Formal Definition of Equilibrium:

Click to expand definition
Consumer's Problem:
Taking prices \(\{w,p\}\) as given, chooses \(\{šŸ‘š,šŸŒ½\}\) to solve \[\max_{šŸ‘š,šŸŒ½} \ln (šŸŒ½) + \gamma \ln (šŸ‘š+1)\] subject to the constraints that \[šŸ‘š,šŸŒ½ \geq 0\] \[pšŸ‘š + šŸŒ½ \leq w + \pi_šŸ‘š + \pi_šŸŒ½\]
Manufacturing Firm's Problem:
Taking prices \(\{w,p\}\) as given, chooses \(\{N_šŸ‘š\}\) to solve \[\max p z_šŸ‘š N_šŸ‘š - w N_šŸ‘š\]
Agricultural Firm's Problem:
Taking prices \(\{w,p\}\) as given, chooses \(\{N_šŸŒ½\}\) to solve \[\max z_šŸŒ½ N_šŸŒ½ - w N_šŸŒ½\]
Market Clearing:
\[N_šŸ‘š + N_šŸŒ½ = 1\] \[ šŸ‘š = z_šŸ‘š N_šŸ‘š \] \[ šŸŒ½ = z_šŸŒ½ N_šŸŒ½ \]

Solution to firms' problems:

For non-zero finite output, it must be that \[p z_šŸ‘š = w = z_šŸŒ½\] and for zero output of šŸ‘š, it must be that \[w \geq p z_šŸ‘š\] In either case, profits will be zero.

Solution to consumer's problem:

In an interior solution where \(šŸ‘š>0\), \[\frac{(šŸ‘š+1)}{\gamma šŸŒ½} = p\] \[ pšŸ‘š + šŸŒ½ = w \] \[šŸŒ½ = \frac{w+p}{\left(1+\gamma\right)}\] \[šŸ‘š = \frac{w\gamma-p}{p\left(1+\gamma\right)}\] And in a solution where \(šŸ‘š=0\), \[šŸŒ½ = w\] Note that interior solution requires that \(p < w\gamma\)

Click to expand work for consumer problem Assume profits are zero. Will be true in equilibrium, so no big deal. Then Assuming interior solution where \(šŸŒ½,šŸ‘š>0\), Lagrangian is: \[\mathcal{L} = \ln (šŸŒ½) + \gamma \ln (šŸ‘š+1) - \lambda [ pšŸ‘š + šŸŒ½ - w ] \] FOCs: \[{1 \over šŸŒ½} = \lambda\] \[{\gamma \over (šŸ‘š+1)} = \lambda p\] \[pšŸ‘š + šŸŒ½ = w \] Implies \[{\gamma \over (šŸ‘š+1)} = {1 \over šŸŒ½} p\] Solve this to express each variable in terms of the other: \[šŸŒ½=šŸ‘š\frac{p}{\gamma}+\frac{p}{\gamma}\] \[šŸ‘š=šŸŒ½\frac{\gamma}{p}-1\] Combine these with budget constraint to solve explicitly for allocation variables: \[šŸŒ½ = \frac{w+p}{\left(1+\gamma\right)}\] \[šŸ‘š = \frac{w\gamma-p}{p\left(1+\gamma\right)}\]

Combine problems to get equilibrium

If \(\gamma z_šŸ‘š > 1\), then we have the interior solution: \[šŸŒ½ = z_šŸŒ½ \cdot \left( \frac{1+z_šŸ‘š}{z_šŸ‘š\left(1+\gamma\right)} \right) \] \[šŸ‘š = \frac{\gamma z_šŸ‘š - 1}{\left(1+\gamma\right)}\] \[N_šŸŒ½ = \frac{šŸŒ½}{z_{šŸŒ½}} = \frac{\left(z_{šŸ‘š}+1\right)}{z_{šŸ‘š}\left(1+\gamma\right)}\] \[N_{šŸ‘š} = \frac{šŸ‘š}{z_{šŸ‘š}}=\frac{\gamma-\frac{1}{z_{šŸ‘š}}}{1+\gamma}\] Otherwise, if \(\gamma z_šŸ‘š \leq 1\), \[N_šŸŒ½ = 1\] \[N_šŸ‘š = 0\] \[šŸŒ½ = w = z_šŸŒ½\]

Click to expand work for solving for the equilibrium allocations For an interior solution, we have the following characterizing equations:
From consumer: \[\frac{(šŸ‘š+1)}{\gamma šŸŒ½} = p\] \[ pšŸ‘š + šŸŒ½ = w \] \[šŸŒ½ = \frac{w+p}{\left(1+\gamma\right)}\] \[šŸ‘š = \frac{w\gamma-p}{p\left(1+\gamma\right)}\] \[šŸŒ½,šŸ‘š>0\] From firms: \[p z_šŸ‘š = w = z_šŸŒ½\] From Market Clearing \[N_šŸ‘š + N_šŸŒ½ = 1\] \[ Y_šŸ‘š = z_šŸ‘š N_šŸ‘š \] \[ Y_šŸŒ½ = z_šŸŒ½ N_šŸŒ½ \]
Note how this implies that \[p=\frac{ z_šŸŒ½ }{ z_šŸ‘š }\] And substitute out the prices to get allocations: \[šŸŒ½ = \frac{ z_šŸŒ½ + {z_šŸŒ½ \over z_šŸ‘š} }{\left(1+\gamma\right)} = z_šŸŒ½ \frac{1+z_šŸ‘š}{z_šŸ‘š\left(1+\gamma\right)} \] \[šŸ‘š = \frac{z_šŸŒ½ \gamma-{z_šŸŒ½ \over z_šŸ‘š}}{{z_šŸŒ½ \over z_šŸ‘š}\left(1+\gamma\right)} = \frac{\gamma z_šŸ‘š - 1}{\left(1+\gamma\right)}\] \[N_šŸŒ½ = \frac{šŸŒ½}{z_{šŸŒ½}} = \frac{\left(z_{šŸ‘š}+1\right)}{z_{šŸ‘š}\left(1+\gamma\right)}\] \[N_{šŸ‘š} = \frac{šŸ‘š}{z_{šŸ‘š}}=\frac{\gamma-\frac{1}{z_{šŸ‘š}}}{1+\gamma}\] Note that the above equations are consistent with \(šŸ‘š>0\) iff \(\gamma z_šŸ‘š > 1\). If this isn't the case, then it must instead be that \(šŸ‘š=0\) and \[N_šŸŒ½ = 1\] \[N_šŸ‘š = 0\] \[šŸŒ½ = w = z_šŸŒ½\]

Key takeaways.

Below a certain threshold of productitivity, none of the "non-essential" good is produced