Question: Suppose that the firm's production function is

\(y=k^{5} l^{25} m^{25}\)

Where \(k\) is capital, \(l\) is labor, and \(m\) materials. Assume that capital is constant at level \(\bar{k}\) and its price is \(r\). The prices of labor and material are \(w\) and \(u\) respectively.

1. Derive the conditional factor demands for \(l\) and \(m\).
2. Derive the cost function.
3. Show that the conditional factor demands are homogeneous of degree 0 in input prices.
4. Show that the cost function is homogeneous of degree 1 in input prices.

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