Consider a resource-based economy which can allocate labor (L) to harvest timber

Question

Consider a resource-based economy which can allocate labor (L) to harvest timber (T) or fish (F). Assume the economy faces constant world prices for timber and fish, denoted Pt and Pf, respectively. Labor is constrained by the following equation: L=T^2+(F^2/2). Further, suppose Pt= $500/ton and Pf=$100/ton and L= 1700 available hours.

How should labor be allocated to timber and fish production to maximize the one- period value (V) of resource production? (Note: PtT+PfF ).

What is the marginal value (shadow price) of an additional unit of labor?

Explanation / Answer

So, T^2+(F^2/2)=1700.

We have to Maximise P = T x 500 + F x 100.

When F is zero, then T^2 = 1700 or T = 41.23 tonn.

and when T = zero F^2/2 = 1700 or F = 58.31.

Assuming that there is no Volume limit to produce either Timber or Fish.

When T = 41.23 P = 41.23 x 500 = $20,615.53

and When F = 58.31 P = $5,831

So, 1,700 hour should be used to produce 41.23 tons of Timber.

Hence price per hour = 20615.53/1700=12.13

Shadow price is 12.13.

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