Public Goods - Voluntary provision

Question

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

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The expression (13.30)

The expression (13.31)

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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 $x_2^h$ ?

Answer 1

Given

$$\max_{x_2^h} \quad U^h(\phi(\bar z + y^h - x_2^h), x_2^h)$$

You have to find FOCs by taking the chain rule twice for the first argument. Denote the argument for function $\phi$ as $(\cdot)$.

$$\frac{\partial U^h}{\partial x_2^h} = \frac{\partial U^h}{\partial \phi} \cdot \frac{\partial \phi}{\partial x_2^h}$$ $$\frac{\partial \phi}{\partial x_2^h} = \frac{\partial \phi}{\partial (\cdot)} \cdot \frac{\partial (\cdot)}{\partial x_2^h}$$ $$\implies \frac{\partial U^h}{\partial x_2^h} = \frac{\partial U^h}{\partial \phi} \cdot \frac{\partial \phi}{\partial (\cdot)} \cdot \frac{\partial (\cdot)}{\partial x_2^h}$$

Thus, letting $U^h_{1,2}$ represent the partial derivative with respect to the first and second arguments of $U^h$ respectively:

$$\begin{align} \frac{\partial U^h}{\partial x_2^h} & = U_1^h(\phi(\bar z + y^h - x_2^h), x_2^h) \cdot \phi_z(\bar z + y^h - x_2^h) \cdot -1 \\ & + U_2^h(\phi(\bar z + y^h - x_2^h), x_2^h) \cdot 1 \end{align}$$

The second argument in $U$ has no $\phi$, so only chain rule should be applied there.

Answer 2

Cant you just apply this

Obviously different values but same principle ?