<p>In the given scenario, the optimal social outcome from a Coasian bargain is not in general the same as the optimal social outcome from a Pigouvian tax, although there is a special case in which this is so.</p>
<p>The scenario outlined in this question has several features that make it rather more complicated than some textbook discussions of policy towards negative externalities.</p>
<p>a)  It focuses not on benefits and costs but on utility functions (cf (A)).</p>
<p>b)  It considers optimality, defined as a state in which the sum of the utility of the two parties is maximised, rather than efficiency  (a state in which the utility of one party cannot be increased without lowering that of the other, also known as Pareto-optimality) (cf (B)).</p>
<p>c)  No direct link is specified between utility and any monetary quantity (cf (C)). Payment or receipt of money by either party affects utility indirectly via the quantity each can afford of a second good not subject to an externality.</p>
<p>d)  It uses general utility functions with no restrictions on their functional form.</p>
<p>e)  Although both parties can consume the second good, as a consequence of (d) above a given quantity of that good does not necessarily affect each party’s utility in the same way.  It is possible therefore that a monetary transfer between the parties (whether via tax revenue paid on to the other party, or a Coasian bargain) could raise or lower the sum of utility not for reasons relating to the externality but simply by transferring spending power to or from the party which gains more utility from the same quantity of the good.</p>
<p>This answer is concerned, as the question asks, with social optima.  Whether a government is likely to set a tax at a rate that yields a social optimum, and whether Coasian bargaining is likely to lead to an agreement that yields a social optimum, are further questions which I do not address. I make one simplifying assumption not in the question, namely, that the utility functions are additively separable, so that cross-partial derivatives can be ignored.</p>
<p><strong>Case 1: <span class="math-container">$a$</span>’s consumption of <span class="math-container">$y$</span> taxed at a rate <span class="math-container">$t$</span>, the use of the tax revenue being considered outside the scope of the optimisation</strong><br />
This is the case considered in the first half of the question.  The analysis of <span class="math-container">$a$</span>’s optimisation is correct, but this is how I would specify the social optimisation:</p>
<p><span class="math-container">$\max\limits_{x_a,y_a,t} \ u_a(x_a,y_a)+u_b(x_b,y_b)$</span></p>
<p>subject to  <span class="math-container">$\ p_xx_a+(p_y+t)y_a=e_a\qquad(1)$</span></p>
<p>and  <span class="math-container">$\ \dfrac{\partial u_a}{\partial y_a}=\dfrac{p_y+t}{p_x}\dfrac{\partial u_a}{\partial x_a}\qquad(2)$</span></p>
<p>The maximisation does not need to be over <span class="math-container">$x_b$</span> and <span class="math-container">$b$</span>’s income constraint <span class="math-container">$p_xx_b=e_b$</span> need not be included since, with no tax or other payment to <span class="math-container">$b$</span>, and no consumption of <span class="math-container">$y$</span> by <span class="math-container">$b$</span>, <span class="math-container">$x_b$</span> is fixed at <span class="math-container">$e_b/p_x$</span>.  But it does need to be over <span class="math-container">$t$</span>, and <span class="math-container">$t$</span> needs to be included in <span class="math-container">$a$</span>’s income constraint (1), since the tax has both an income and a substitution effect on <span class="math-container">$a$</span>, with consequences for both parties’ utility.  The first-order condition (2) from <span class="math-container">$a$</span>’s optimisation also needs to be included as a constraint: we want to maximise welfare only over those combinations of the goods that <span class="math-container">$a$</span> would actually choose in response to given tax rates.  Also, <span class="math-container">$x_a$</span>, <span class="math-container">$y_a$</span> and <span class="math-container">$t$</span> must be non-negative.</p>
<p>The Lagrangian is then:</p>
<p><span class="math-container">$L=u_a+u_b+\lambda_1[p_xx_a+(p_y+t)y_a-e_a]+\lambda_2\Bigg[\dfrac{\partial u_a}{\partial y_a}-\dfrac{p_y+t}{p_x}\dfrac{\partial u_a}{\partial x_a}\Bigg]$</span></p>
<p>This leads to first-order conditions:</p>
<p><span class="math-container">$\dfrac{\partial L}{\partial x_a}=\dfrac{\partial u_a}{\partial x_a}+\lambda_1p_x-\lambda_2\dfrac{p_y+t}{p_x}\dfrac{\partial^2u_a}{\partial x_a^2}=0\qquad(3)$</span></p>
<p><span class="math-container">$\dfrac{\partial L}{\partial y_a}=\dfrac{\partial u_a}{\partial y_a}+\dfrac{\partial u_b}{\partial y_a}+\lambda_1(p_y+t)+\lambda_2\dfrac{\partial^2u_a}{\partial y_a^2}=0\qquad(4)$</span></p>
<p><span class="math-container">$\dfrac{\partial L}{\partial t}=\lambda_1y_a-\dfrac{\lambda_2}{p_x}\dfrac{\partial u_a}{\partial x_a}=0\qquad(5)$</span></p>
<p>For this case only I will outline a method of solution.  Using (3) to (5) it is possible to eliminate <span class="math-container">$\lambda_1$</span> and <span class="math-container">$\lambda_2$</span> and obtain the relation:</p>
<p><span class="math-container">$(p_y+t)\Bigg[y_a\dfrac{\partial^2u_a}{\partial x_a^2}\bigg(\dfrac{\partial u_a}{\partial y_a}+\dfrac{\partial u_b}{\partial y_a}\bigg)+\bigg(\dfrac{\partial u_a}{\partial x_a}\bigg)^2\Bigg]=p_x \dfrac{\partial u_a}{\partial x_a}\Bigg[\bigg(\dfrac{\partial u_a}{\partial y_a}+\dfrac{\partial u_b}{\partial y_a}\bigg)-y_a\dfrac{\partial^2u_a}{\partial y_a^2}\Bigg]\qquad(6)$</span></p>
<p>This can be rearranged as a formula for <span class="math-container">$t$</span> (note the difference from the formula in the question):</p>
