **Income**: The total amount of money available for spending, which is £50 in Frank's case.- **Prices of goods**: Knowing the prices helps us form the equation. Here, the price of X is £2, and the price of Y is £1.- **Decision-making**: Frank needs to choose a combination on or within this line based on his preferences.Understanding this concept is crucial for identifying which consumption bundles are within a consumer's reach.

Marginal Rate of Substitution

The Marginal Rate of Substitution (MRS) is a key concept in understanding consumer preferences. It measures the rate at which a consumer is willing to substitute one good for another while maintaining the same level of satisfaction. Mathematically, the MRS is usually the derivative of the indifference curve, representing the slope at any given point.In Frank's example, the MRS between goods X and Y is given by the formula:\[ MRS = \frac{X}{Y} \]This means Frank is willing to give up Y units for X at a rate that keeps him equally happy. To achieve an optimal consumption choice, the absolute value of the MRS must equal the absolute value of the slope of the budget constraint.Importance:
The MRS takes into consideration personal preferences, indicating which good Frank prefers to increase versus the other.- **Optimizing**: At the optimal bundle, the MRS equals the negative of the price ratio, leading us to the condition: \[ \left| \frac{X}{Y} \right| = \left| -\frac{P_X}{P_Y} \right| \]- **Significance**: Here, this tells us Frank's rate at which he can trade between X and Y remains viable within his budget.Grasping the MRS helps predict changes in consumption patterns based on shifts in prices or preferences.

Optimal Consumption Bundle

The optimal consumption bundle is the combination of goods that provides the consumer with the maximum possible satisfaction while staying within their budget constraints. For it to be valid, two conditions must be satisfied:1. The budget constraint must hold
2. The Marginal Rate of Substitution equals the price ratio between the goodsFor Frank, solving the system of equations conforms to these criteria. The equations were:Budget Constraint:\[ 2X + Y = 50 \]MRS Condition:\[ \frac{X}{Y} = 2 \]By solving this system, we found:- **Optimal Values**: Frank's ideal consumption is 20 units of good X and 10 units of good Y.Verification showed:- **Maintaining Balance**: At X = 20 and Y = 10, both the budget constraint and MRS conditions are satisfied.Why this matters:
The optimal bundle reflects Frank's best choice based on his preferences, prices, and budget. It represents balance, where his marginal rate of substitution aligns perfectly with his budget limitations.Understanding this concept equips students with the analytical tools to predict how rational consumers behave in real-world scenarios.