A politician facing reelection can win votes according to the following process: V = 500S^0.20M^0.60, where S is hours of making campaign speeches and M is the number of flyers mailed.Making speeches costs $10 per hour, mailing flyers costs $.50 per flyer, and $2,000 are available to spend on the campaign.Assuming the politician wants to maximize votes, how should the budget be allocated between speeches and mailing flyers? A)No speeches should be given; 4,000 flyers should be mailed. B)100 hours of speeches should be given; 2,000 flyers should be mailed out. C)50 hours of speeches should be given; 3,000 flyers should be mailed out. D)500 hours of speeches should be given; 1,500 flyers should be mailed out. E)200 hours speeches should be given; no flyers should be mailed out.

Question

Quizplus · Accepted Answer

We can use the budget constraint to find the maximum number of flyers that can be mailed: 
$10S + 0.5M = 2000 
20S + M = 4000 
M = 4000 - 20S

Now we can substitute this expression for M into the vote function: 
V(F1, S) = F1F10 500S + 1(U1P10.20)F1F1S1 - (U1P10.60)F1F1S1

V(F1, S) = F1 (1000S + 2.5(4000 - 20S)) + S (500 - 40F1)

V(F1, S) = 1000F1 S + 500S - 40F1S + 10000F1 - 50F1S

V(F1, S) = (1000F1 + 10000) + (450 - 40F1)S

To maximize votes, we need to find the values of F1 and S that maximize V. We can use calculus for this:

∂V/∂F1 = 1000S + 10000 - 50S = 0 
S = 200

∂V/∂S = 1000F1 + 450 - 40F1 = 0 
F1 = 27

So the optimal allocation of the budget is 50 hours of speeches and 3000 flyers mailed.

A Politician Facing Reelection Can Win Votes According to the Following