<p><span class="math-container">$t=\dfrac{p_x\frac{\partial u_a}{\partial x_a}\Big[\big(\frac{\partial u_a}{\partial y_a}+\frac{\partial u_b}{\partial y_a}\big)- y_a\frac{\partial^2u_a}{\partial y_a^2}\Big]}{y_a\frac{\partial^2u_a}{\partial x_a^2}\big(\frac{\partial u_a}{\partial y_a}+\frac{\partial u_b}{\partial y_a}\big)+\big(\frac{\partial u_a}{\partial x_a}\big)^2}-p_y\qquad(7)$</span></p>
<p>From (2) we also have:</p>
<p><span class="math-container">$t=\dfrac{p_x\frac{\partial u_a}{\partial y_a}}{\frac{\partial u_a}{\partial x_a}}-p_y\qquad(8)$</span></p>
<p>Setting the fractions in (7) and (8) equal, as they must be, and rearranging, we can infer:</p>
<p><span class="math-container">$y_a=\dfrac{\big(\frac{\partial u_a}{\partial x_a}\big)^2\frac{\partial u_b}{\partial y_a}}{\frac{\partial u_a}{\partial y_a}\frac{\partial^2u_a}{\partial x_a^2}\big(\frac{\partial u_a}{\partial y_a}+\frac{\partial u_b}{\partial y_a}\big)+\big(\frac{\partial u_a}{\partial x_a}\big)^2\frac{\partial^2u_a}{\partial y_a^2}}\qquad(9)$</span></p>
<p>At this level of generality it’s hard to make further progress, but clearly if the utility functions were specified and we could express the various derivatives as functions of <span class="math-container">$x_a$</span> and <span class="math-container">$y_a$</span>, then (9) would yield an equation in <span class="math-container">$x_a$</span> and <span class="math-container">$y_a$</span>.  Together with (1) and (7) this would give three equations which could be solved for the three variables <span class="math-container">$x_a$</span>, <span class="math-container">$y_a$</span> and <span class="math-container">$t$</span>.   All of the above of course is subject to the caveat that these variables must be non-negative.</p>
<p><strong>Case 2: <span class="math-container">$a$</span>’s consumption of <span class="math-container">$y$</span> taxed at rate <span class="math-container">$t$</span>, the tax revenue being paid on by the government to <span class="math-container">$b$</span></strong></p>
<p>In this case <span class="math-container">$x_b$</span> depends on how much tax revenue <span class="math-container">$b$</span> receives, so we need to optimise over <span class="math-container">$x_b$</span>  as well as <span class="math-container">$x_a$</span> and <span class="math-container">$y_a$</span>, and include <span class="math-container">$b$</span>’s income constraint.  So the social optimisation should be specified as:</p>
<p><span class="math-container">$\max\limits_{x_a,y_a,t,x_b}u_a+u_b$</span></p>
<p>subject to  <span class="math-container">$\ p_xx_a+(p_y+t)y_a=e_a\qquad(10)$</span></p>
<p>and  <span class="math-container">$\ \dfrac{\partial u_a}{\partial y_a}=\dfrac{p_y+t}{p_x}\dfrac{\partial u_a}{\partial x_a}\qquad(11)$</span></p>
<p>and  <span class="math-container">$\ p_xx_b=e_b+ty_a\qquad(12)$</span></p>
<p>The Lagrangian is:</p>
<p><span class="math-container">$L=u_a+u_b+\lambda_1[p_xx_a+(p_y+t)y_a-e_a]+\lambda_2\Bigg[\dfrac{\partial u_a}{\partial y_a}-\dfrac{p_y+t}{p_x}\dfrac{\partial u_a}{\partial x_a}\Bigg]+\lambda_3[p_xx_b-e_b-ty_a]$</span></p>
<p>This leads to first-order conditions:</p>
<p><span class="math-container">$\dfrac{\partial L}{\partial x_a}=\dfrac{\partial u_a}{\partial x_a}+\lambda_1p_x-\lambda_2\dfrac{p_y+t}{p_x}\dfrac{\partial^2u_a}{\partial x_a^2}=0\qquad(13)$</span></p>
<p><span class="math-container">$\dfrac{\partial L}{\partial y_a}=\dfrac{\partial u_a}{\partial y_a}+\dfrac{\partial u_b}{\partial y_a}+\lambda_1(p_y+t)+\lambda_2\dfrac{\partial^2u_a}{\partial y_a^2}-\lambda_3t=0\qquad(14)$</span></p>
<p><span class="math-container">$\dfrac{\partial L}{\partial t}=\lambda_1y_a-\dfrac{\lambda_2}{p_x}\dfrac{\partial u_a}{\partial x_a}-\lambda_3y_a=0\qquad(15)$</span></p>
<p><span class="math-container">$\dfrac{\partial L}{\partial x_b}=\dfrac{\partial u_b}{\partial x_b}+\lambda_3x_b=0\qquad(16)$</span></p>
<p>Using these four equations it should be possible to eliminate <span class="math-container">$\lambda_1$</span>, <span class="math-container">$\lambda_2$</span> and <span class="math-container">$\lambda_3$</span> and then, using (11) and (12) and if the utility functions are specified, solve for <span class="math-container">$x_a$</span>, <span class="math-container">$y_a$</span>, <span class="math-container">$t$</span> and <span class="math-container">$x_b$</span>.</p>
<p><strong>Case 3: No tax, a Coasian bargain in the context that <span class="math-container">$b$</span> is the right-holder and the parties agree to negotiate only on the basis of a flat rate <span class="math-container">$p_z$</span> per unit of <span class="math-container">$y$</span> consumed by <span class="math-container">$a$</span></strong></p>
<p>If the parties agree to negotiate only on the basis of a flat rate <span class="math-container">$p_z$</span> per unit of <span class="math-container">$y$</span>, the optimal rate will be obtainable exactly as for Case 2, albeit replacing <span class="math-container">$t$</span> by <span class="math-container">$p_z$</span> throughout.  Why?  Because for any <span class="math-container">$y_a$</span>, transfer of the sum <span class="math-container">$p_zy_a=ty_a$</span> from <span class="math-container">$a$</span> to <span class="math-container">$b$</span> will have exactly the same effect on the behaviour and therefore the utility of each party regardless of whether it is a tax collected from <span class="math-container">$a$</span> by the governemnt and then paid to <span class="math-container">$b$</span>, or a direct payment from <span class="math-container">$a$</span> to <span class="math-container">$b$</span>.</p>
<p>Note what is not being said here.  It is not claimed that the social optimum that can be achieved via any Coasian bargain on any basis the parties may choose is the same as for Case 2.  Such a claim would be false, as is shown in considering Case 4 below.</p>
<p><strong>Case 4: No tax, a Coasian bargain in the context that <span class="math-container">$b$</span> is the right-holder and the parties agree to negotiate on the basis of a lump sum payment for consumption by <span class="math-container">$a$</span> of specific quantity of <span class="math-container">$y$</span>.</strong></p>
