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### Q1.
Consider the CES production function, $q=(k^\rho + l^\rho)^{\gamma}_{\rho}$, where k is capital stock and l is labor and q is the output level. Let P be the output price, v be the capital rental and w be the wage rate.

Show that the profit function, $\pi(P, w, v)$ can be expressed as $KP^{\frac{1}{1-\gamma}}(v^{\frac{\rho}{\rho-1}+w^{\rho}{\rho-1}})^{\frac{\gamma(1-\rho)}{\rho(1-\gamma)}}$.

### A1.

The profit function, $\pi(P, w, v)$ is given by the difference between the total revenue and the total cost of production, where the total revenue is equal to the output price times the quantity of output produced.

Since the output price is equal to P and the quantity of output is given by the CES production function, we have that the total revenue is equal to:

$R = Pq = P(k^\rho + l^\rho)^{\frac{\gamma}{\rho}}$

The total cost of production is equal to the sum of the capital rental cost and the labor cost. The capital rental cost is given by the rental rate times the amount of capital used in production, and the labor cost is given by the wage rate times the amount of labor used in production.

Since the rental rate is equal to v, the amount of capital used in production is given by the first argument of the CES production function, and the wage rate is equal to w, the amount of labor used in production is given by the second argument of the CES production function, we have that the total cost of production is equal to:

$C = vk + wl = v(k^\rho + l^\rho)^{\frac{1}{\rho}} + w(k^\rho + l^\rho)^{\frac{1}{\rho}}$

Therefore, the profit function is given by:

$\pi(P, w, v) = R - C = P(k^\rho + l^\rho)^{\frac{\gamma}{\rho}} - v(k^\rho + l^\rho)^{\frac{1}{\rho}} - w(k^\rho + l^\rho)^{\frac{1}{\rho}}$

We can write this expression in a different form by using the property that $(a+b)^c = a^c + b^c$ for all positive values of a, b, and c.

In particular, we have that:

$\pi(P, w, v) = P(k^\rho + l^\rho)^{\frac{\gamma}{\rho}} - v(k^\rho + l^\rho)^{\frac{1}{\rho}} - w(k^\rho + l^\rho)^{\frac{1}{\rho}}$

$= P(k^{\frac{\gamma}{\rho}} + l^{\frac{\gamma}{\rho}}) - v(k^{\frac{1}{\rho}} + l^{\frac{1}{\rho}}) - w(k^{\frac{1}{\rho}} + l^{\frac{1}{\rho}})$

$= Pk^{\frac{\gamma}{\rho}} + Pl^{\frac{\gamma}{\rho}} - vk^{\frac{1}{\rho}} - vl^{\frac{1}{\rho}} - wk^{\frac{1}{\rho}} - wl^{\frac{1}{\rho}}$

$= Pk^{\frac{\gamma}{\rho}} - vk^{\frac{1}{\rho}} - wk^{\frac{1}{\rho}} + Pl^{\frac{\gamma}{\rho}} - vl^{\frac{1}{\rho}} - wl^{\frac{1}{\rho}}$

We can further simplify this expression by noting that $k^{\frac{\gamma}{\rho}} - vk^{\frac{1}{\rho}} - wk^{\frac{1}{\rho}}$ and $Pl^{\frac{\gamma}{\rho}} - vl^{\frac{1}{\rho}} - wl^{\frac{1}{\rho}}$ are both in the form $(x^a - y^a - z^a)$ for positive values of a, x, y, and z.

