plotting – Manipulate and Plot of Tangent Point in Optimization Problem: Solve Problems

I want to illustrate how changes in the values of exogenous variables and parameters (T,w,(Alpha)) are changing the optimal values of two endogenous variables (f,c)=(f*,c*). The solution is with a tangency condition and a constraint.

Changes in alpha should move the U-graph along the Bcon-graph; changes in T and w change the Bcon-graph and therefore the optimal values of f and c as well as the U-graph.

U = f^(Alpha)*c^(1 - (Alpha)) 
Bcon = c - (T - f)*w
MRS = D(U, f)/D(U, c)
AbsSlpCon = D(Bcon, f)
TC = MRS - AbsSlpCon
sols = Solve({TC == 0, Bcon == 0}, {f, c})
{SuperStar(f), SuperStar(c)} = {f, c} /. Last(sols)
c1(T_, w_) := c /. Solve(c - (T - f)*w == 0, c)
c2(T_, w_, (Alpha)_) := c /. Solve(U(SuperStar(f), SuperStar(c)) == U(f, c), c)
Manipulate(Plot({c1(T, w), c2(T, w, (Alpha))}, {f, 0, 24}, PlotRange -> {25, 3000}), {T, 8, 24}, {w, 100, 500}, {(Alpha), 0, 1})

Unfortunately,

  1. I cannot use Bcon in line 8 to describe c1(T_,w_) but have to copy the function there to get a linear graph in the plot;

  2. get no output for c2(T_, w_, (Alpha)_) in line 9, which is showing the tangent U-graph on the Bcon-graph.

“Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.”

Any hints or suggestions?

Thanks!