<p>This is one example of a bargain not involving a flat rate charge per unit of <span class="math-container">$y$</span> (others are possible, eg an increasing rate per unit, but will not be considered here).</p>
<p>It is convenient to illustrate this case by a numerical example.  Suppose that: <span class="math-container">$u_a= \ln(x_a+1)+\ln(y_a+1)$</span>; <span class="math-container">$\ u_b=\ln(x_b+1)-0.1y_a^2$</span>; <span class="math-container">$\ p_x=p_y=1$</span>; <span class="math-container">$\ e_a=e_b=4$</span>.  Suppose further that the parties agree that <span class="math-container">$a$</span> will consume <span class="math-container">$y_a=0.9$</span> in return for payment to <span class="math-container">$b$</span> of <span class="math-container">$0.4$</span>.  Because the quantity <span class="math-container">$y_a$</span> is specified in the agreement, there is no opportunity for <span class="math-container">$a$</span> to optimise within the scope of the agreement, other than by spending on <span class="math-container">$x_a$</span> all the funds not required to pay the lump sum and to purchase the <span class="math-container">$0.9$</span> <span class="math-container">$y_a$</span>.</p>
<p>The quantities of <span class="math-container">$x$</span> consumed by each party will then be:</p>
<p><span class="math-container">$x_a=4-0.4-0.9=2.7\qquad(17)$</span></p>
<p><span class="math-container">$x_b=4+0.4=4.4\qquad(18)$</span></p>
<p>and social utility will be:</p>
<p><span class="math-container">$u_a+u_b=\ln(3.7)+\ln(1.9)+\ln(5.4)-0.1(0.9^2)\approx1.308+0.642+1.686-0.081=3.555\quad(19)$</span></p>
<p>I shall show that that is more social utility than can be obtained via any flat rate charge.  This may seem surprising since the lump sum payment of <span class="math-container">$0.4$</span> for consumption of <span class="math-container">$0.9$</span> can be viewed as equivalent to a per unit charge <span class="math-container">$p_z$</span>  of <span class="math-container">$0.4/0.9\approx 0.444$</span>.  The point however is that once the parties agree to the lump sum, <span class="math-container">$a$</span> is committed to consuming <span class="math-container">$0.9$</span> of <span class="math-container">$y$</span>, whereas if the parties agree to a rate per unit, <span class="math-container">$a$</span> can still choose to consume the quantity of <span class="math-container">$y$</span> that will maximise <span class="math-container">$a$</span>’s utility given that rate.  In fact, substituting into (1) and (2) with <span class="math-container">$p_z$</span> replacing <span class="math-container">$t$</span> and <span class="math-container">$p_z=0.444$</span> we have:</p>
<p><span class="math-container">$x_a+1.444y_a=4\qquad(20)$</span></p>
<p><span class="math-container">$\dfrac{1}{y_a+1}=\dfrac{1.444}{x_a+1}=\dfrac{1.444}{5-1.444y_a}\qquad(21)$</span></p>
<p>from which we can infer:</p>
<p><span class="math-container">$5-1.444y_a=1.444y_a+1.444\qquad(22)$</span></p>
<p><span class="math-container">$3.556=2.888y_a$</span>   and so   <span class="math-container">$\ y_a\approx 1.231\qquad(23)$</span></p>
<p>In response to a flat rate of <span class="math-container">$0.444$</span>, therefore, <span class="math-container">$a$</span> will choose to consume not <span class="math-container">$0.9$</span> but <span class="math-container">$1.231$</span> of <span class="math-container">$y$</span>.  Furthermore, we can infer:</p>
<p><span class="math-container">$x_a=4-(1+0.444)(1.231)=2.222\qquad(24)$</span></p>
<p><span class="math-container">$x_b=4+0.444(1.231)=4.547\qquad(25)$</span></p>
<p><span class="math-container">$u_a+u_b=\ln(3.222)+\ln(2.231)+\ln(5.547)-0.1(1.231^2)\approx1.170+0.802+1.713-0.152=3.533\qquad(26)$</span></p>
<p>Comparing (26) with (19), social utility is somewhat less with the flat rate.</p>
<p>Making a similar calculation for various values of <span class="math-container">$p_z$</span> I obtained the results in the table below.</p>
<p><a href="https://i.sstatic.net/JTvbM.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/JTvbM.png" alt="enter image description here" /></a></p>
<p>It can be seen that, with a flat rate charge, the social optimum is achieved when the charge <span class="math-container">$p_z$</span> is approximately <span class="math-container">$0.50$</span>, with social utility approximately <span class="math-container">$3.536$</span> which is less than the <span class="math-container">$3.555$</span> achieved with a lump sum of <span class="math-container">$0.4$</span> for consumption of <span class="math-container">$0.9$</span> of <span class="math-container">$y$</span>.</p>
<p>Thus for the specific example illustrated, a lump sum payment for a specific level of consumption of y can yield more social utility than any flat rate.  Given the equivalence of Cases 2 and 3, it follows that the socially optimal outcome from Coasian bargaining will not in general equal the socially optimal outcome from a tax paid on to <span class="math-container">$b$</span>.</p>
<p><strong>References</strong></p>
<p>A.  Hanley N, Shogren J F &amp; White B (2nd edn 2007) <em>Environmental Economics in Theory and Practice</em>  (pp 45-8 discuss Coasian bargaining in the context of given marginal benefit and marginal cost curves, without mentioning utility).</p>
<p>B.  <a href="https://en.wikipedia.org/wiki/Coase_theorem" rel="nofollow noreferrer">Wikipedia – Coase Theorem</a> (introduces the Coase Theorem in terms of economic efficiency).</p>
<p>C.  Perman R, Ma Y, McGilvray &amp; Common M (3rd edn 2003) <em>Natural Resources and Environmental Economics</em> (pp 137-9 discuss Coasian bargaining in the context of marginal benefit and marginal cost curves supported by utility functions which include a monetary variable (wealth)).</p>

