(*Basic Model*)

n[t_]:=Exp[λ t]
bigI[t_]:= I0 Exp[γ t]
i[t_]:=D[bigI[t],t]
p[t_]:=p0(bigI[t]/I0)^α n[t]^β

bigI[t]

(*Total Cost*)

c0*n[tf] + Integrate[n[t](c-p[t]+i[t]),{t,0,tf}]

(*Solve for tf*)

Solve[c==p[t],t]

(*Plug in tf to total cost T*)

FullSimplify[(c0 (E^(tf λ))+((c (-1+(E^(tf λ))))/λ)+(((-1+(E^(tf (γ+λ)))) I0 γ)/(γ+λ))-(((-1+((((E^(tf γ))))^α) ((((E^(tf λ))))^(1+β))) p0)/(α γ+λ+β λ))/.tf->((Log[(c/p0)])/(α γ+β λ))/.c->p0*r)]

(*Approximate by eliminating small quantities associated with startup*)

FullSimplify[((p0 r ((r^(λ/(α γ+β λ)))))/λ)+((I0 ((r^((γ+λ)/(α γ+β λ)))) γ)/(γ+λ))+((-p0 ((((r^(γ/(α γ+β λ)))))^α) ((((r^(λ/(α γ+β λ)))))^(1+β)))/(α γ+λ+β λ))]

(*Simplify in a way Mathematica doesn't want to do*)

FullSimplify[(r^(λ/(α γ+β λ))) (((p0 r)/λ)+((I0 (r^(γ/(α γ+β λ))) γ)/(γ+λ))-((p0 r)/(α γ+λ+β λ)))]

Simplify[(r^(λ/(α γ+β λ))) (((I0 (r^(1/(α ))) γ)/(γ+λ))+p0 r ((1/λ)-(1/(α γ+λ))))/.γ->q*λ/.β->ω α]

Manipulate[Plot[k^(1/q)(a q/(1+a q))+w k(q/(1+q)),{q,0,10}],{k,10,100},{a,0,1},{w,0,100}]