Question

Suppose that a consumer's utility function is U = x1x2 and the corresponding marginal rate of substitution between x1 and x2, where x1 is on the horizontal axis and x2 is on the vertical axis, is given by MRS = x2/x1. The budget constraint is given by 2x1 + 4x2 = 80. What is the optimal consumption bundle for this consumer?

Consider the consumer in question 7. If the price of x

1

increases,

A)

consumption of x

1

increases

B)

he/she gets better off

C)

consumption of x

1

decreases

D)

consumption of x

1

does not change

E)

the indi

fference

curve shifts out

9. Consider the consumer in question 7. If the sellers of x

1

offer a discount of $10 with the

purchase above $50,

A)

the consumer does not get a

ffected by this discount

B)

the consumer gets better o

ff

C)

the consumer gets worse off

D)

the consumer's

budget line becomes steeper

E)

we need more information to answer this question

Answer 1

In the given problem, we need to maximize utility U=x₁x₂ subject to budget constraint 2x₁+4x₂-80=0 .

Let , L = x₁x₂+(2x₁+4x₂-80)

To maximize utility, first order conditions require,

dL/dx₁= x₂ + 2 = 0 .....(1)

dL/dx₂=x₁ + 4 = 0 .......(2)

and dL/d= 2x₁+4x₂-80 = 0 .....(3)

From equations (1) and (2),

x₂/2 = x₁/4

or, x₁=2x₂ ......(4)

Using (4) in the budget equation,

2x₁+4x₂=80

or, 2x₁+2x₁=80

or, 4x₁ = 80

or, x₁=20

and x₂=x₁/2 = 20/2 = 10

Thus, the optimum consumption bundle for the consumer is x₁=20 and x₂=10.

1. If the price of x₁ increases, at the same budget, the consumer will reduce the consumption of x₁. The only valid answer is option c .

2. If the sellers of x₁ offers a discount of $10 with the purchase above $50, then the consumers will be better off.This is because consumers can now buy 25 units of x₁ at $40 (with discount of $10) and 10 units of x₂.

Thus, the new consumption bundle becomes x₁=25 and x₂=10. Thus, the consumer is better off and option B is the correct answer.