CC BY-SA 4.0
2021-09-11T12:53:29.257Z
2021-09-11T13:03:59.833Z
2021-09-11T13:03:59.833Z
Adam Bailey
<p>Consider an economy with two goods, <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. Suppose person a's consumption of good <span class="math-container">$y$</span> imposes a negative externality on person b. Person a's utility maximisation problem is
<span class="math-container">$$\max_{x_a,y_a} \ u_a(x_a,y_a),$$</span>
subject to
<span class="math-container">$$p_xx_a+(p_y+t)y_a=e_a,$$</span>
where <span class="math-container">$t$</span> is a Pigouvian tax on consuming <span class="math-container">$y_a$</span>.</p>
<p>The first-order condition is
<span class="math-container">$$\frac{\partial u_a}{\partial{y_a}}=(p_y+t)\left[\frac{1}{p_x}\frac{\partial u_a }{\partial x_a}\right]$$</span></p>
<p>Let's assume that a benign socal planer would like to maximise a utilitiarian social welfare function. Then the social welfare maximisation problem is
<span class="math-container">$$\max_{x_a,\ y_a, \ x_b} \ u_a(x_a,y_a)+u_b(x_b,y_a),$$</span>
subject to
<span class="math-container">$$\begin{align*}
p_xx_a+p_yy_a&amp;=e_a,\\
p_xx_b&amp;=e_b.
\end{align*}$$</span>
The first-order condition is
<span class="math-container">$$\frac{\partial u_a}{\partial y_a}-\frac{p_y}{p_x}\frac{\partial u_a}{\partial x_a}+ \frac{\partial u_b}{\partial y_a}=0,\\\\$$</span>
where <span class="math-container">$\partial u_b/\partial y_a&lt;0$</span>. We can rewrite this condion as
<span class="math-container">$$\frac{\partial u_a}{\partial y_a}=\left[p_y + p_x\frac{-\partial u_b/\partial y_a}{\partial u_a/\partial x_a} \right]\left[\frac{1}{p_x}\frac{\partial u_a }{\partial x_a}\right].$$</span></p>
<p>With the optimal Pigouvian tax, <span class="math-container">$t^*$</span>, the competitive equilibrium will be socially optimal. Then the first-order conditon for person a's will be the same as the first-order condition for social welfare maximisation, which means that
<span class="math-container">$$
t^* = p_x\frac{-\partial u_b/\partial y_a}{\partial u_a/\partial x_a}
$$</span></p>
<p>Consider, instead of a Pigouvian tax, a Coasian bargain. Suppose that I allocate property rights to person b. Let <span class="math-container">$z$</span> be the rights to impose this external cost on person b, and <span class="math-container">$p_z$</span> be the price of such rights. Then person a's problem is <span class="math-container">$$\max_{x_a,\ y_a} \ u_a(x_a,y_a)$$</span> subject to <span class="math-container">\begin{align*} p_xx_a+p_yy_a+p_zz_a&amp;=e_a,\\ z_a&amp;=y_a. \end{align*}</span>
Assuming that the market for rights clears (<span class="math-container">$z_a=z_b=y_a$</span>), the first-order condition is
<span class="math-container">$$-\frac{p_y+p_z}{p_x}\frac{\partial u_a}{\partial x_a}+\frac{\partial u_a}{\partial y_a}=0.$$</span>
Person b's problem is
<span class="math-container">$$\max_{x_b,z_b} \ u_b(x_b,y_a)$$</span> subject to <span class="math-container">\begin{align*} p_xx_b&amp;=e_a+p_zz_b,\\ z_b&amp;=y_a. \end{align*}</span>
The first-order condition is
<span class="math-container">$$\frac{p_z}{p_x}\frac{\partial u_b}{\partial x_b}+\frac{\partial u_b}{\partial y_a}=0,$$</span>
remembering that <span class="math-container">$y_a=z_b$</span>.</p>
<p>In equilibrium, person a's and person b's first-order conditions are equal to <span class="math-container">$\frac{p_z}{p_x}$</span>. I have tried rearranging to find a condition similar to that for social welfare maximisation, however, I am unsure how I solve for <span class="math-container">$p_z$</span>. Should <span class="math-container">$p_z$</span> be the same as the Pigouvian tax <span class="math-container">$t^*$</span>? How can I show equivalence between them?</p>

