Wondering how this total differentiation is arrived at

Econ 4021 Dana Galizia, Carleton University investment, so that the total output available to be consumed by households is zzf (n2, k2), while these households demand c2 units. Again, in equilibrium these must equal one another. Substituting nt = 1 into the goods market-clearing conditions and into the firm's FOC for k2 (i.e., (13)), we can summarize the equilibrium of our model with the following three equations: c1 = zIf (1, k1) - k2 , (14) C2 = zzf (1, k2) (15) 1 = Bu' (C2) u' (CI) 2 z2fk (1, k2) * Total Diff. (16) This is three equations in three endogenous variables (C1, C2, and k2), which can in principle be solved. Given such a solution, we can then obtain the remaining endogenous variables as follows. (12) gives wt, t = 1,2. (9) and (10) then give di and d2, and then (7) then gives p1. 3 Comparative statics also : LN3 ( if discount see rate changes Totally differentiating (14)-(16), allowing only 21, z2, and the endogenous variables (C1, (2, k2) to change, we get de1 = f (1, ki) dzi - dk2 , (17) dc2 = f (1, k2) dz2 + zzfk (1, k2) dkz , (18) 0 = 0 (c1) de1 - o (c2) dez + - dzz + Jkk (1, k2) dkiz , 2 2 fk ( 1, k2 ) (19) where as in LN3 o(c) = -u"(c)/u'(c) > 0, and we've simplified (19) by using (16). Substituting (17) and (18) into (19) for ci and c2, respectively, we can then solve to obtain dk, = (ci) f (1,kidz + 22 1 - 0 (c2 ) f ( 1 , k2 ) 5 5 -dzz , (20) where we've defined S= 0 (C1) to (c2) zzfk (1, k2) _fkk (1, k2) fke (1, k2) > 0 , where the sign of ( follows from the fact that o(c) > 0, fk > 0, and fkk 0), with no change in z2. Setting dz2 = 0 in (20), we get dk2 = " (ci) f (1, k1) dzi , 5 (21)