How do you make two interactive sliders dependent on eachother?

Question

I have this code.

Definition of q is:

α=0.86;

q[x_,y_,t_,u_,f_]:=(((1-α) (y-t*x-f))/(((α^α)((1-α)^(1-α)) (y-t*x-f))/u)^(1/(1-α)));

Manipulate[
 Plot[{q[x, y, t, u, f]}*30000, {x, 0, 40}, PlotRange -> {0, 1000}, 
  PlotStyle -> {Thickness[0.008]}, AxesLabel -> {"x", "q(x)"}, 
  Filling -> Bottom, 
  PlotLabel -> 
   "Housing Consumption (floor space per dwelling in sqm) "], {y, 
  70000, 100000}, {t, 600, 800}, {u, 5633.58, 6000}, {f, 0, 50000}]

What do I do, if I want f and t to be dependent on each other? For example, if I move up one, the other goes down using a function.

Answer 1

You may use TrackingFunctionand InverseFunction in Manipulate.

Say you have some function w that maps f to t

w[f_] := 800 - 200/50000 f
Plot[w[x], {x, 0, 50000}]

When with definitions in OP

Manipulate[
 Plot[{q[x, y, t, u, f]}*30000
  , {x, 0, 40}
  , PlotRange -> {0, 1000}
  , PlotStyle -> {Thickness[0.008]}
  , AxesLabel -> {"x", "q(x)"}
  , Filling -> Bottom
  , PlotLabel -> 
   "Housing Consumption (floor space per dwelling in sqm)"
  ]
 , {{y, 100000}, 70000, 100000}
 , {{t, 800}, 600, 800
  , TrackingFunction -> (t = #; f = InverseFunction[w][#]; &)
  , Appearance -> "Labeled"}
 , {{u, 6000}, 5633.58, 6000}
 , {f, 0, 50000
  , TrackingFunction -> (f = #; t = w[#]; &)
  , Appearance -> "Labeled"}
 ]

In fact, you can evaluate multiple relationships in the manipulate with the addition of a relationship function paramerter g.

Say there is another relationship, w2.

w2[f_] := 600 + (100/π) (Sin[(2 π f)/50000] + (2 π f)/50000) 
Plot[w2[x], {x, 0, 50000}, ImageSize -> 250]

Then

Manipulate[
 Plot[{q[x, y, t, u, f]}*30000
  , {x, 0, 40}
  , PlotRange -> {0, 1000}
  , PlotStyle -> {Thickness[0.008]}
  , AxesLabel -> {"x", "q(x)"}
  , Filling -> Bottom
  , PlotLabel -> 
   "Housing Consumption (floor space per dwelling in sqm)"
  ]
 , {{y, 100000}, 70000, 100000}
 , {{t, 800}, 600, 800
  , TrackingFunction -> (t = #; f = InverseFunction[g][#]; &)
  , Appearance -> "Labeled"}
 , {{u, 6000}, 5633.58, 6000}
 , {f, 0, 50000
  , TrackingFunction -> (f = #; t = g[#]; &)
  , Appearance -> "Labeled"}
 , {{g, w}, {w, w2}}
 ]

Hope this helps.

Answer 2

Clear["Global`*"]

$Version

(* "12.3.1 for Mac OS X x86 (64-bit) (June 19, 2021)" *)

α = 0.86;

q[x_, y_, t_, u_, 
   f_] := (((1 - α) (y - t*x - 
        f))/(((α^α) ((1 - α)^(1 - α)) (y - t*x - 
           f))/u)^(1/(1 - α)));

Assume that the relation between f and t is linear

f == a*t + b /. 
 Solve[f == a*t + b /. {{f -> 0, t -> 800}, {f -> 50000, t -> 600}}, {a, 
    b}][[1]]

(* f == 200000 - 250 t *)

Since they move in opposite directions, the starting values are at opposite sides of their respective range.

fOld = 0;
tOld = 800;

Manipulate[
 If[f == fOld && t != tOld,
  f = 200000 - 250 t; fOld = f; tOld = t,
  If[t == tOld && f != fOld,
   t = (200000 - f)/250; tOld = t; fOld = f]];
 Plot[{q[x, y, t, u, f]}*30000, {x, 0, 40},
  PlotRange -> {0, 1000},
  PlotStyle -> {Thickness[0.008]},
  AxesLabel -> {"x", "q(x)"},
  Filling -> Bottom,
  PlotLabel ->
   "Housing Consumption (floor space per dwelling in sqm)"],
 {{y, 77000}, 70000, 100000, 100, Appearance -> "Labeled"},
 {{t, 800}, 600, 800, 1, Appearance -> "Labeled"},
 {{u, 5633.58}, 5633.58, 6000, 0.01, Appearance -> "Labeled"},
 {{f, 0}, 0, 50000, 100, Appearance -> "Labeled"},
 TrackedSymbols :> All]