CC BY-SA 4.0
2021-08-15T09:23:55.910Z
What happens with the tax-money?
Michael Greinecker
CC BY-SA 4.0
2021-08-15T12:53:26.327Z
It could either:

- be paid in a transfer to person b, or

- be paid into some government fund (this is a partial equilibrium).

I'm wondering whether this matters for equivalence between a Pigouvian tax and Coasian bargaining?
dotpad
CC BY-SA 4.0
2021-08-15T13:50:03.357Z
In what you call Coasian bargaining, it becomes part of the consumption of b, so yes.
Michael Greinecker
CC BY-SA 4.0
2021-08-15T15:22:54.987Z
Could you please clarify the status of the prices $p_x$ and $p_y$.  Are they to be taken as given (eg determined within a larger market with many more participants than $a$ and $b$?  Or are they variables to be determined within your system?  I think you mean the former, but it would be good to be sure.
Adam Bailey
CC BY-SA 4.0
2021-08-15T17:38:24.717Z
Adam, yes exactly! Prices $p_x$ and $p_y$ are determined in Walrasian markets with many more consumers of $x$ and $y$. This is just a partial equilibrium description with two individuals.
dotpad
CC BY-SA 4.0
2021-09-10T10:26:06.157Z
I have made an edit to correct what appear to be typos in a few of the subscripts.  Please feel free to reverse my edit if I've misunderstood.  I hope to post an answer soon.
Adam Bailey
<microeconomics><mathematical-economics><public-economics><externalities><pigouvian-taxes>
Pigouvian tax equivalent to Coasian payments
dotpad
# Pigouvian tax equivalent to Coasian payments

(Asked by: username_0 on 2021-08-14)

Consider an economy with two goods, $x$ and $y$. Suppose person a's consumption of good $y$ imposes a negative externality on person b. Person a's utility maximisation problem is
$$\max\_{x\_a,y\_a} \ u\_a(x\_a,y\_a),$$
subject to
$$p\_xx\_a+(p\_y+t)y\_a=e\_a,$$
where $t$ is a Pigouvian tax on consuming $y\_a$.

The first-order condition is
$$\frac{\partial u\_a}{\partial{y\_a}}=(p\_y+t)\left[\frac{1}{p\_x}\frac{\partial u\_a }{\partial x\_a}\right]$$

Let's assume that a benign socal planer would like to maximise a utilitiarian social welfare function. Then the social welfare maximisation problem is
$$\max\_{x\_a,\ y\_a, \ x\_b} \ u\_a(x\_a,y\_a)+u\_b(x\_b,y\_a),$$
subject to
$$\begin{align\*}
p\_xx\_a+p\_yy\_a&=e\_a,\\
p\_xx\_b&=e\_b.
\end{align\*}$$
The first-order condition is
$$\frac{\partial u\_a}{\partial y\_a}-\frac{p\_y}{p\_x}\frac{\partial u\_a}{\partial x\_a}+ \frac{\partial u\_b}{\partial y\_a}=0,\\\\$$
where $\partial u\_b/\partial y\_a<0$. We can rewrite this condion as
$$\frac{\partial u\_a}{\partial y\_a}=\left[p\_y + p\_x\frac{-\partial u\_b/\partial y\_a}{\partial u\_a/\partial x\_a} \right]\left[\frac{1}{p\_x}\frac{\partial u\_a }{\partial x\_a}\right].$$

With the optimal Pigouvian tax, $t^\*$, the competitive equilibrium will be socially optimal. Then the first-order conditon for person a's will be the same as the first-order condition for social welfare maximisation, which means that
$$
t^\* = p\_x\frac{-\partial u\_b/\partial y\_a}{\partial u\_a/\partial x\_a}
$$

Consider, instead of a Pigouvian tax, a Coasian bargain. Suppose that I allocate property rights to person b. Let $z$ be the rights to impose this external cost on person b, and $p\_z$ be the price of such rights. Then person a's problem is $$\max\_{x\_a,\ y\_a} \ u\_a(x\_a,y\_a)$$ subject to \begin{align\*} p\_xx\_a+p\_yy\_a+p\_zz\_a&=e\_a,\\ z\_a&=y\_a. \end{align\*}
Assuming that the market for rights clears ($z\_a=z\_b=y\_a$), the first-order condition is
$$-\frac{p\_y+p\_z}{p\_x}\frac{\partial u\_a}{\partial x\_a}+\frac{\partial u\_a}{\partial y\_a}=0.$$
Person b's problem is
$$\max\_{x\_b,z\_b} \ u\_b(x\_b,y\_a)$$ subject to \begin{align\*} p\_xx\_b&=e\_a+p\_zz\_b,\\ z\_b&=y\_a. \end{align\*}
The first-order condition is
$$\frac{p\_z}{p\_x}\frac{\partial u\_b}{\partial x\_b}+\frac{\partial u\_b}{\partial y\_a}=0,$$
remembering that $y\_a=z\_b$.

