2. (40 points) Consider the simple model of firm choice developed in class. The firm maximizes profits \(\pi\) by choosing labor N, taking the wage \(w\) as given. The production function used by the firm is
\(zK^\alpha N^{1-\alpha}\), \(\alpha \in (0, 1)\)
where capital K is exogenously fixed and z is total factor productivity.
(a) (20) Solve for the optimal labor demand of the firm. In a diagram, show the firm's labor demand curve. In the same diagram, show what happens to the labor demand curve if an earthquake destroys part of the firm's stock of capital.
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(b) (20) Imagine there are two firms using the same production function and the same quantity of capital, but the TFP of one firm is \(z_H\) while the TFP of the other firm is \(z_L < z_H\). Which firm will hire more labor? Derive an expression showing the factor difference in labor demanded between the two firms (ie, calculate the ratio of labor demanded by firm H over firm L). If \(z_H = 4 \cdot z_L\) and \(\alpha = 1/2\), how much more labor will the larger firm hire?