A company estimated that the relationship between the unit price and demand per month for a potential new product is approximated by P = $ 100 – $ 0.1D. The company can produce the product by increasing fixed costs $ 17,500 per month, and the estimated variable costs is $ 40 per unit. What is the optimal demand, D*, and based on this demand, should the company produce new product? Why? a) Work out the complete solution by differential calculus, starting with formula for profit or loss per month b) Solve graphically for an approximate answer

Suppose that the Travnikar Corporation manufactures widgets. Analysts that work for the company have found a function C(x) which reports the cost to produce x widgets and a function R(x) which reports the total revenue from selling x widgets. Currently, the company is producing 500 widgets, but the analysts find that R'(500)=85 and C'(500)=79. In order to maximize profits, should the analysts recommend increasing or decreasing production?

Introduction to Calculus in Economics (continued): In the previous Problem Set question, we started looking at the cost function C (æ), the cost of a firm producing z items. An important microeconomics concept is the marginal cost, defined in (non- mathematical introductory) economics as the cost of producing one additional item. If the current production level is æ items with cost C (z), then the cost of computing h additionial (C(z+h)-C(z)) items is C (z + h). The average cost of those h items is . As we analyze the cost of just the last item produced, this can be made into a mathematical model by taking the limit as h → 0, i.e. the derivative C' (z). Use this function in the model below for the Marginal Cost function MC (x). Problem Set question: The cost, in dollars, of producing z units of a certain item is given by C (z) = 0.02a3 – 10z + 450. (a) Find the marginal cost function. MC (z) (b) Find the marginal cost when 50 units of the item are produced. The marginal cost when 50…

Given a cost function: C(x) = x² - 200x - 3500 And the following revenue function: R(x) = 6x2-400x What level of production will yield a profit of $6,000? 40 or 50 units of production 60 units of production 30 or 60 units of production 30 units of production None of these options 50 units of production

ATV is a price-setting firm and estimates the demand for its cement using a demand function in the linear form: Q = f( P, M, PR) where Qc =  demand for cement/month (in yards) Pc = the price of cement per yard, M = country’s tax revenues per capita, and PR = the price of asphalt per yard. DEPENDENT VARIABLE Qc R- SQUARE P- VALUE ON F     64 0.8093 0.0001             INDEPENDENTVARIABLE PARAMETER ESTIMATE STANDARD ERROR T-RATIO P-VALUE INTERCEPT 8.20  4.01 2.04 0.0461           PC -3.54 1.64 -2.16 0.0357 M 0.64287 0.19 3.38 0.0014 PA 0.7854 0.38 2.07 0.0439 10. Write the resulting regression equation.

In a certain store, cashiers can serve 45 customers per hour on average. If a customers arrive at the store in a given hour, then the average number of customers C waiting in line can be modeled by the function x² C(x) = 2025 - 45x where x < 45. Answer the following questions with a graphing calculator. Round your answers to integers. 1) If 40 customers arrived in the store in the past hour, there are approximately 7 2) If there are 2 customers waiting in line, approximately customers waiting in line. customers arrived in the past hour.

The function that relates the inverse of speed (1/S) (i.e., the time in minutes that it takes to travel 1 mile) to traffic (T) on a highway is 1/S = 4 + 0.05T. Demand for traffic (measure as traffic volume minute) on this road is T=6880-70 (1/S) Find the equilibrium traffic volume in the absence of any congestion toll. Find the marginal cost function for traffic volume, where cost is expressed as travel time per mile in minutes. per Find the optimal level of traffic volume. Find the optimal congestion toll, expressed in minutes. (Imagine that the congestion toll consists of sitting in the “penalty box" for this period of time.) The function that relates the inverse of speed (1/S) (i.e., the time in minutes that it takes to travel 1 mile) to traffic (T) on a highway is 1/S = 4 + 0.05T. Demand for traffic (measure as traffic volume per minute) on this road is T=6,880- 70-(1/S). a. Find the equilibrium traffic volume in the absence of any congestion toll. b. Find the marginal cost…

3.  Suppose that the feasible region of a cost minimization linear programming problem has three corners points of (5,8), (10,5), and (4,10). If the objective function is given as: Minimize Z = 2X + Y     Which of the following represents an iso-cost line?     Select one: a.   X + 2Y  =  10   b.   X ­– Y  =  10   c.   2X ­– Y  =  10   d.   2X + Y  =  10   e. none of the other options.