In equilibrium, person a's and person b's first-order conditions are equal to $\frac{p\_z}{p\_x}$. I have tried rearranging to find a condition similar to that for social welfare maximisation, however, I am unsure how I solve for $p\_z$. Should $p\_z$ be the same as the Pigouvian tax $t^\*$? How can I show equivalence between them?

**Tags:** microeconomics, mathematical-economics, public-economics, externalities, pigouvian-taxes

> **Comments:**
> **username_1**: What happens with the tax-money?  
> **username_0**: It could either:  - be paid in a transfer to person b, or  - be paid into some government fund (this is a partial equilibrium).  I'm wondering whether this matters for equivalence between a Pigouvian tax and Coasian bargaining?  
> **username_1**: In what you call Coasian bargaining, it becomes part of the consumption of b, so yes.  
> **username_2**: Could you please clarify the status of the prices $p_x$ and $p_y$.  Are they to be taken as given (eg determined within a larger market with many more participants than $a$ and $b$?  Or are they variables to be determined within your system?  I think you mean the former, but it would be good to be sure.  
> **username_0**: username_2, yes exactly! Prices $p_x$ and $p_y$ are determined in Walrasian markets with many more consumers of $x$ and $y$. This is just a partial equilibrium description with two individuals.  
> **username_2**: I have made an edit to correct what appear to be typos in a few of the subscripts.  Please feel free to reverse my edit if I've misunderstood.  I hope to post an answer soon.  

---

## 1 answers

### Answer by username_2 on 2021-09-11. Score: 2

In the given scenario, the optimal social outcome from a Coasian bargain is not in general the same as the optimal social outcome from a Pigouvian tax, although there is a special case in which this is so.

The scenario outlined in this question has several features that make it rather more complicated than some textbook discussions of policy towards negative externalities.

a) It focuses not on benefits and costs but on utility functions (cf (A)).

b) It considers optimality, defined as a state in which the sum of the utility of the two parties is maximised, rather than efficiency (a state in which the utility of one party cannot be increased without lowering that of the other, also known as Pareto-optimality) (cf (B)).

c) No direct link is specified between utility and any monetary quantity (cf (C)). Payment or receipt of money by either party affects utility indirectly via the quantity each can afford of a second good not subject to an externality.

d) It uses general utility functions with no restrictions on their functional form.

e) Although both parties can consume the second good, as a consequence of (d) above a given quantity of that good does not necessarily affect each party’s utility in the same way. It is possible therefore that a monetary transfer between the parties (whether via tax revenue paid on to the other party, or a Coasian bargain) could raise or lower the sum of utility not for reasons relating to the externality but simply by transferring spending power to or from the party which gains more utility from the same quantity of the good.

This answer is concerned, as the question asks, with social optima. Whether a government is likely to set a tax at a rate that yields a social optimum, and whether Coasian bargaining is likely to lead to an agreement that yields a social optimum, are further questions which I do not address. I make one simplifying assumption not in the question, namely, that the utility functions are additively separable, so that cross-partial derivatives can be ignored.

**Case 1: $a$’s consumption of $y$ taxed at a rate $t$, the use of the tax revenue being considered outside the scope of the optimisation**  
This is the case considered in the first half of the question. The analysis of $a$’s optimisation is correct, but this is how I would specify the social optimisation:

$\max\limits\_{x\_a,y\_a,t} \ u\_a(x\_a,y\_a)+u\_b(x\_b,y\_b)$

subject to $\ p\_xx\_a+(p\_y+t)y\_a=e\_a\qquad(1)$

and $\ \dfrac{\partial u\_a}{\partial y\_a}=\dfrac{p\_y+t}{p\_x}\dfrac{\partial u\_a}{\partial x\_a}\qquad(2)$

The maximisation does not need to be over $x\_b$ and $b$’s income constraint $p\_xx\_b=e\_b$ need not be included since, with no tax or other payment to $b$, and no consumption of $y$ by $b$, $x\_b$ is fixed at $e\_b/p\_x$. But it does need to be over $t$, and $t$ needs to be included in $a$’s income constraint (1), since the tax has both an income and a substitution effect on $a$, with consequences for both parties’ utility. The first-order condition (2) from $a$’s optimisation also needs to be included as a constraint: we want to maximise welfare only over those combinations of the goods that $a$ would actually choose in response to given tax rates. Also, $x\_a$, $y\_a$ and $t$ must be non-negative.

