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# Indifference curves

## Introduction

Indifference curves are used to help understand how consumers allocate their scarce income between competing uses.

Before applying the theory to consumer decision-making we need to understand the main assumptions behind indifference curves, including:

## Assumptions

### Limited budget

Firstly, we start by assuming that an individual's budget is fixed – later, we can change the budget to see its effect on spending.

### Completeness of choices

'Completeness' means that, when presented with alternatives, there are only two choices possible – to prefer one good or basket of goods to another one, or to be indifferent to two goods or two baskets of goods.

### Logic of choosing

This assumption (often called the assumption of ‘transitivity’) states that consumers are logical in how they demonstrate their preferences. For example, if I prefer tea to coffee, and prefer coffee to cola, then I must also prefer tea to cola!

This assumption states that consumers are consistent in how they make their choices. Hence, if I prefer tea to coffee today, then – unless something related to the choice changes – I will also prefer tea to coffee tomorrow.

### More is preferred to less

Clearly, the assumption that more is preferred to less only relates to a beneficial ‘good’, rather than a harmful one. Although the marginal utility derived from increased consumption may fall, the assumption still holds that more of something good is preferred to less of it.

## An indifference curve

A basic indifference curve is formed from a series of combinations of two goods or baskets of goods that provide the consumer with the same utility – in other words, they are indifferent to them.

For example, (and assuming an individual likes burgers and apples), over a one week period an individual may be indifferent to combinations of four burgers and seven apples, and to three burgers and eleven apples. If asked to sacrifice one additional burger, from three to two, the individual would only be indifferent to a combination with only two burgers so long as he or she could consume sixteen apples.

The complete set of combinations that this individual is indifferent to is shown below:

All these combinations yield equal utility given that the consumer is indifferent to them. These can be plotted, as follows, with burgers on the X axis and apples on the Y axis:

An indifference curve is convex to the origin, reflecting the fact that when there are fewer bananas in the basket, the consumer requires an increasing number of apples to compensate for the loss of bananas.

This is referred to as the diminishing marginal rate of substitution (MRS) (in this case, of apples for burgers). This is shown below:

The MRS measures how much of good y an individual is prepared to give up to get one more unit of x. Indifference curve analysis does not attempt to attach a ‘cardinal’ value to utility, but rather attaches an ‘ordinal’ value to utility, and in this way resolves a problem with marginal utility analysis.

## A map of indifference

If we now change the assumption of a fixed income (budget) we can then suggest that more than one indifference curve exists for an individual facing a choice between baskets of two goods. For example, with an increased income it is likely that more of both would be in the revised basket.

Over a range of incomes numerous indifference curves would exist - in theory, an infinite number. The principle of logical choice would clearly suggest that these indifference curves cannot cross each other.

For example, at income b1, the individual is indifferent to 4x + 7y. At b2, the individual is indifferent to 8x (more of x) and 17y (more of y). It would be illogical to be indifferent to 4x + 7y, and 8x and 4y.

This combination would result in a coordinate/combination inside and to the left of the original indifference curve, and if the sequence is completed, the two curves would cross. Hence, the logic of indifference analysis is that indifference curves cannot cross.

## Applications of indifference analysis

Indifference curve analysis can be used to explain several micro-economic concepts, including demand and income and consumer equilibrium.

### The budget line

We can now attach prices to goods, and a budget to the consumer to see how they are related, and then derive equilibrium, a price consumption line, and then a demand curve.

Firstly, we can identify the consumer’s budget line for a specific good. If, in our example, we assume the consumer’s budget is \$80, and the prices of burgers is \$10, and apples is \$1, then all the following combinations are possible.

## Consumer equilibrium

If we now combine the data for indifference curve map, and the budget line, we can calculate when the consumer is in equilibrium.

Given the prices of the products, the budget and the consumer's indifference, we can establish that this specific consumer is in equilibrium when he or she purchases four burgers and forty apples over a two week period. Equilibrium will occur when the consumer is both indifferent to a basket, and when that basket is affordable with the fixed budget. In this case it will be at 4y+40x. At this single point, the gradients of the relevant indifference curve, and the budget line (which will be at tangent to each other) with be identical.

## A change in price

We can now derive a demand curve for burgers (or apples) by holding the price of one constant, and then changing the price of the other one. In this case we will hold apples at \$1 each, keep the budget at \$80, and increase the price of burgers to \$12 each. The following combinations are now possible:

A rise in the price of burgers will rotate the budget line inwards, indicating fewer burgers can be afforded. The actual reduction will be determined by the relevant indifference curve, which in this example will be 3, and reduction of one burger. There has also been a change in demand for apples – how demand for apples changes depends upon the gradient of the indifference curve.

