Question

Consider a consumer who budgets $60 per month for the consumption of beer (denoted by ?) and pizza (denoted by ?). For the purpose of this question, all other budgeting and consumption decisions by the consumer are to be ignored. To the consumer, beer and pizza are perfect complements such that he drinks two bottles of beer with each pizza eaten. The prices of beer and pizza are $4/bottle and $12/pizza, respectively.

a. State a utility function representing the consumer’s preferences.

b. Determine whether the consumer’s preferences are (i) complete, (ii) transitive, (iii) monotonic, (iv) convex, (v) continuous and (vi) rational.

c. Let ? ∗ and ? ∗ and denote the utility-maximizing levels of consumption. Derive the values of ? ∗ and ? ∗ .

Answer 1

Given that,

Beer and pizza are complimentary goods.

Income available for the consumption of beer and pizza = $60

Price of pizza = $12 per pizza

Price of beer = $4 per bottle

(a) Perfect compliment goods refers to the goods which are used together to satisfy a given want.

Let x be the bottles of beer and y denotes the pizza. It is given that this consumer drinks 2 bottles of beer with each pizza.

Therefore, the utility function representing the consumer preferences as follows:

U (x, y) = min (x, 2y)

(b) The Indifference Curves framed are convex to the origin, these are L shaped indifference curves which satisfy the condition of monotonicity on the grounds that if there should arise an occurrence of perfect complements then more of the good is largely preferred to the less of it, likewise the preference relation also fulfils reflexive, complete and transitive inclinations.

(c) From the utility function,

x = 2y …….(1)

Budget constraint:

Px*x + Py*y = Income

4x + 12y = 60…….(2)

From (1) and (2), we get

4(2y) + 12y = 60

8y + 12y = 60

20y = 60

y* = 3 and x* = 6

x* and y* are the utility maximization consumption level.