The Lagrangian is then:

$L=u\_a+u\_b+\lambda\_1[p\_xx\_a+(p\_y+t)y\_a-e\_a]+\lambda\_2\Bigg[\dfrac{\partial u\_a}{\partial y\_a}-\dfrac{p\_y+t}{p\_x}\dfrac{\partial u\_a}{\partial x\_a}\Bigg]$

This leads to first-order conditions:

$\dfrac{\partial L}{\partial x\_a}=\dfrac{\partial u\_a}{\partial x\_a}+\lambda\_1p\_x-\lambda\_2\dfrac{p\_y+t}{p\_x}\dfrac{\partial^2u\_a}{\partial x\_a^2}=0\qquad(3)$

$\dfrac{\partial L}{\partial y\_a}=\dfrac{\partial u\_a}{\partial y\_a}+\dfrac{\partial u\_b}{\partial y\_a}+\lambda\_1(p\_y+t)+\lambda\_2\dfrac{\partial^2u\_a}{\partial y\_a^2}=0\qquad(4)$

$\dfrac{\partial L}{\partial t}=\lambda\_1y\_a-\dfrac{\lambda\_2}{p\_x}\dfrac{\partial u\_a}{\partial x\_a}=0\qquad(5)$

For this case only I will outline a method of solution. Using (3) to (5) it is possible to eliminate $\lambda\_1$ and $\lambda\_2$ and obtain the relation:

$(p\_y+t)\Bigg[y\_a\dfrac{\partial^2u\_a}{\partial x\_a^2}\bigg(\dfrac{\partial u\_a}{\partial y\_a}+\dfrac{\partial u\_b}{\partial y\_a}\bigg)+\bigg(\dfrac{\partial u\_a}{\partial x\_a}\bigg)^2\Bigg]=p\_x \dfrac{\partial u\_a}{\partial x\_a}\Bigg[\bigg(\dfrac{\partial u\_a}{\partial y\_a}+\dfrac{\partial u\_b}{\partial y\_a}\bigg)-y\_a\dfrac{\partial^2u\_a}{\partial y\_a^2}\Bigg]\qquad(6)$

This can be rearranged as a formula for $t$ (note the difference from the formula in the question):

$t=\dfrac{p\_x\frac{\partial u\_a}{\partial x\_a}\Big[\big(\frac{\partial u\_a}{\partial y\_a}+\frac{\partial u\_b}{\partial y\_a}\big)- y\_a\frac{\partial^2u\_a}{\partial y\_a^2}\Big]}{y\_a\frac{\partial^2u\_a}{\partial x\_a^2}\big(\frac{\partial u\_a}{\partial y\_a}+\frac{\partial u\_b}{\partial y\_a}\big)+\big(\frac{\partial u\_a}{\partial x\_a}\big)^2}-p\_y\qquad(7)$

From (2) we also have:

$t=\dfrac{p\_x\frac{\partial u\_a}{\partial y\_a}}{\frac{\partial u\_a}{\partial x\_a}}-p\_y\qquad(8)$

Setting the fractions in (7) and (8) equal, as they must be, and rearranging, we can infer:

$y\_a=\dfrac{\big(\frac{\partial u\_a}{\partial x\_a}\big)^2\frac{\partial u\_b}{\partial y\_a}}{\frac{\partial u\_a}{\partial y\_a}\frac{\partial^2u\_a}{\partial x\_a^2}\big(\frac{\partial u\_a}{\partial y\_a}+\frac{\partial u\_b}{\partial y\_a}\big)+\big(\frac{\partial u\_a}{\partial x\_a}\big)^2\frac{\partial^2u\_a}{\partial y\_a^2}}\qquad(9)$

At this level of generality it’s hard to make further progress, but clearly if the utility functions were specified and we could express the various derivatives as functions of $x\_a$ and $y\_a$, then (9) would yield an equation in $x\_a$ and $y\_a$. Together with (1) and (7) this would give three equations which could be solved for the three variables $x\_a$, $y\_a$ and $t$. All of the above of course is subject to the caveat that these variables must be non-negative.

**Case 2: $a$’s consumption of $y$ taxed at rate $t$, the tax revenue being paid on by the government to $b$**

In this case $x\_b$ depends on how much tax revenue $b$ receives, so we need to optimise over $x\_b$ as well as $x\_a$ and $y\_a$, and include $b$’s income constraint. So the social optimisation should be specified as:

$\max\limits\_{x\_a,y\_a,t,x\_b}u\_a+u\_b$

subject to $\ p\_xx\_a+(p\_y+t)y\_a=e\_a\qquad(10)$

and $\ \dfrac{\partial u\_a}{\partial y\_a}=\dfrac{p\_y+t}{p\_x}\dfrac{\partial u\_a}{\partial x\_a}\qquad(11)$

and $\ p\_xx\_b=e\_b+ty\_a\qquad(12)$

The Lagrangian is:

$L=u\_a+u\_b+\lambda\_1[p\_xx\_a+(p\_y+t)y\_a-e\_a]+\lambda\_2\Bigg[\dfrac{\partial u\_a}{\partial y\_a}-\dfrac{p\_y+t}{p\_x}\dfrac{\partial u\_a}{\partial x\_a}\Bigg]+\lambda\_3[p\_xx\_b-e\_b-ty\_a]$

This leads to first-order conditions:

$\dfrac{\partial L}{\partial x\_a}=\dfrac{\partial u\_a}{\partial x\_a}+\lambda\_1p\_x-\lambda\_2\dfrac{p\_y+t}{p\_x}\dfrac{\partial^2u\_a}{\partial x\_a^2}=0\qquad(13)$

$\dfrac{\partial L}{\partial y\_a}=\dfrac{\partial u\_a}{\partial y\_a}+\dfrac{\partial u\_b}{\partial y\_a}+\lambda\_1(p\_y+t)+\lambda\_2\dfrac{\partial^2u\_a}{\partial y\_a^2}-\lambda\_3t=0\qquad(14)$

$\dfrac{\partial L}{\partial t}=\lambda\_1y\_a-\dfrac{\lambda\_2}{p\_x}\dfrac{\partial u\_a}{\partial x\_a}-\lambda\_3y\_a=0\qquad(15)$

$\dfrac{\partial L}{\partial x\_b}=\dfrac{\partial u\_b}{\partial x\_b}+\lambda\_3x\_b=0\qquad(16)$

Using these four equations it should be possible to eliminate $\lambda\_1$, $\lambda\_2$ and $\lambda\_3$ and then, using (11) and (12) and if the utility functions are specified, solve for $x\_a$, $y\_a$, $t$ and $x\_b$.

