Consider a two good economy where a denotes consumption of apricots and b denotes consumption of bananas. Anu's utility function is UAnu (a, b) = a + 2b, and Binu's utility function is UBinu (a, b) = min{a, 2b}. Anu initially has no apricots and 12 bananas. Binu initially has 12 apricots and no bananas. In the competitive equilibrium, which one of the following will be Anu's optimal consumption bundle ?
- 6 apricots and 9 bananas
- 9 apricots and 9 bananas
- 4 apricots and 10 bananas
- 0 apricots and 12 bananas
The Correct Option is A
Solution and Explanation
Step 1: Utility functions
Anu's utility function: \[ U^{Anu}(a,b) = a + 2b \] Binu's utility function: \[ U^{Binu}(a,b) = \min\{a,\,2b\} \] Initial endowments: - Anu: \((0,\,12)\) - Binu: \((12,\,0)\) Total resources: \((12,\,12)\).
Step 2: Anu’s preferences
For Anu, the marginal utilities are: \[ MU_a = 1, \quad MU_b = 2 \] So, her marginal rate of substitution (MRS) is: \[ MRS_{Anu} = \frac{MU_a}{MU_b} = \frac{1}{2}. \] This means Anu values bananas twice as much as apricots.
Step 3: Binu’s preferences
Since \[ U^{Binu}(a,b) = \min\{a,2b\}, \] Binu will consume in the fixed proportion: \[ a = 2b. \] Thus her consumption always lies on the line \(a=2b\).
Step 4: Competitive equilibrium
In equilibrium, markets clear: \[ a^{Anu} + a^{Binu} = 12, \quad b^{Anu} + b^{Binu} = 12. \] Also, Binu’s bundle must satisfy \(a^{Binu} = 2b^{Binu}\).
Step 5: Testing feasible allocations for Anu
Candidate allocations for Anu (a,b):
- (6,9) → \(U^{Anu} = 6 + 2(9) = 24\)
- (9,9) → \(U^{Anu} = 9 + 2(9) = 27\)
- (4,10) → \(U^{Anu} = 4 + 20 = 24\)
- (0,12) → \(U^{Anu} = 24\)
Among these, (9,9) gives the highest utility (27), but violates Binu’s consumption rule since that leaves (3,3) for Binu, which does not satisfy \(a=2b\). The feasible bundle that satisfies Binu’s proportion condition is **(6,9)**, leaving (6,3) for Binu, which lies on her \(a=2b\) line.
Final Answer:
Anu’s optimal consumption bundle in competitive equilibrium is: \[ \boxed{(6 \;\; \text{apricots}, \; 9 \;\; \text{bananas})} \]
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