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  • We assume a two person economy, denoted by person A and B. Good y is an ordinary private good, and each person begins with an allocation of this good given by $ y^{A*} $ & $ y^{B*} $ respectively.
  • Each person may consume some y directly or devote some portion of it to the production of a single public good, x. The amounts contributed are given by $y_{s}^{A}$ and $y_{s}^{B}$, and the public good has production function defined by $$ x=f\left(y_{s}^{A}+y_{s}^{B}\right) $$

  • Resulting utilities for these two people are given: $$ \begin{array}{c}{U^{A}\left(x, y^{A *}-y_{s}^{A}\right)} \\ {U^{B}\left(x, y^{B *}-y_{s}^{B}\right)}\end{array} $$

  • The necessary conditions for efficient resource allocation in this problem consist of choosing the levels of public goods subscriptions ($ y^{A*} $ & $ y^{B*} $) that maximize, say, A’s utility for any given level of B’s utility. The Lagrangian expression is $$ \mathscr{L}=U^{A}\left(x, y^{A *}-y_{s}^{A}\right)+\lambda\left[U^{B}\left(x, y^{B *}-y_{s}^{B}\right)-K\right] $$ where K is a constant level of B's utility. The first-order conditions for max are $$ \begin{aligned} \frac{\partial \mathscr{L}}{\partial y_{s}^{A}} &=U_{1}^{A} f^{\prime}-U_{2}^{A}+\lambda U_{1}^{B} f^{\prime}=0 \\ \frac{\partial \mathscr{L}}{\partial y_{s}^{B}} &=U_{1}^{A} f^{\prime}-\lambda U_{2}^{B}+\lambda U_{1}^{B} f^{\prime}=0 \end{aligned} $$
  • A comparison of these two equations yields $$ \lambda U_{2}^{B}=U_{2}^{A} $$ and when substituting into the preceding equation the end result is: $$ \frac{U_{1}^{A}}{U_{2}^{A}}+\frac{\lambda U_{1}^{B}}{\lambda U_{2}^{B}}=\frac{1}{f^{\prime}} $$ or more simply, $$ M R S^{A}+M R S^{B}=\frac{1}{f^{\prime}} $$

For such goods, the MRS in consumption must reflect the amount of y that all consumers would be willing to give up to get one more x, because everyone will obtain the benefits of the extra x output. Hence it is the sum of each individual’s MRS that should be equated to dy/dx in production (here given by $ \frac{1}{f'} $).

Q: I follow the order of operations mathematically but don't understand the bolded statement.

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You can take MRS as the willingness to pay (Take the other good in MRS as money). If the good is a private good that costs 3, your willingness to pay should be 3 because you are buying it for yourself. However, a public good is used by all payers, so we only need the sum of the willingnesses to pay to be equal to 3. (You are simply splitting the bill)

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  • $\begingroup$ I understand thank you. If I can ask, I seem to have confused myself with some of the math. In differentiating the Lagrangian function, how is this the result: $$ \frac{\partial \mathscr{L}}{\partial y_{s}^{A}}=U_{1}^{A} f^{\prime}-U_{2}^{A}+\lambda U_{1}^{B} f^{\prime} $$? The first term is a utility function composed of two variables: $$ \left(x, y^{A *}-y_{s}^{A}\right) $$. But for example the first term is $$ U_{1}^{A} f^{\prime} $$ $\endgroup$ – aisync Feb 19 at 22:46
  • $\begingroup$ My best understanding is that it's the derivative of the objective utility function w/ respect to the 1st variable (x), multiplied by the derivative of the production function via chain rule? I don't know where the other terms come from. $\endgroup$ – aisync Feb 19 at 22:49

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