Question

When time is an important constraint on a consumer’s choices, the consumer’s choice problem (when 3 activities x₁ , x₂ , and x₃ are chosen) is

MaxU (x₁, x₂, x₃) + λ (M - p₁ x₁ - p₂ x₂ - p₃ x₃) + µ (T - x₁ - x₂ - x₃)

a)Interpret the multipliers λ and µ. What is each measuring?

b)Derive the first order condition for best choice for one of the activities, say 1 x .Arrange this so that it has the form marginal benefit equals marginal cost, and explain what the marginal benefit and marginal cost terms refer to.

c)What is the value of the consumer’s time in this model?

Answer 1

Given the consumer choice problem among the three activities, λ represents the lagrangian multiplier for the budget constraint and µ represents the lagrangian multiplier for the time constraint.  λ measures the rate of change in the optimum utility resulting from change in the budget. On the other hand, µ represents the rate of change in the optimum utility resulting from variation in time.

The first order conditions for best choice of activity 1 is given by

The marginal benefit is represented by the left hand side that is the change in the utility due to change in activity 1. In other words, it can be interpreted as an increase in the utility due to additional unit of first activity. The marginal cost is represented by the right hand side which means that in order to increase utility of undergoing the first activity the consumer has to sacrifice a budget of λp1 in terms of price. Additionally, the consumer has to sacrifice µ units of time.

Total time available with the consumer is T.

x1 + x2 + x3 = T

The time spend on each activity depends upon the value of λ which determines the value of consumer's time.