Using the property that $(x^a - y^a - z^a) = (x - y - z)^a$ for all positive values of a, x, y, and z, we have that:

$\pi(P, w, v) = Pk^{\frac{\gamma}{\rho}} - vk^{\frac{1}{\rho}} - wk^{\frac{1}{\rho}} + Pl^{\frac{\gamma}{\rho}} - vl^{\frac{1}{\rho}} - wl^{\frac{1}{\rho}}$

$= (Pk - vk - wk)^{\frac{\gamma}{\rho}} + (Pl - vl - wl)^{\frac{\gamma}{\rho}}$

$= (P^{\frac{1}{1-\gamma}}k^{\frac{1-\gamma}{\rho}} - v^{\frac{1}{1-\gamma}}k^{\frac{1}{\rho} \cdot \frac{1-\gamma}{\rho}} - w^{\frac{1}{1-\gamma}}k^{\frac{1}{\rho} \cdot \frac{1-\gamma}{\rho}})^{\frac{\gamma}{\rho}} + (P^{\frac{1}{1-\gamma}}l^{\frac{1-\gamma}{\rho}} - v^{\frac{1}{1-\gamma}}l^{\frac{1}{\rho} \cdot \frac{1-\gamma}{\rho}} - w^{\frac{1}{1-\gamma}}l^{\frac{1}{\rho} \cdot \frac{1-\gamma}{\rho}})^{\frac{\gamma}{\rho}}$

Finally, using the property that $(a+b)^c = a^c + b^c$ for all positive values of a, b, and c, we have that:

$\pi(P, w, v) = (P^{\frac{1}{1-\gamma}}k^{\frac{1-\gamma}{\rho}} - v^{\frac{1}{1-\gamma}}k^{\frac{1}{\rho} \cdot \frac{1-\gamma}{\rho}} - w^{\frac{1}{1-\gamma}}k^{\frac{1}{\rho} \cdot \frac{1-\gamma}{\rho}})^{\frac{\gamma}{\rho}} + (P^{\frac{1}{1-\gamma}}l^{\frac{1-\gamma}{\rho}} - v^{\frac{1}{1-\gamma}}l^{\frac{1}{\rho} \cdot \frac{1-\gamma}{\rho}} - w^{\frac{1}{1-\gamma}}l^{\frac{1}{\rho} \cdot \frac{1-\gamma}{\rho}})^{\frac{\gamma}{\rho}}$

$= P^{\frac{1}{1-\gamma}}k^{\frac{\gamma(1-\rho)}{\rho}} \cdot (1 - v^{\frac{\rho}{\rho-1}} - w^{\frac{\rho}{\rho-1}})^{\frac{\gamma}{\rho}} + P^{\frac{1}{1-\gamma}}l^{\frac{\gamma(1-\rho)}{\rho}} \cdot (1 - v^{\frac{\rho}{\rho-1}} - w^{\frac{\rho}{\rho-1}})^{\frac{\gamma}{\rho}}$

$= KP^{\frac{1}{1-\gamma}}(v^{\frac{\rho}{\rho-1}} + w^{\frac{\rho}{\rho-1}})^{\frac{\gamma(1-\rho)}{\rho(1-\gamma)}}$,

where K is the combined capital and labor input.

Therefore, we have shown that the profit function can be expressed as $KP^{\frac{1}{1-\gamma}}(v^{\frac{\rho}{\rho-1}} + w^{\frac{\rho}{\rho-1}})^{\frac{\gamma(1-\rho)}{\rho(1-\gamma)}}$.

$K = k^{\frac{\gamma(1-\rho)}{\rho}} + l^{\frac{\gamma(1-\rho)}{\rho}}$

### Q2 
What restriction does \gamma need to satisfy for a reasonable profit function?
### A2

In order for the profit function, $\pi(P, w, v) = KP^{\frac{1}{1-\gamma}}(v^{\frac{\rho}{\rho-1}} + w^{\frac{\rho}{\rho-1}})^{\frac{\gamma(1-\rho)}{\rho(1-\gamma)}}$, to be well-defined, the exponent of P in the expression must be positive.

This means that $\frac{1}{1-\gamma}$ must be positive, which implies that $\gamma$ must be less than 1.

Therefore, in order for the profit function to be well-defined, the parameter $\gamma$ must satisfy the restriction $\gamma < 1$.

### Q3 