- Two goods: manufactured goods and \(š\) agricultural goods \(š½\)
- Consumer
*needs*to have non-zero food, but can have zero manufactured goods and still survive.
\[ U=\ln (š½) + \gamma \ln (š+1) \]
- Both goods are produced with CRS technology. \[ Y_š = z_š N_š \] \[ Y_š½ = z_š½ N_š½ \]
- No leisure decisions, I guess, just to keep things easy. Set price of š½ as numeraire?

For non-zero finite output, it must be that \[p z_š = w = z_š½\] and for zero output of š, it must be that \[w \geq p z_š\] In either case, profits will be zero.

In an interior solution where \(š>0\), \[\frac{(š+1)}{\gamma š½} = p\] \[ pš + š½ = w \] \[š½ = \frac{w+p}{\left(1+\gamma\right)}\] \[š = \frac{w\gamma-p}{p\left(1+\gamma\right)}\] And in a solution where \(š=0\), \[š½ = w\] Note that interior solution requires that \(p < w\gamma\)

If \(\gamma z_š > 1\), then we have the interior solution: \[š½ = z_š½ \cdot \left( \frac{1+z_š}{z_š\left(1+\gamma\right)} \right) \] \[š = \frac{\gamma z_š - 1}{\left(1+\gamma\right)}\] \[N_š½ = \frac{š½}{z_{š½}} = \frac{\left(z_{š}+1\right)}{z_{š}\left(1+\gamma\right)}\] \[N_{š} = \frac{š}{z_{š}}=\frac{\gamma-\frac{1}{z_{š}}}{1+\gamma}\] Otherwise, if \(\gamma z_š \leq 1\), \[N_š½ = 1\] \[N_š = 0\] \[š½ = w = z_š½\]

From consumer: \[\frac{(š+1)}{\gamma š½} = p\] \[ pš + š½ = w \] \[š½ = \frac{w+p}{\left(1+\gamma\right)}\] \[š = \frac{w\gamma-p}{p\left(1+\gamma\right)}\] \[š½,š>0\] From firms: \[p z_š = w = z_š½\] From Market Clearing \[N_š + N_š½ = 1\] \[ Y_š = z_š N_š \] \[ Y_š½ = z_š½ N_š½ \]

Note how this implies that \[p=\frac{ z_š½ }{ z_š }\] And substitute out the prices to get allocations: \[š½ = \frac{ z_š½ + {z_š½ \over z_š} }{\left(1+\gamma\right)} = z_š½ \frac{1+z_š}{z_š\left(1+\gamma\right)} \] \[š = \frac{z_š½ \gamma-{z_š½ \over z_š}}{{z_š½ \over z_š}\left(1+\gamma\right)} = \frac{\gamma z_š - 1}{\left(1+\gamma\right)}\] \[N_š½ = \frac{š½}{z_{š½}} = \frac{\left(z_{š}+1\right)}{z_{š}\left(1+\gamma\right)}\] \[N_{š} = \frac{š}{z_{š}}=\frac{\gamma-\frac{1}{z_{š}}}{1+\gamma}\] Note that the above equations are consistent with \(š>0\) iff \(\gamma z_š > 1\). If this isn't the case, then it must instead be that \(š=0\) and \[N_š½ = 1\] \[N_š = 0\] \[š½ = w = z_š½\]

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