I'm studying public economics but my question here is purely mathematical in nature. I have a function:

$$ V(1-\tau, R) = \max_zu((1-\tau)z+R,z) $$

I need to take the total derivative of this, in my notes, I'm told it is equal to

$$ dV=u_c [-zd\tau + dR] + dz[(1-\tau)u_c + u_z] $$

where $c^i=(1-\tau)z^i+R$, $\tau$ is the tax rate, R is government revenue, z is income, V is the indirect utility function.

and the notation $u_c$ denotes the derivative of u with respect to c.

Is the first bracket the partial derivative with respect to c and the second the partial derivative with respect to z?

  • 2
    $\begingroup$ Hint: $\mathrm df(x,y,z)=f_x(x,y,z)\mathrm dx+f_y(x,y,z)\mathrm dy+f_z(x,y,z)\mathrm dz$. The first bracket results from the first two summands. $\endgroup$
    – Herr K.
    Jan 6 at 17:12

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