Question

Consider the utility function of a consumer who obtains utility from consuming only two goods, ?1 and ?2 , in fixed proportions. Specifically, the utility of the consumer requires the consumption of two units of ?2 for each unit of ?1.

i. Report the mathematical expression of the utility function of the consumer.

ii. Provide a diagram of the corresponding indifference curves.

iii. Provide at least one example and economic intuition.

Suppose that the consumer has available income equal to Y, and the price of the goods is respectively ?1 and ?2.

i. Derive the expressions of the quantities that maximise the utility of the consumer under the budget constraint.

ii. Provide a diagram of the optimal solution.

Answer 1

Two Goods: q₁ and q₂

Utility requires the consumption of two units of q₂ for each unit of q₁

(i)

Consumer is consuming the goods only in a fixed proportion. This means q₁ and q₂ are perfect complements.

Mathematical exprssion for this would be:

This mathematical expression leads to the following optimal consumtion pattern:

2q₁ = q₂

Which is in line with the statement provided in the question.

(ii)

Consumer will always consume along the red line.

Equation of the red line: 2q₁ = q₂

(iii)

Lets take example of One cup coffee and 2 cubes of sugar.

This is the example of perfect comlementary goods.

If you always consume 2 cubes of sugar with one cup of coffee extra coffee or sugar will not given you any extra utility.

NOW,

Consumer's Income: I

Price of q₁ = p₁

Price of q₂ = p₂

1)

Budget line will be following:

Consumer always consumes in proportion : 2q₁ = q₂

Putting this in the Budget line to get the optimal solution subject to the budget constraint:

2)