Question

A representative firm which only can survive for one period. It has the following technology.

Y=zF(K,N^d),

Where K is the given capital stock ( the representative firm owns K but no market value if the firm sells it), N^d is a labor demand by paying competitive market wage rate w, and z is total factor productivity (TFP). Let’s further assume the production functions be a continuous concave function. Without loss of generality, let’s assume the output good price equals to 1.

1 show the marginal product of labor MP_N , is a downward sloping curve respects to N^d

2. Write down the representative firm’s profit function and graphically determine or ( not both) mathematically prove the profit maximization condition. MP_N =w

3. Determine the labor demand curve and graph it. Suppose the representative firm experience a negative productivity shock. i.e. z decreases. Graph the “new” (after z changed) labor demand curve in the same graph.

Answer 1

1. A continuous concave function by definition is a function which is upwards sloping curve and reaches a maximum value before it starts to fall after reaching the maximum value, Which suggests that returns to factor of productions are decreasing because if weren't the case then production function wouldn't be a continously concave function. Which means marginal product with respect to N is downwards sloping which means the additional labor contributes less to total output. And when the marginal product with respect to Nd reaches zero production function reaches its maximum value.

2.  in the image above I have solved for profit maximizing condition both mathematically and graphically. The profit is equal to revenue - cost. Here revenue is P times Y that is output and cost is the wages given to labor. Using first order of calculas and partially differentiating both side with respect to Nd we get that profit maximizing condition is where value of marginal product of labor = wage rate.

We can understand it graphically also, see in the diagram the downwards sloping line is the demand curve for labor Nd and it is equal to the MPN (marginal product of labor N). The profit is maximized when N is at Nd* where marginal product of labor is equal to wage rate. Which is point A in the diagram. After this point the value of marginal product of labor is less than the wage rate, which means the additional revenue is less than the additional cost which reduces the profit. So the firm will employ labor upto Nd*.

3. The labor demand curve is nothing but the value of marginal product of labor(MPN) and since we know the marginal product of labor is decreasing with Nd therefore the demand curve for labor is downwards sloping line. Which i have drawn below. In the above digram the initial labor demand curve is given by the line zMPN and at wage rate W the equilibrium is at point A where equilibrium number of labors is Nd* where the profit is being maximized. After the productivity which which decreases the z to z' and the new labor demand curve is given by dotted line z'MPN. You can see there is a leftwards shift in labor demand curve because reduction in productivity z reduces the marginal product of labor MPN which cause the value of MPN to reduces at every level of N. The new equilibrium is at point B where the equilibrium number of labor is Nd'.