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Appendix 1— Derivation of model III from model II 



We start with model II 



"SP_DEPM ~ K, )""•-> + e r 



For brevity, y is used to denote PPS. We first note that 

 y'~=yy"'~ 1 '- Now we obtain the Taylor series expansion of 

 yU-D about. v=l giving 



Because y is a proportion, it satisfies 0<y<l so that 

 the higher powers of {y— 1) will be individually and col- 

 lectively small. If y is close to 0.5, a further algebraic 

 simplification of the second term is possible, giving the 

 identity 



v(y-l) = -0.25 + (v-0.5) 2 . 



v i; '"= 1 + (A-1)(v-1)-mA-1)(A-2i/2'<y-I) : 

 +(A-l)(A-2)(A-3)/3!(.v-l)-' + ... . 



Multiplying this expression through by y gives 



y l = y + _y(_v-l)(A-l) 

 + ytterms involving second and higher powers of t v— 1 )]. 



When v is in the range 0.25<v<0.75, the right-hand side 

 of this identity remains close to -0.25. Thus model II 

 may be simplified to 



B sp pps = a„(PPS) - 0.25a„( A-l ) + <?, 



which is the form of model III. 



