Profit max Varian workbook qn

[L14]

Pls can someone help me understand how to solve qn the below

Varian workbook 19.3 d)
Brother Jed takes heathens and reforms them into righteous
individuals. There are two inputs needed in this process: heathens (who
are widely available) and preaching. The production function has the
following form: rp = min{h, p}, where rp is the number of righteous
persons produced, h is the number of heathens who attend Jed’s sermons,
and p is the number of hours of preaching. For every person converted,
Jed receives a payment of s from the grateful convert. Sad to say, heathens
do not flock to Jed’s sermons of their own accord. Jed must offer heathens
a payment of w to attract them to his sermons. Suppose the amount of
preaching is fixed at ¯p and that Jed is a profit-maximizing prophet.

(d) If w < s, how many heathens will be converted? If w > s,
how many heathens will be converted? .

[Arushi :))]

First thing to note here is that p is fixed,
So this person will employ heathens till it reaches the fixed amount of p.
Let p be fixed at 3.
then r= min {h,3}
We see that increasing heathen is increasing output till h=3.
r= h in each case till h=3
we can therefore write profit= h*s-h*w = h(s-w)
Now, if w<s , then he employs heathens upto the point where h= fixed no. of preachings because increasing h increases profit. At that point,
r = fixed amount of p. so that fixed amount of p number of heathens are converted.
So , maximum profit is achieved when h = fixed no of preachings.
But if w>s then profit is always negative, hence the profit maximizing level is zero in this case .

[L14]

Thanks Arushi ..very good intution :)

[Arushi :))]

You are welcomeeeeee :)