Splitting a Grant Between Two Labs
A university research office has 120,000 soles to divide between two laboratories. Lab A promises a return modeled by R_A(x) = 4√x thousand soles, where x is the amount allocated to A in thousands of soles. Lab B promises R_B(y) = 3√y thousand soles, where y is the amount allocated to B in thousands. The total allocation must satisfy x + y = 120, with x and y both non-negative. A committee member proposes an equal split of 60,000 soles each. Another proposes maximizing total return. A third argues that because the square-root function grows more slowly as x increases, the smaller lab should receive everything.
1. Compare the total return under the equal split with the allocation that maximizes total return.
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Equal split (x = y = 60): R = 4√60 + 3√60 = 7√60 ≈ 7 × 7.746 ≈ 54.2 thousand soles.
Maximization: let f(x) = 4√x + 3√(120 − x). Differentiate: f'(x) = 2/√x − 1.5/√(120 − x). Set to zero: 2/√x = 1.5/√(120 − x), so 2√(120 − x) = 1.5√x, giving 4(120 − x) = 2.25x, i.e., 480 − 4x = 2.25x, so 6.25x = 480 and x = 76.8. Then y = 43.2.
Check: R = 4√76.8 + 3√43.2 ≈ 4(8.764) + 3(6.573) ≈ 35.06 + 19.72 ≈ 54.78 thousand soles.
Conclusion: the optimal split (about 76.8 to A and 43.2 to B) yields roughly 54.8 thousand versus 54.2 thousand for the equal split - a modest but real gain.
2. Evaluate the third committee member's argument that the smaller lab should receive everything.
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The argument is incorrect. The square-root function is concave, so marginal return falls as allocation grows, but it never becomes zero or negative for positive x. Giving everything to one lab forces the other's return to zero and abandons a positive marginal contribution.
The correct condition for maximizing a sum of concave functions under a fixed total is equal marginal returns, not equal allocations: set the derivatives equal, as done above. Because the coefficients differ (4 versus 3), the optimum is not an even split either - Lab A gets more because its coefficient is larger.
Conclusion: the optimum lies strictly between the extremes (x ≈ 76.8, y ≈ 43.2); giving all funds to one lab is dominated.