sanity check

ClearAll[y, yd, td, cd, hs, cs, gs, ts, hh, dHs, dHh];
eqns = {y[t] == cs[t] + gs[t],dHs[t] == hs[t] - hs[t - 1],dHh[t] == hh[t] - hh[t - 1],cs[t] == cd[t],gs[t] ==     gd[t],ts[t] == td[t],cd[t] == c1 * yd[t] + c2 * hh[t - 1],td[t] == \[Theta] * y[t - 1],yd[t] == y[t] - \[Theta] * y[t - 1],y[0] == 0, cs[0] == 0,yd[0] == 0, hh[0] == 0,hs[0] == 0, td[0] == 0, gs[0] == 0,gs[1] == g };(*all your recurrences*)
vars = {y, yd, td, cd, hs, cs, gs, ts, hh, dHs, dHh};
init = {y[0] == 0, cs[0] == 0, yd[0] == 0, hh[0] == 0, hs[0] == 0, td[0] == 0, gs[0] == 0, dHh[0] == 0, gs[1] == 20};
RecurrenceTable[Join[eqns, init], vars, {t, 1, 10}]

Still gives -

There are more dependent variables than equations, so the system is underdetermined.

So I rewrote it, think I got it correct (, but hope it helps).

ClearAll[y, taxes, yd, c, h, hg, g];
Manipulate[
 Module[{sim},
  sim =
   RecurrenceTable[{
     g[t] == Piecewise[{{g1, t <= switchTime}, {g2, t > switchTime}}],
     taxes[t] == taxRate y[t],
     yd[t] == y[t] - taxes[t],
     c[t] == income yd[t] + wealth h[t - 1],
     y[t] == c[t] + g[t],
     h[t] == h[t - 1] + yd[t] - c[t],
     hg[t] == hg[t - 1] + g[t] - taxes[t],
     h[0] == 0,
     hg[0] == 0},
    {y, c, yd, taxes, h, hg, g},
    {t, 1, periods}];
  ListLinePlot[
   sim[[All, #]] & /@ {1, 2, 5, 7},
   PlotLegends -> {"y", "c", "h", "g"},
   PlotRange -> All,
   AxesLabel -> {"Time", "Value"},
   ImageSize -> Large]]
 , {{g1, 20, "initial g"}, 1, 30, 1, Appearance -> "Labeled"},
      {{g2, 30, "new g"}, 1, 50, 1, Appearance -> "Labeled"},
      {{taxRate, 0.2, "tax rate \[Theta]"}, 0., 1., 0.05, 
  Appearance -> "Labeled"},
      {{income, 0.6, "income \[Alpha]1"}, 0., 0.99, 0.05, 
  Appearance -> "Labeled"},
      {{wealth, 0.4, "wealth \[Alpha]2"}, 0., 1., 0.05, 
  Appearance -> "Labeled"},
      {{switchTime, 20, "switch time"}, 1, 49, 1, 
  Appearance -> "Labeled"}, {{periods, 50, "periods"}, 20, 100, 1, 
  Appearance -> "Labeled"}]

Thanks to all. I haven't been using Show Expression. This gives me some critical information I haven't used before, not only this issue but some from past modules where I fat-fingered some entry.

You use g as a function in RecurrenceTable and as a number in Manipulate. I tried this:

RecurrenceTable[{y[t] == Subscript[c, s][t] + Subscript[g, s][t], 
    Subscript[\[CapitalDelta]h, s][t] == 
     Subscript[h, s][t] - Subscript[h, s][t - 1], 
    Subscript[\[CapitalDelta]h, h][t] == 
     Subscript[h, h][t] - Subscript[h, h][t - 1], 
    Subscript[c, s][t] == Subscript[c, d][t], 
    Subscript[g, s][t] == Subscript[g, d][t], 
    Subscript[c, d][t] == c1 yd[t] + c2 Subscript[h, h][t - 1], 
    td[t] == \[Theta] y[t - 1], yd[t] == y[t] - \[Theta] y[t - 1], 
    y[0] == 0, Subscript[c, s][0] == 0, yd[0] == 0, 
    Subscript[h, h][0] == 0, Subscript[h, s][0] == 0, 
    td[0] == 0} /. {Subscript[g, s] -> g, \[Theta] -> taxRate, 
    c1 -> consumeRate, c2 -> overspend} /. {taxRate -> 1, 
   consumeRate -> 1, overspend -> 1}, {g, td, y, yd, Subscript[c, d], 
  Subscript[c, s], Subscript[g, d], Subscript[h, h], Subscript[h, s], 
  Subscript[\[CapitalDelta]h, h], Subscript[\[CapitalDelta]h, s]},
 {t, 1, 50}]

and it gives this warning:

RecurrenceTable::underdet: There are more dependent variables than equations, so the system is underdetermined.