Question 6: Suppose Connie has preferences over bread (good 1) and butter (good 2) 1/2 1/2 U (q1, 42) %3D The price of bread is p1 = 2 and the price of butter is p2 = 2 and her income is Y = 20. Due to a wheat shortage, the government has stipulated a quota Q which denotes the maximum number of units of bread that any one consumer may purchase. (1) Draw Connie's budget set for the case that Q = 8 and for the case that Q = 4. (2) Suppose first that Q = 8. Find Connie's optimal choice. (3) Now suppose that Q 4. Find Connie's optimal choice. Use graphs to illustrate your answer.
The microeconomic theory of utility maximization states how consumer chooses his consumption bundle such that his highest satisfaction level is maximized. However, while making such choice, an individual is restricted by his money income as economic goods are to be purchased at market prices. In such theory, the utility level is represented graphically by the indifference (IC) curve and the budget constraint represents the affordability of the consumer.
In this case, the consumer’s budget constraint is given by,
Y = p1*q1 + p2*q2
20 = 2q1 + 2q2
The utility maximization problem is,
Max U(q1,q2) subject to 20 = 2q1 + 2p2
The objective function is,
U = q1^(1/2)*q2^(1/2) + λ [20 - 2q1 - 2q2]
The FOC is,
∂U/∂q1 = 1/2*(q2/q1)^(1/2) -2 λ = 0 ⇒ 1/2*(q2/q1)^(1/2) = 2 λ ..........(i)
∂U/∂q2 = 1/2*(q1/q2)^(1/2) -2 λ = 0 ⇒ 1/2*(q1/q2)^(1/2) = 2 λ ..........(ii)
∂U/∂λ = 20 - 2q1 - 2p2 = 0 ⇒ 20 = 2q1 + 2p2 ....................................(iii)
From (i)&(ii),
[1/2*(q2/q1)^(1/2)] / [1/2*(q1/q2)^(1/2)] = 1
⇒ q2 = q1 ...............(iv)
Putting q2 = q1 in (iv),
20 = 2q1 + 2q1
⇒ 20/4 = q1
⇒ q1 = 5 units
So, q1 (bread) = q2 (butter) = 5 units. This implies a utility-maximizing consumer (Connie) purchases 5 units of each good to maximize his utility subject to the income constraint.
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