**Case 3: No tax, a Coasian bargain in the context that $b$ is the right-holder and the parties agree to negotiate only on the basis of a flat rate $p\_z$ per unit of $y$ consumed by $a$**

If the parties agree to negotiate only on the basis of a flat rate $p\_z$ per unit of $y$, the optimal rate will be obtainable exactly as for Case 2, albeit replacing $t$ by $p\_z$ throughout. Why? Because for any $y\_a$, transfer of the sum $p\_zy\_a=ty\_a$ from $a$ to $b$ will have exactly the same effect on the behaviour and therefore the utility of each party regardless of whether it is a tax collected from $a$ by the governemnt and then paid to $b$, or a direct payment from $a$ to $b$.

Note what is not being said here. It is not claimed that the social optimum that can be achieved via any Coasian bargain on any basis the parties may choose is the same as for Case 2. Such a claim would be false, as is shown in considering Case 4 below.

**Case 4: No tax, a Coasian bargain in the context that $b$ is the right-holder and the parties agree to negotiate on the basis of a lump sum payment for consumption by $a$ of specific quantity of $y$.**

This is one example of a bargain not involving a flat rate charge per unit of $y$ (others are possible, eg an increasing rate per unit, but will not be considered here).

It is convenient to illustrate this case by a numerical example. Suppose that: $u\_a= \ln(x\_a+1)+\ln(y\_a+1)$; $\ u\_b=\ln(x\_b+1)-0.1y\_a^2$; $\ p\_x=p\_y=1$; $\ e\_a=e\_b=4$. Suppose further that the parties agree that $a$ will consume $y\_a=0.9$ in return for payment to $b$ of $0.4$. Because the quantity $y\_a$ is specified in the agreement, there is no opportunity for $a$ to optimise within the scope of the agreement, other than by spending on $x\_a$ all the funds not required to pay the lump sum and to purchase the $0.9$ $y\_a$.

The quantities of $x$ consumed by each party will then be:

$x\_a=4-0.4-0.9=2.7\qquad(17)$

$x\_b=4+0.4=4.4\qquad(18)$

and social utility will be:

$u\_a+u\_b=\ln(3.7)+\ln(1.9)+\ln(5.4)-0.1(0.9^2)\approx1.308+0.642+1.686-0.081=3.555\quad(19)$

I shall show that that is more social utility than can be obtained via any flat rate charge. This may seem surprising since the lump sum payment of $0.4$ for consumption of $0.9$ can be viewed as equivalent to a per unit charge $p\_z$ of $0.4/0.9\approx 0.444$. The point however is that once the parties agree to the lump sum, $a$ is committed to consuming $0.9$ of $y$, whereas if the parties agree to a rate per unit, $a$ can still choose to consume the quantity of $y$ that will maximise $a$’s utility given that rate. In fact, substituting into (1) and (2) with $p\_z$ replacing $t$ and $p\_z=0.444$ we have:

$x\_a+1.444y\_a=4\qquad(20)$

$\dfrac{1}{y\_a+1}=\dfrac{1.444}{x\_a+1}=\dfrac{1.444}{5-1.444y\_a}\qquad(21)$

from which we can infer:

$5-1.444y\_a=1.444y\_a+1.444\qquad(22)$

$3.556=2.888y\_a$ and so $\ y\_a\approx 1.231\qquad(23)$

In response to a flat rate of $0.444$, therefore, $a$ will choose to consume not $0.9$ but $1.231$ of $y$. Furthermore, we can infer:

$x\_a=4-(1+0.444)(1.231)=2.222\qquad(24)$

$x\_b=4+0.444(1.231)=4.547\qquad(25)$

$u\_a+u\_b=\ln(3.222)+\ln(2.231)+\ln(5.547)-0.1(1.231^2)\approx1.170+0.802+1.713-0.152=3.533\qquad(26)$

Comparing (26) with (19), social utility is somewhat less with the flat rate.

Making a similar calculation for various values of $p\_z$ I obtained the results in the table below.

[![enter image description here](https://i.sstatic.net/JTvbM.png)](https://i.sstatic.net/JTvbM.png)

It can be seen that, with a flat rate charge, the social optimum is achieved when the charge $p\_z$ is approximately $0.50$, with social utility approximately $3.536$ which is less than the $3.555$ achieved with a lump sum of $0.4$ for consumption of $0.9$ of $y$.

Thus for the specific example illustrated, a lump sum payment for a specific level of consumption of y can yield more social utility than any flat rate. Given the equivalence of Cases 2 and 3, it follows that the socially optimal outcome from Coasian bargaining will not in general equal the socially optimal outcome from a tax paid on to $b$.

**References**

A. Hanley N, Shogren J F & White B (2nd edn 2007) *Environmental Economics in Theory and Practice* (pp 45-8 discuss Coasian bargaining in the context of given marginal benefit and marginal cost curves, without mentioning utility).

B. [Wikipedia – Coase Theorem](https://en.wikipedia.org/wiki/Coase_theorem) (introduces the Coase Theorem in terms of economic efficiency).

C. Perman R, Ma Y, McGilvray & Common M (3rd edn 2003) *Natural Resources and Environmental Economics* (pp 137-9 discuss Coasian bargaining in the context of marginal benefit and marginal cost curves supported by utility functions which include a monetary variable (wealth)).


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