Consider the model from but slightly modified, Rt+1 = Rteρ(1−Rt/k). Consider the model from Q1 but slightly modified, Rt+1 = Rteρ(1−Rt/k).

Natural Resource Mathematics

Consider the model from Q1 but slightly modified,
    Rt+1 = Rteρ(1−Rt/k). (2)
(a) Write down an expression for ρ in terms of r, such that the model from Q1
is mathematically equivalent to the model.
(b) Now we introduce noise, in the following way. Consider
Rt+1 = ztRteρ(1−Rt/k), (3)
where zt is a random variable such that log (zt) ∼ N(0,σ2). Assume that k
is fixed. Find the maximum likelihood estimate of ρ, in terms of k and the
given time series data, {R0,R1,…,Rn}.
(c) From parts (a) and (b), write down an estimate for r. Call it ˆrMLE