The firm determines the best level of output and price for products that are jointly produced in fixed proportion

Joint products result from production processes that naturally yield multiple products. A decision to run the production process automatically produces the entire product group. For example, the processing of sugarcane results in a by-product called bagasse, which is the residue of the cane stalk after the juice has been squeezed out. By-products may be used, or sold or otherwise disposed of. Bagasse, for example, is burned to make steam to generate electricity. By-products that cannot be used or sold create a problem (and a cost) of disposal. The proportion of joint products may be either fixed or variable.

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__Joint Products in Fixed Proportions of One to One__

Since there is only one production process, there is no economically sound to allocate costs to the individual products. The demand curves, however, can be and usually are quite different for the main product and the by-products. Determination of the optimal output and prices involves optimization of the total marginal revenue from all products in relation to marginal cost. We shall explain the procedure with the help of figure.

In figure, it depicts two cases of the joint products A and B produced in the fixed proportion 1:1. In order to maximize profit, we must find the level of production at which MRT = MC; however, the is a complication; neither product may be sold beyond the quantity where its individual MR < 0. This is because of negative marginal revenue means we would be losing money on each unit sold. In both panels, MRs = 0 at QM units of joint production.

The step-by-step procedure for determining optimal output and price goes as follows:

**Step 1:**Develop or obtain the demand function for Product A (DA on the figure) and its related marginal revenue (MRA).

**Step 2:**Develop or obtain the demand function for By-product B (DB on the graphs) and its related marginal revenue (MRB).

**Step 3:**Add MRA and MRB to obtain MRT.

**Step 4:**Obtain the total cost function and take its derivate to get marginal cost, MC.

**Step 5:**Observe the value of Q at which MRB = 0 (QM on the graphs).

**Step 6:**Observe the level of Q at which MRT = MC (QC on the graphs).

**Step 7:**Compare QC to QM. If QC ≤ QM, then the optimal output and sales level for both products is QC. This condition is illustrated on Panel A. Go to step 9.

**Step 8:**If QC > QM, as illustrated on Panel B, then the maximum quantity of By-product B that can be sold is QM. To find the optimal quantity of the main product A, find the level at which MRA = MC. This is at Q3 on Panel B. Of course, Q3 units of By-product B will also be produced but the quantity Q3 – QM will be dumped, destroyed, or otherwise disposed of, because selling it means losing money on every unit sold. In the past, the cheapest method of disposal has too often been indiscriminate dumping. Legal and environmentally sound disposal methods may incur additional costs. These additional costs can be quite substantial, and thus provide a powerful incentive to find new uses and new markets for the unwanted product.

**Step 9:**Use the demand functions of A and B to find the prices at which the optimal quantities may be sold. Thus, in both panels, Product A would sell for PA, and By-product B would sell for PB.

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__Joint Product in Fixed Proportions Other Than a One-to-one Ratio__

If the joint products are produced in fixed proportions other than 1:1, we must first remember that the cost function pertains to output of the main product. Therefore, we want to establish a ratio of 1”x, where x is the number of units of By-product per unit of the main product. For example, suppose that the demand and cost functions remain the same for Product A and By-product B of the previous example, but the production technology changes so that the production ratio becomes QA:QB = 2:3.

We note that the ratio 2:3 is the same as the ratio 1.0:1.5. Hence QA = Q and QB = 1.5QA. By making appropriate substitutions of 1.5 QA for QB and vice versa, the procedure described above for a 1:1 ratio can be followed.

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