tailieunhanh - Basic Mathematics for Economists phần 6

cho việc quản lý ra quyết định. Tuy nhiên, cần lưu ý rằng, từ một quan điểm lý thuyết kinh tế thuần túy, lập trình tuyến tính không thể thực hiện bất kỳ dự đoán chung về giá cả hay đầu ra cho một số lượng lớn của các công ty. Tính hữu dụng của nó nằm trong lĩnh vực quản lý | Figure In Figure the rate of change of total revenue between points B and A is ATR AC the slope of the line AB AỔ BC p which is an approximate value for marginal revenue over this output range. Now suppose that the distance between B and A gets smaller. As point B moves along TR towards A the slope of the line AB gets closer to the value of the slope of TT which is the tangent to TR at A. A tangent to a curve at any point is a straight line having the slope at that point. Thus for a very small change in output MR will be almost equal to the slope of TR at A. If the change becomes infinitesimally small then the slope of AB will exactly equal the slope of TT . Therefore MR will be equal to the slope of the TR function at any given output. We know that the slope of a function can be found by differentiation and so it must be the case that dTR MR -7- dq Example Given that TR 80q - 2q2 derive a function for MR. Solution dTR MR 80 - 4q dq This result helps to explain some of the properties of the relationship between TR and MR. The linear demand schedule D in Figure represents the function p 80 2q 1 1993 2003 Mike Rosser p Figure We know that by definition TR pq. Therefore substituting 1 for p TR 80 - 2q q 80q - 2q2 which is the same as the TR function in Example above. This TR function is plotted in the lower section of Figure and the function for MR already derived is plotted in the top section. You can see that when TR is rising MR is positive as one would expect and when TR is falling MR is negative. As the rate of increase of TR gets smaller so does the value of MR. When TR is at its maximum MR is zero. With the function for MR derived above it is very straightforward to find the exact value of the output at which TR is a maximum. The TR function is horizontal at its maximum point and its slope is zero and so MR is also zero. Thus when TR is at its maximum MR 80 - 4q 0 80 4q 20 q 1993 2003 Mike Rosser One can also see that the MR .