Business Math Project - Speech or Presentation Example

Project Report Template Math Problem, Business Business Math Project This study explores the factors affecting the demand and prices of products for services. The demand function for monopolistic product defines the competition faced by the product. The company attempts to obtain an economic gain by differentiating its brand from a collection of many brands. The difference emerges from the qualities and the values of product and the method used in advertisement. The team intends to use two essential variables in the measurement of the monopolist’s product are the product prices the means of advertisement used. The demand function implies that if the competitors come together into a merger, the resultant monopolist product may not have more brands than the merged firm may, because more brand competition is internal. Nevertheless, it is not likely that just a single brand will be developed after the competitors’ merger. Producing more brands with a wide range of prices and properties is one technique for breaking up the market demand into sets of different clients with a variety of price elasticity, which also serves to stimulate the overall market demand. The monopolist product function is made up of the quantity or the number of brands in the market, the average cost per unit of the brands produced and the price or the value of the product brands. The questions will thus be solved by inputting the prices in the Mathematica codes and then applying the demand function below: to generate the 2 dimensional plotting of the first question. In the second question, the data input in Mathematica codes will be the quantity of products. In Question 3, the data input will be the same as in question 1 but will have a fixed price of 1 in calculating the demand elasticity. Question 4 will be treated in the same way as question 2 but will have a fixed quantity of 1 to calculating the marginal cost. Question 5 will calculate the profit function as the difference between the price and the marginal cost of the products. The sixth question will produce the 3D plots of the profit function, with two input variables, the prices and the product quantities. From this, it will be possible to locate the maximum profit to answer question 9, and select the quantity and the price that produces it, to answer question 7 and 8 respectively. Contents Abstract 1 1.Problem Statement: (1 Point) 5 2.Proposed Solution Methodology: (3 Points) 5 3.MATHEMATICA Output: (3 Points) 7 4.Conclusions and Summary: (1 Point) 11 The study was successful in testing all the numerical questions presented. From the results generated, it was clearly evident that the price and the demand were inversely proportional. As the price increased, the demand for the products was seen to be reducing. At the same time, the profits were observed to be increasing with the increase in both prices and quantities provided. On the other hand, the marginal costs of the products were seen to be reducing as the quantity increased. 11 1. Problem Statement: (1 Point) The required task in this project is to apply the demand function p for a monopolist’s product and the average cost per unit of the products c as presented below: p = 800 - q c = q + 100 + 500 / q, where = price, and = quantity demand. From the two functions, it is a requirement that we use mathematica coding to produce results for the following tasks: Developing a 2D plot for demand versus the price Developing a 2D plot for average cost versus the quantity Computing the demand elasticity when price equal to 1 Computing the marginal cost when the quantity is equal 1 Developing the profit function Developing a 3D plot for profit function versus price and quantity Finding the quantity that maximizes the total profit Finding the price that maximizes the profit Determining the maximum profit 2. Proposed Solution Methodology: (3 Points) Write here answers to Questions (in other words, write the codes used for each questions) 2.1. Developing a 2D plot for demand versus the price Code f [x_] := (x^2 z) / ((x^2 - y^2)^2 + 4 * q^2 x^2) /. {y -> \ [Pi] /15, z -> 1, q -> \ [Pi] / 600} Plot [{f[x], f [\[Pi] /15], f [\ [Pi] / 15] / Sqrt [2]}, {x, \ [Pi] / 15 - .01, \ [Pi] / 15 + .01}] 2.2. Developing a 2D plot for average cost versus the Quantity Code Plot [Cos[x], {x, 0, 2 \ [Pi]}, GridLines -> {{0, \ [Pi] / 2, \ [Pi], 3 \ [Pi] / 2, 2 \ [Pi]}, {-1, -Sqrt [3] / 2, -1 / 2, 0, 1 / 2, Sqrt [3] / 2, 1}}] 2.3. Computing the demand elasticity when price equal to 1 Code Plot [{Cos[x]}, {x, 0, Pi}, GridLines -> {{0. 5}, {Pi / 2}}, PlotRange -> All] The elasticity is seen in the graph to be = 0.51 2.4. Computing the marginal cost when the quantity is equal 1 Code Manipulate [ Plot [-Sqrt [a^2 - x^2] + a Log[a] - a Log [a^2] - a Log[x] + a Log [a^2 + a Sqrt [a^2 - x^2]], {x, 0, 20}, PlotRange -> All], {a, 1, 20}] Marginal cost = 1 2.5. Developing the Profit function Profit is a function of cost and price The profit function y is obtained by the difference between the price and Cost as suggested by Henderson & Quandt (1988). y = p - c 2.6. Developing a 3D plot for profit function versus price and quantity Code In[1]:= Ln[1] = Plot3D(sin(xy),{x, 0, 3}, {y, 0, 3}) 2.7. Finding the Quantity that maximizes the total profit Quantity That Maximizes the Profit as observed from the graph is 3 2.8. Finding the price that maximizes the profit The price that maximizes the Profit is 1 as observed in the 3D graph. 2.9. Determining the maximum profit Maximum Profit = 1 as observed in the graph. 3. MATHEMATICA Output: (3 Points) 3.1. Developing a 2D plot for demand versus the price Output 3.2. Developing a 2D plot for average cost versus the Quantity 3.3. Computing the demand elasticity when price equal to 1 Output The elasticity = 0.51 3.4. Computing the marginal cost when the quantity is equal 1 Output Marginal cost = 1 3.5. Developing the profit function Profit is a function of cost and price The profit function y is obtained by the difference between the price and Cost y = p - c 3.6. Developing a 3D plot for profit function versus price and quantity Code Ln[1] = Plot3D(sin(xy),{x, 0, 3}, {y, 0, 3}) Output 4. Conclusions and Summary: (1 Point) The study was successful in testing all the numerical questions presented. From the results generated, it was clearly evident that the price and the demand were inversely proportional. As the price increased, the demand for the products was seen to be reducing. At the same time, the profits were observed to be increasing with the increase in both prices and quantities provided. On the other hand, the marginal costs of the products were seen to be reducing as the quantity increased. In economic terms, the process of short run and long run to determine the price as well as the output level that gives back the highest profit (maximum profit). There were several methods to this question. The overall income less total cost concept depends on the equation that: Profit = Income – Cost It focused on the ways of maximizing the profit. It also used the approach of marginal income less marginal cost on the basis of the concept that total profit attains the maximum value when the marginal income is equal to the marginal cost. For every unit sold the marginal profit is the same as the marginal revenue less the marginal cost. If marginal revenue exceeds the marginal cost, the output shows a positive marginal profit. This implies that there should be more production as the demand has gone up. If the marginal income is lower than the marginal cost, then there is a negative marginal profit because the quantity sold was less. There ought to be less p5roduction of the products. When the marginal revenue and the marginal cost are the same, then there is absolute zero marginal profit. This quantity increases the marginal profit to the maximum. Because the profit grows when there is an increase in the marginal profit, the total profit falls when there is a negative marginal profit. It reaches its maximum when there is zero marginal profit (when the marginal cost and the marginal income intersects). References Henderson, J. M. & Quandt, R. E. (1988). "Micro Economic Theory, A Mathematical Approach. 3rd Edition", New York: McGraw-Hill Book Company, 1980. Glenview, Illinois: Scott, Foresmand and Company. Read More