• 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.


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)

  • $\begingroup$ 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 '20 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 '20 at 22:49

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