I am currently reading the book "Microeconomics: Principles and Analysis" by Cowell (2006), page 452-453. e have a two-commodity world, in which there are ${n_h}$ agents (households): commodity 1 is a pure public good and commodity 2 is purely private.
Each agent has an exogenously given income ${y^h}$ , denominated in units of the private good 2. We imagine that the public good is to be financed voluntarily, each household makes a contribution ${z^h}$ which leaves
$$x_2^h = {y^h} - {z^h}$$ of the private good available for h’s own consumption and $${x_1} = \phi (\bar z + {y^h} - x_2^h)$$ where z is the total input of the good 2 used in the production process. Each agent realises that the total output of the public good depends upon his or her own contribution and upon that made by others
The expression (13.30)
The expression (13.31)
Did he come to expression (13.33) by applying the chain rule as
$$\eqalign{ & z = U({x_1},x_2^h) \cr & {x_1} = g({x_2}) \cr & {{dz} \over {d{x_2}}} = {{\partial f} \over {\partial {x_1}}}{{d{x_1}} \over {d{x_2}}} + {{\partial f} \over {\partial {x_2}}}{{d{x_2}} \over {d{x_2}}} = {{\partial f} \over {\partial {x_1}}}{{d{x_1}} \over {d{x_2}}} + {{\partial f} \over {\partial {x_2}}} \cr} $$
since ${x_1}$ is a function of $x_2^h$ ?