It is possible that demand for apples could rise (the most obvious answer), given that there is likely to be some level of substitution between burgers and apples, but demand could also fall because the increase the price of burgers results in a fall in real income and leaves less available to spend on apples.

Changing the price of a single good creates a ‘price consumption line’. Increasing the budget If the budget is increased (say to \$160), more of both can be consumed (double the quantities at a budget of \$80), and the budget line shifts outwards, and remains parallel to the original budget line. A reduction in income sifts the budget line inwards. If we include the indifference curve map, we can identify the ‘income consumption’ line, as shown.

### Converting the Y axis

It makes sense to convert the Y axis to ‘all other goods’ so that a single good can be placed on the X axis. For example, we could bundle all the other goods an individual could purchase, keep their prices and budget constant, and simply vary the price of burgers (or any single product).

We can then isolate the price consumption line for the single product, as shown. This tracks how demand will change in response to changes in price, and from this we can derive a demand curve.

We can also identify the income consumption line for a single product, as shown. The income consumption line track the effect of changes in income on demand, and from this we can derive an Engel curve.

## Deriving a demand curve

By altering the price of good x we can construct a price-consumption line, and derive a demand curve, as shown:

## Deriving an Engel curve

By altering the budget (or income) we can construct an income-consumption line, and derive an Engel curve, as shown.

## Price elasticities

Price consumption lines can be used to illustrate the idea of price elasticity of demand. The two price consumption lines show an elastic and inelastic demand.

This would correspond to steeper and flatter demand curves, as shown:

## Income elasticities

By varying income, we can illustrate the difference between normal goods (with a positive YED) and inferior goods (with a negative YED). In the following we can see two difference income consumption lines, with corresponding Engel curves (after German statistician, Ernst Engel).

## Income and substitution effect of a price change

By altering the price of good x we can separate out the effect of the price change on the purchasing power of a given budget – the income effect on demand for x - and on the relative change in demand as consumers switch towards (or away from) a substitute – the substitution effect on demand for x.

In our example, the overall effect of the price reduction is for demand to increase from 3 to 5 units - the movement, M to K.

The substitution effect, which relates to the change in demand following a price change, assumes the level of utility remains constant. This means that the gradient of the budget line changes, which creates a movement around the existing indifference curve (rather than moving to a new indifference curve.). In the case of burgers, the movement from M to L represents the substitution effect. This movement corresponds to a small increase in demand from 3x to 3.5x.

The remainder of the effect is the income effect (which increases demand from 3.5x to 5x) and can be seen by the shift in the rotated budget line (1a) to budget line 2. A positive income effect is demonstrated by assuming relative prices remain the same, while purchasing power has increased. If prices remain constant, the gradient of the budget line must remain unchanged, but shifts to the right.

## The case of Giffen goods

In the case of a Giffen good (after Sir Robert Giffen), such as rice in a developing country, the good is an inferior one, where a rise in real incomes leads to an increase in demand. However, a price rise for an inferior good is most likely to lead to a reduction in the quantity demanded through the substitution effect - but, not so for a Giffen good.

For the Giffen good, a rise in price leads to an increase in demand because the substitution effect (which reduced demand) is very weak (as there are few substitutes for a staple food) while the income effect (which increases demand) is stronger, with the 'net' effect being that demand increases following the price rise.

This can be seen diagrammatically.

The price rise of \$5 to \$10 for 10 pounds of rice pivots the budget line from line 1 to line 1a. Demand is initially at 3 (tangent K), and the substitution effect (the movement around indifference curve, IC1) results in a fall in demand to 2, and a new tangent at point M. However, when the income effect is added (with the budget line shifting to budget line 2), the quantity demanded increases from 2 to 4, with the new tangent at L (tangent budget line 2/IC2). So, the income effect (which increases demand) outweighs the substitution effect (which reduces demand).

For many, the Giffen good remains a mathematical curiosity, but it is clear that in low income countries, with a single staple diet, supplemented occasionally with more expensive foods, a rise in the price of the staple might well see a rise in demand.

Research has tended to confirm the existence of real world 'Giffen behaviour'. For example, Robert Jensen and Nolan Miller (20071) confirmed that they 'find strong evidence of Giffen behavior with respect to rice in Hunan province (of China)....and in Gansu (province) with respect to wheat.

1. Giffen Behavior: Theory and Evidence, Robert T. Jensen, Nolan H. Miller, NBER Working Paper No. 13243, Issued in July 2007, Revised in December 2007, NBER Program(s):Economics of Aging.