We know that, for q < 1, the optimal level of monitoring is q = rL (12- k)rD. Substituting in
the value of rL above we get q = R (14ck)rD. We therefore have that
∂SW
∂k
rD RR +(1 - k)rD∖
4Ц-----2-----++ rD - rE = 0.
The solution we obtain is
re RrD + rD2 - 8c (rE - rD)
k g = min < -------D—2— -------,, 1 > .
rD2
From this expression, we obtain that kreg < 1 for c > 8(rRrD,c) ⇔ R < 8c (rEr-rD). Otherwise,
for R > 8c (rEr^-rD), we have that kreg = 1.
The previous solution assumed that q< 1. To get the bounds on when q =1, substitute
the solution for kreg, assuming kreg < 1, into
R — (1 — kreg)rD RrD — 4c (rE — rD)
4c 2crD
From here, we see that for c > ↑rR'2r ⇔ R < 2c2rEr-rD, q < 1. Otherwise, for R >
2c2rEr-rD, q = 1 and k should be set such that q(k) = 1 ⇔ kreg = 4c+rD R. Note, however,
that for R<4c this solution would imply that kreg > 1, which is not feasible. Therefore, for
R < 4c, we obtain that kreg = 1, which implies that q = R < 1.
One final point that needs to be verified is that, for R < 2c2rEr-rD, then q = R~(1-k)rD <
1, but that for R > 8c (rEr^-rD ), we have that kreg = 1, which would imply that q = R.
Note, however, that for both of these conditions to be true at the same time requires that
8c(rE-RD) < 2c2rEr-rD. This will be satisfied if and only if 4 (rE — rD) < 2rE — rD ⇔
rE < 3rD. We can now use this in the necessary condition for q < 1, which is R < 2c2rEr-rD.
Given the restriction on rE and rD, the right hand side must be less than 2c2(2 rrD) rD = 4c.
Therefore, the joint assumption that R < 2c2rEr-rD and R > 8c (rEr^-rD) implies that R < 4c,
and consequently that q = R < 1, as desired. □
29
More intriguing information
1. he Effect of Phosphorylation on the Electron Capture Dissociation of Peptide Ions2. Linkages between research, scholarship and teaching in universities in China
3. The name is absent
4. Spectral calibration of exponential Lévy Models [1]
5. The name is absent
6. The name is absent
7. Macroeconomic Interdependence in a Two-Country DSGE Model under Diverging Interest-Rate Rules
8. TWENTY-FIVE YEARS OF RESEARCH ON WOMEN FARMERS IN AFRICA: LESSONS AND IMPLICATIONS FOR AGRICULTURAL RESEARCH INSTITUTIONS; WITH AN ANNOTATED BIBLIOGRAPHY
9. The name is absent
10. Multifunctionality of Agriculture: An Inquiry Into the Complementarity Between Landscape Preservation and Food Security