Factor Demand/Monopsony Problem

I'm a little bit confused as to how to calculate the factor demand for labour for an individual firm. The question is as follows:
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Firms in a competitive industry sell output at price p=10 and use labour as their only input. One unit of labour is required to produce one unit of output. The supply curve of labour is L=w-1 (where w is the wage)

(a) Find the equilibrium wage, the profits of firms and the surplus enjoyed by
workers.

(b) If the government taxes wages at a rate of 50% find how your answers to part (a) change, and find the revenue raised by the government.

Suppose now instead that there is a single firm which acts as a monopsonist.

(c) Find how your answers to (a) and (b) change.

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From the question:
q = L
p = 10
L = w – 1
w = L + 1
My working for (a):
π = R - wL – F
π = pq – (L+1)L - F
dπ/dL = (dR/dq)(dq/dL) – 2L -1 = 0
2L = 10 – 1 (since dR/dq = p for competitive firm, and dq/dL = 1 given in question).
L = 4.5.
W = 5.5.
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But then an alternative route, where I don’t substitute w for L before differentiating gives a different answer:
π = R - wL – F
dπ/dL = (dR/dq)(dq/dL) – w = 0
w = 10
L = 9.
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I swear the first route is the calculation for Marginal Expenditure = Marginal Benefit, and hence, the calculation for monopsony demand? But then the alternative route doesn’t really seem right either?