Sarah's Pretzel plant has the following short-run cost function: where q is Sarah's output level, w is the cost of a labor hour, and K is the number of pretzel machines Sarah leases. Sarah's short-run marginal cost curve is At the moment, Sarah leases 10 pretzel machines, the cost of a labor hour is $6.85, and she can sell all the output she produces at $35 per unit. If the cost per labor hour rises to $7.50, what happens to Sarah's optimal level of output and profits?

First, we need to determine Sarah's optimal output and profits before the increase in the wage rate. The profit maximizing output level is where the market price equals marginal cost (providing the price exceeds the average variable cost). To determine the optimal output level, we need to first equate marginal cost to the market price. That is, The average variable cost at this output level is: Since Sarah will maximize profits at 232.07 units. Sarah's profits are: To determine the optimal output level at the higher wage rate, we need to first equate marginal cost to the market price. That is, The average variable cost at this output level is: Since Sarah will maximize profits at 221.79 units. Sarah's profits are: The higher wage rate causes Sarah to reduce output and her profits also fall. In this case, profits fall by 4.9% when the wage rate rises by 9.5%.