<p> Profit Maximization: Firm Behavior</p><p>The profits of a firm are defined as Profits = Total Revenue – Total Costs. The goal of the firm is to choose the level of production that maximizes profits (Recall that for any given level of production, the choice of inputs can be determined from the isocost/isoquant analysis). </p><p>To determine the optimal level of production requires an understanding of Revenue and Costs. Given this material has been covered in earlier courses; we will proceed quickly with a review. We begin with costs.</p><p>Costs</p><p>Definition of Total Costs: </p><p>Total Costs = Fixed costs + Variable costs </p><p> or </p><p>TC= FC+VC.</p><p>Fixed costs are the costs that do not change as production changes, while variable costs are the costs that change with production. Fixed costs may include the cost of land or buildings. Another very important part of fixed costs is “opportunity costs”. Opportunity costs represent the profits that could be earned in another industry; more on this later. </p><p>Definition of Average Costs per unit:</p><p>Average Costs = Total Costs/Total Production</p><p>Or</p><p>AC = TC/Q</p><p>Substituting for TC we have</p><p>AC = (FC+VC)/Q = FC/Q + VC/Q = AFC +AVC Where AFC is average fixed costs and AVC is average variable costs. </p><p>Since AC = AFC +AVC, the graph of AC is determined by the graphs of AFC and AVC pictured below. Because fixed costs do not change as quantity produced increases, the average fixed costs is continually falling. And because we will assume variable costs are rising (this is actually the result of decreasing marginal product of inputs), the average variable costs are continually rising. Putting these two graphs together yields the U-shaped average cost curve in the second graph.</p><p>Definition of Marginal Costs</p><p>Marginal cost is the increase in total cost when production rises by one unit. That is,</p><p>To see the relationship between MC and marginal product, suppose labor is the only input. Then . But this implies . Substituting in the MC equation and we have</p><p>.</p><p>But the term in brackets is just the inverse of the marginal product of labor. Hence we have .</p><p>Hence one can see that a diminishing marginal product will give rise to an increasing marginal cost. This is pictured in the graph below.</p><p>Combining MC and AC on the Graph</p><p>The graph below combines MC and AC. Notice that the MC intersects the AC when AC is at its minimum point. This is because if MC is below AC, AC would be falling. And if MC is above AC, then AC would be rising. Hence whenever AC is falling, MC must be below AC. And whenever AC is rising, MC must be above AC. The only way this is possible, is for MC to intersect AC at its minimum point.</p><p>Revenue</p><p>Total revenue is defined as</p><p>,</p><p>Where p is the price.</p><p>Average Revenue is defined as</p><p>Because average revenue always equals the price, it is equivalent to the (inverse) demand curve faced by the firm. Marginal Revenue is defined as</p><p>Consider the term . Since price may depend on Q (due to a downward sloping demand curve), we have</p><p>.</p><p>So marginal revenue is given by</p><p>The first term captures how price may change if Q changes and the second term captures the effect of Q changing. At this point we must distinguish between two possible cases a firm may be in. </p><p>Case 1 will be a situation in which the firm can change Q with no resulting change in the price. This corresponds to a small firm in a competitive industry. In this case, =0, so we have</p><p>Case 2 will be a situation in which if the firm increases the quantity, the price must fall to sell the extra unit, so in this case <0. We thus have</p><p>But note that since <0, it must be that MR < p. But since p corresponds to the demand curve, one can say marginal revenue is less than the price indicated by the demand curve. The graphs for Case 1 and Case 2 can be seen below.</p><p>Profit Maximization</p><p>Profits are the difference between total revenue and total cost. However in deciding the optimal amount to produce to maximize profits it is better to focus on marginal revenue and marginal cost. </p><p>Notice that as long as marginal revenue of a unit exceeds the marginal cost of a unit, then the production of that unit will increase profits. Hence as long as MR > MC, the firm should produce a unit. Given MR is constant and MC is rising, this implies the firm should continue to produce until MR = MC. Call this quantity Q*. For any amount below Q*, MR is bigger than MC and those units add to profits. For any amount greater than Q*, MR is less than MC, and those units would therefore reduce profits. </p><p>Hence profits are maximized at Q*, which is where MR = MC.</p><p>This can more easily be demonstrated using a graph. Recall we have two cases to consider with MR. These are given below. Notice in case 1, since MR = p, one can say the rule for optimization is where p=MC. Also, notice in case 2 price is determined by the choice of Q*. That is, one chooses Q* such that MR = MC, and based on that Q*, one reads from the demand curve the corresponding price, p*.</p>