Public Economics Taxation

If the utility function is: Consumption Function:
U(c, `) = c ? ` 1-? . Y = (l + k) ah 1-a ,

Labor Supply:
Demand for a commodity x is D(q) with a decreases in q = p + t
Supply for commodity x is S(p) with an increases in p
Equilibrium is satisfied under the condition: Q = S(p) = D(p + t)
Begin from t = 0 and S(p) = D(p). So as to characterize dp/dt: the result of a tax increase on price, which regulates the load of tax:
Adjust dt to causes change in dp so that equilibrium holds:
S(p + dp) = D(p + dp + dt) ?
S(p) + S’(p)dp = D(p) + D’(p)(dp + dt) ?
S’(p)dp = D’(p)(dp + dt) ?
dp /dt = D’(p) /S’(p) – D’(p)

Therefore with the derivation above:

c ? ` 1-? = v(l + k) ah 1-a