Economics

A product called ClickClacks are manufactured by ClickClack Inc. In this project, you will use the Profit, Revenue, and Cost functions for producing and selling ClickClacks to demonstrate your knowledge of the course. Answer each of these questions with a calculation or formula and one or more English sentences to explain to the reader what the calculation does. Everything should be interpreted in such a way that it relates to the production and sale of ClickClacks.1. Choose a reasonable function for the Demand function (price as a function of quantity) and a reasonable function for Cost as a function of quantity. For this project, "reasonable" means that the following calculations can be done easily. Consider using a polynomial, square root, logarithmic, exponential, or trigonometric function or something similar for each.2. Demonstrate with your revenue function how to calculate each of the following and explain their differences:average rate of changerelative rate of changerelative changeinstantaneous rate of changemaximum rate of changetotal changemaximum change3.Use your chosen functions from Question 1 to demonstrate how to calculate each of the following. Explain in words what each calculation means for the production and/or sale of ClickClacks.average cost as a function of quantity (q)minimum average costtotal variable costfixed costtotal cost of producing the first 10 unitsaverage cost for producing 10-20 units4.In economics, they talk about "Marginal Cost" as the cost of producing one more unit. Mathematically, these are not the same. However, "the cost of producing one more unit" is an approximation for marginal cost.Explain what Marginal Cost actually is and why "the cost of producing one more unit" is a good approximation for it.Explain the relationship between Marginal Cost (MC), Marginal Revenue (MR), and Marginal Profit (MP). How can you tell which quantity maximizes profit, using MC and MR?5. (a) Marginal Revenue Product is the instantaneous rate of change with respect to an input such as labour. Let the quantity of ClickClacks (q) as a function of hours of labour per week (m) be given by q = m^2 + 3m. Use your revenue function and this information to find the marginal revenue product at 100 hours of labour per week. Show your work.(b) What does your answer to part (a) mean for ClickClack Inc?6. Choose an interval of q values on which the minimum cost is not a local minimum for the cost function. Explain why your interval accomplishes this, including all calculations.7. In words, explain the difference between the method for finding local extrema of a function f(x,y) and the Lagrange Multiplier method of finding extrema of a function f(x,y) subject to a constraint g(x,y) = c, where c is a constant. Use the specific example of f(x,y) where f is the dollars of revenue gained from a company, x is money invested in that company, and y is hours of labour put in by the people who work for the company. Describe the setup that would lead to each method of optimization (finding max/min). You can invent any details that are missing.