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The Wheat Farmer: chooses a production method that delivers a maximum profit of $\Pi_W =8$.
The Rancher: chooses herd size $x$ to maximize profits $\Pi_C(x) = P \cdot F(x) - c \cdot x^2$
Example: If $F(x) = x$, $c=\frac{1}{2}$.
The FOC are $x^{*} = P_c$
With $P_c=4$ and $c=\frac{1}{2}$, the rancher's privately optimal herd size of $x^* = 4$
With no effective barrier separating the fields cattle sometimes strays into the wheat farmer's fields, damaging crops and reducing wheat farmer's profits.
Assume that if the rancher keeps a herd size $x$ net profits in wheat are reduced from $\Pi_W$ to:
$$\Pi_W(x) = \Pi_W - d \cdot x^2$$The external cost
Suppose $d=\frac{1}{2}$
At the rancher's private optimum herd size of $x^*=4$, the farmer's profit is reduced from 8 to zero:
$$\begin{align} \Pi_W(x) &= \Pi_W - d \cdot x^2 \\ & = 8 - \frac{1}{2} \cdot 4^2 = 0 \end{align}$$
In [1]:
%load_ext autoreload
%autoreload 2
In [2]:
from coase import *
At private optimum Rancher earns \$8 but imposes external costs that drive the farmer's earnings to zero.
In [3]:
coaseplot1()
Private and social marginal benefits and costs can be plotted to see deadweight loss (DWL) differently:
In [4]:
coaseplot2()
Scenario 1: Farmer is given the right to enjoin (i.e. limit or prohibit) cattle herding.
If the farmer enforces a prohibition on all cattle herding:
Scenario 1: Farmer is given the right to enjoin (i.e. limit or prohibit) cattle herding.
Rancher reasons that if she were permitted to herd 2 cattle she'd earn $\$6$ while imposing \$2 in damage.
Scenario 2: Rancher is granted right to graze with impunity.
Farmer reasons that if herd size could be reduced from 4 to 2
"Costless bargaining is efficient tautologically; if I assume people can agree on socially efficient bargains, then of course they will... In the absence of property rights, a bargain establishes a contract between parties with novel rights that needn’t exist ex-ante." Cooter (1990)
In the Farmer and Rancher example there was a missing market for legal entitlements.
Once the market is made complete (by an assumed third party) then the First Welfare Theorem applies: complete competitive markets will lead to efficient allocations, regardless of initial allocation of property rights.
The "Coase Theorem" makes legal entitlements tradable.
In this view insuring efficiency is matter or removing impediments to free exchange of legal entitlements. However,
"The interesting case is when transaction costs make bargaining difficult. What you should take from Coase is that social efficiency can be enhanced by institutions (including the firm!) which allow socially efficient bargains to be reached by removing restrictive transaction costs, and particularly that the assignment of property rights to different parties can either help or hinder those institutions."
Good further discussions from D. Mcloskey and here:
Suppose there is an open field. In the absence of a land market whoever gets to the land first (possibly the more powerful in the the village) will prepare/clear land until the marginal value product of the last unit of land is equal to the clearing cost. We contrast two situations:
(1) Open frontier: where land is still abundant
(2) Land Scarcity.
There will be a misallocation in (2) shown by DWL in the diagram... but also an incentive for the parties to bargain to a more efficient outcome. A well functionining land market would also deliver that outcome.
Land demand for each farmer is given by $P\cdot F_T(T_i) = r$. So for this production $P \frac{1}{\sqrt T_i} = r$ or $P \frac{1}{\sqrt T_i} = cl$ so we can write
$$T^*_i(r) = (P/r)^2$$If there is an open frontier the sum or demands falls short of total land supply and the marginal cost of land is the cost of clearing $r=c_l$.
'Land scarcity' results on the other hand when there is an equilibrium price of land $r>c_l$ where $r$ is found from
$$\sum T^*_i(r) = \bar T$$Now land rent $r-c$ can be charged on the right to access and use land. Trade in these legal entitlements can raise output and efficiency. But there may be conflict and a 'scramble' to establish those rights of first access.
In [5]:
landmarket(P=5, cl = 3, title = 'Open Frontier')
In [6]:
landmarket(P=8, cl = 3, title = 'Land Scarcity')
We can solve for the equilibrium rental rate $r$ given environmental paramters including the price $P$, land endowment $\bar T$, population size $N$ and technology parameters $A)
To do: (things to still do in this notebook)