Question

Suppose that Carl cannot tell the differences between a pack of British and a pack of Danish bacon. In a graph with British bacon on the vertical axis, plot some of Carl's indifference curves for British and Danish bacon, Suppose that Carl has an income of \(£ 20\). The price of Danish bacon is \(£ 2\) per pack, while the price of British bacon is \(£ 4\) per pack. Using the same graph, draw Carl's budget constraint and show his optimal bundle choice.

Short answer

Carl's optimal bundle is 10 packs of Danish bacon and 0 packs of British bacon.

Step-by-step solution

Understand the Problem

Carl cannot distinguish between British and Danish bacon, implying they are perfect substitutes for him. We know his income and the prices of both types of bacon. We need to plot indifference curves and find the optimal consumption point.

Identify the Budget Constraint

With Carl's income of £20, and prices of £2 and £4 for Danish and British bacon respectively, his budget constraint can be described by the equation: \[ 2D + 4B = 20 \] where \( D \) is the number of packs of Danish bacon and \( B \) is the number of packs of British bacon.

Plot the Budget Line

To find the intercepts, set \( B = 0 \): \( 2D = 20 \Rightarrow D = 10 \) (Danish intercept); set \( D = 0 \): \( 4B = 20 \Rightarrow B = 5 \) (British intercept). Draw a straight line through these points (10,0) and (0,5).

Draw Indifference Curves

Since Carl views these goods as perfect substitutes, his indifference curves will be straight lines with a slope of -1, meaning when he exchanges one pack of British bacon, he needs exactly one additional pack of Danish bacon.

Find the Slope of the Budget Line

The slope of the budget line is \(-\frac{P_D}{P_B} = -\frac{2}{4} = -\frac{1}{2}\). This implies that for each pack of British bacon he gives up, he can get half a pack of Danish bacon.

Determine the Optimal Bundle

Since indifference curves have a slope of -1 and the budget line has a slope of -0.5, the optimal point will be where Carl consumes only Danish bacon because it is cheaper per equivalent satisfaction level due to the lower price.

Mark the Optimal Bundle on the Graph

At the optimal point, Carl will spend all his income on Danish bacon. Therefore, his optimal bundle is at point \( D = 10 \) and \( B = 0 \). Plot this point on the graph.

Key concepts

Budget Constraint

A budget constraint is a visual representation of all the combinations of goods and services that a consumer can purchase with a given income and prices. For Carl, his budget constraint is determined by his £20 income and the costs of British bacon (£4 per pack) and Danish bacon (£2 per pack). To express his budget constraint mathematically, we use the formula:\[ 2D + 4B = 20 \]Here, \(D\) represents the number of packs of Danish bacon, while \(B\) represents the number of packs of British bacon. The equation simply states that the total money spent on both types of bacon cannot exceed £20.When plotted on a graph, with British bacon on the vertical axis and Danish bacon on the horizontal axis, we identify the intercepts to visualize the budget line:

• Setting \(B = 0\) gives \(D = 10\), meaning Carl can buy up to 10 packs of Danish bacon if he buys no British bacon.
• Setting \(D = 0\) gives \(B = 5\), indicating he can buy 5 packs of British bacon with his entire income.

The resulting line drawn through these intercepts represents all possible combinations of bacon he can afford.

Perfect Substitutes

Perfect substitutes refer to goods that a consumer perceives as perfectly interchangeable. For Carl, British and Danish bacon are perfect substitutes since he cannot tell them apart. This means that when Carl consumes these goods, his level of satisfaction remains unchanged if he exchanges one pack of British bacon for one pack of Danish bacon, or vice versa. On a graph, indifference curves for perfect substitutes are straight lines with a slope of -1, indicating that the consumer values these goods equally. This equal valuation is crucial in determining how Carl views his options under his budget constraint. The slope signifies that the trade-off between the two types of bacon remains constant. In Carl's case, since he sees no difference between the types of bacon, the choice between them is purely based on price.

Optimal Bundle Choice

The optimal bundle choice is the combination of goods that maximizes a consumer's satisfaction while staying within budget. For Carl, the optimal bundle can be found where his budget line and the highest possible indifference curve touch. Due to the concept of perfect substitutes, Carl's indifference curves are lines with a slope of -1. Meanwhile, his budget line has a slope of -0.5, determined by the prices of the two types of bacon. The crucial aspect here is that the slope of the indifference curve (representing satisfaction trading off between British and Danish bacon) does not match the budget line's slope. This indicates that Carl can achieve maximum satisfaction by spending less money per pack of bacon. Given that Danish bacon is cheaper, it allows him to gain more satisfaction per unit of currency. Thus, Carl's optimal bundle is located at the point where he buys only Danish bacon: 10 packs (using up all his income of £20), which reflects the most efficient allocation of his budget under these circumstances.