Question

Suppose that indifference curves are described by straight lines with a slope of $-b.$ Given arbitrary prices and money income $p_1,p_2,$ and $m,$ what will the consumer's optimal choices look like?

Short answer

The consumer buys only the cheaper good based on the relative price compared to slope $-b$.

Step-by-step solution

Understanding Indifference Curves

Indifference curves represent points of equal satisfaction to the consumer. If these curves are straight lines with slope $-b$, it implies that the consumer perceives the two goods as perfect substitutes.

Calculate Relative Prices

The relative price of the two goods is given by $\frac{p_1}{p_2}$. This represents the trade-off between the two goods in terms of how many units of good 2 are sacrificed for an additional unit of good 1.

Determine the Optimal Choice

The consumer's optimal choice occurs where the slope of the budget line (relative price) equals the slope of the indifference curve ($-b$). Therefore, the condition for the optimal choice is: $\frac{p_1}{p_2}=b$.

Identify Consumption Bundles

If $\frac{p_1}{p_2}<b$, the consumer will purchase only good 1. Conversely, if $\frac{p_1}{p_2}>b$, the consumer will purchase only good 2. If $\frac{p_1}{p_2}=b$, the consumer is indifferent between the two goods and can choose any combination along his budget.

Key concepts

Consumer Choice

Consumer choice theory helps explain how individuals make decisions about what to buy, given their limited resources. This is all about maximizing satisfaction or 'utility' from the goods or services they consume. Every consumer aims to get the best possible mix of goods without exceeding their budget.

An indifference curve, a key concept here, depicts different combinations of two goods that provide the same level of satisfaction to the consumer. Indifference curves are typically downward sloping, which illustrates that if one good decreases, the other must increase to maintain the same utility. This highlights the idea of substitutability between goods within the realm of consumer choice.

Budget Line

A budget line represents all the combinations of two goods that a consumer can purchase with their budget, given current prices. It's like a boundary of consumption possibilities, dictated by income and the prices of the two goods.

The equation of a budget line is usually written as:$$p_1{x}_1+p_2{x}_2=m$$Here, $p_1$ and $p_2$ represent the prices of goods 1 and 2, $x_1$ and $x_2$ are quantities of these goods, and $m$ symbolizes the total budget.

• The slope of the budget line is $\frac{-p_1}{p_2}$, indicating how much of one good must be foregone to purchase an additional unit of the other.
• Shifts in the budget line can occur due to changes in income or the prices of goods.

This concept is essential for understanding how consumption choices are limited by purchasing power.

Perfect Substitutes

Perfect substitutes are goods that can replace each other entirely in consumption without any change in utility. This is represented by indifference curves that are straight lines.

• When two goods are perfect substitutes, you don't mind replacing one with the other because they offer the same level of satisfaction.
• The slope, denoted by '-b', of these straight indifference curves shows how much of one good a consumer is willing to trade for the other.

If the slope of the consumer's budget line is different from the slope of the indifference curves, consumers will opt entirely for the cheaper option, contributing to a distinct choice pattern where either of the goods is chosen solely, depending on their relative cost.

Relative Prices

Relative prices tell us the ratio of one good's price to another's and exhibit the rate at which consumers can exchange one good for another in markets without changing their budget expenditure.

• Mathematically, it's expressed as $\frac{p_1}{p_2}$.
• This ratio gives you the opportunity cost of purchasing one more unit of good 1, expressed in terms of good 2.

Relative prices are crucial in choices where consumers face perfect substitutes. When the slope of the indifference curve ($-b$) matches the relative price ($\frac{p_1}{p_2}$), consumers are indifferent to the combination of goods consumed. Deviations from this equality signal which good should be bought more or less in order to maximize utility given the budget constraint.