/* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/
/* [ Created by wxMaxima version 0.7.3a ] */

/* [wxMaxima: comment start ]
Clear all symbols

   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
Kill(all)$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Definitions (1) - (3)

Relabel variables since maxima offers no support for greek letters
alpha: a
rho: r
mu: m
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
profit(w,p):= p*a - w -c;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
r:q*(1+m*log(w/W));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
a:r*x + (1-r)*y;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Zero profit price (6)

Set w=W
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
w:W;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
solve([profit(w,p)=0],p);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Clear w=W
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
kill(W,w);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Check equation (6)
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
-(W+c)/((q-1)*y-q*x) = (W+c)/(q*(x-y)+y);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Check if both expressions are equal after simplifying
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
is(ratsimp(%));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Selection wage curve (7) - (9)
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
W:w;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
Dprofit(w,p):= diff(profit(w,p),w);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
D2profit(w,p):=diff(profit(w,p),w,2);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
Dprofit(w,p) = (p*q*m*(x-y)*1/w-1);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
is(ratsimp(%));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
D2profit(w,p) = (-p*q*m*(x-y)*1/(w^2));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
is(ratsimp(%));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
solve([Dprofit(w,p)=0],p);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
-w/(m*q*y-m*q*x) = w/(m*q*(x-y));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
is(ratsimp(%));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Equilibrium (10), (11)
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
solve([profit(w,p)=0, Dprofit(w,p)=0],[w,p]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Check w
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
-(c*m*q*y-c*m*q*x)/(((m-1)*q+1)*y+(1-m)*q*x) =
 (m*q*(x-y))/(q*(x-y)*(1-m)+y)*c;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
is(ratsimp(%));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Check p
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
c/(((m-1)*q+1)*y+(1-m)*q*x) = c/(q*(x-y)*(1-m)+y);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
is(ratsimp(%));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Stability (12)
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
kill(w,W);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
solve([p*q*m*(x-y)*1/w-1=0],w);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
p:(W+c)/(q*(x-y)+y);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
w:m*p*q*x-m*p*q*y;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
w = (m*(x-y)*q)/((x-y)*q+y)*(W+c);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
is(ratsimp(%));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
W_:(m*q*(x-y))/(q*(x-y)*(1-m)+y)*c;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
w-W= -((1-m)*(x-y)*q+y)/((x-y)*q+y)*(W-W_);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
is(ratsimp(%));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Inequality (13)
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
kill(W,p);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Condition is given wrongly in the manuscript. 
It should relate to mediocre productivity y 
rather than prolific productivity x and read

py > W

and inequalities (13) should read

y > qxm/(1+qm) and mq < y/(x-y)

Replacing the inequality with equality, 
I check whether these inequalities are equivalent to py > W.



   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
p_:c/(y+q*(x-y)*(1-m));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
solve([p_*y=W_],y);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
solve(p_*y=W_,q);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Quality mix (14) and (15)
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
diff(W_, q);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
is(ratsimp(%=(c*m*(x-y)*y)/(y+(1-m)*q*(x-y))^2));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
diff(p_,q);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
is(ratsimp(%=-(1/(q*(x-y)*(1-m)+y))^(2)*c*(x-y)*(1-m)));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
(c*q*(x-y)*(q*(x-y)+y)*m)/((c+W)*(y-q*(x-y)*(1-m)));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]

   [wxMaxima: comment end   ] */

/* [wxMaxima: comment start ]
*****APPENDIX: Performance Pay*****;
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
kill(p,W,w);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
The profit function (17) is 
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
prof(w,p):=p*(r*x+(1-r)*y)-r*w-(1-r)*(y/x)*w-c;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
r:q*(1+m*log(w/W));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
prof(w,p);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
The derivative of prof(w,p) with respect to w is
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
diff(prof(w,p),w);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
Dprof(w,p):=%o47;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
Dprof(w,p);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
W:w$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
solve([prof(w,p)=0, Dprof(w,p)=0],[w,p]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Check that expression (18) for w is correct

   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
-(c*m*q*x*y-c*m*q*x^2)/((q^2-2*q+1)*y^2+(2*q-2*q^2)*x*y+q^2*x^2)
=(c*m*q*x*(x-y))/((1-q)*y+q*x)^2;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
is(ratsimp(%));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Compare flat pay with performance pay
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
kill(W);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
W_t:c*q*x*m*((x-y)/(q*x+(1-q)*y)^(2));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
V_t:(y/x)*W_t;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
W_:((m*q*(x-y))/(q*(x-y)*(1-m)+y))*c;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
(W_-(q*W_t+(1-q)*V_t))=
(c*q^(2)*(x-y)^(2)*m^(2))/((q*x+(1-q)*y)*(q*(x-y)*(1-m)+y));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
is(ratsimp(%));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
Wage Compression
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
kill(all);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
The Lagrangean is
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
L(w,v,l):=q*(1+m*log(w/W))*(p*x-w)+(1-q)*(1+m*log(v/V))*
(p*y-v)-c-l*(q*log(w/W)+(1-q)*log(v/V));
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
The derivatives are  
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
L(w,v,l);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
diff(L(w,v,l),w);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
DLw(w,v,l):=%o3;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
diff(L(w,v,l),v);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
DLv(w,v,l):=%o5;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: comment start ]
We evaluate these expressions at W = w and V = v
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
W:w;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
V:v;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
solve([DLw(w,v,l)=0], w);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
w_:(m*p*x-l)/(m+1);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
solve([DLv(w,v,l)=0], v);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
v_:(m*p*y-l)/(m+1);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
ratsimp(w_-v_)=(p*(x-y)*m)/(1+m);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
is(ratsimp(%));
/* [wxMaxima: input   end   ] */

/* Maxima can't load/batch files which end with a comment! */
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