Questions and Answers of Numerical Mathematical Economics

A company shows for a given month the following customer order data (Table 3):a. Assume a normal distribution construct an empirical histogram and an interval at 95% probability in which the possible
A company is planning the launch of a new product, and they project for this new product the following data for unit price and quantities. The company has employed various marketing techniques to
Inventory. A merchant knows that the number of items he sells in a month is distributed according to a Poisson distribution. On average, he sells 48 items per year. If at the beginning of a month he
Inventory. A merchant knows that the number of a certain kind of item that he can sell in a given period of time is Poisson distributed. How many of these items should the merchant stock, so that the
Some cars arrive at a queue with an average rate of occurrence of four per minute. Assume the cars arrive at the queue with a Poisson distribution, and determine the probability that at least two
Suppose that the average number of calls arriving at the switchboard of a corporation is 30 calls per hour. Assume that the number of calls arriving during any time period has a Poisson distribution
Plot three Poisson distributions with:a. l ¼ 4:0b. l ¼ 1:0c. l ¼ 0:5
We are required to conduct a poll in a town where on average we know that 40% of the population agrees and 60% disagrees with a new town decree. If we interview 10 people (randomly chosen) solve for
Consider a fair coin and run n ¼ 5 Bernoulli independent trials of coin flip.a. What is the probability of obtaining exactly x ¼ 2 heads, considering p ¼ 1 /2 ?b. Plot the binomial distribution
Run a three-coin digital flip Monte Carlo experiment and calculate the probability of having two heads, in case you throw the three fair coins eight times. See Table 1 below which represents the
Table 6 shows for a group of companies the relation between their profits and some explanatory variables.a. Run a multiple regression with ANOVA table in all three independent variables, and using
a. Run a linear regression of the branch revenues versus the parent company revenues (the independent variable) and make an estimate for the budget for Q1 and Q2 of Year 4.b. Obtain the ANOVA table
A company shows the following historical sales data (Table 4):a. Run the linear regression model and obtain the ANOVA table.b. Make a prediction of the quantity sold for year 2019 and build the
Table 3 puts together the excess (over a risk-free rate) monthly returns of a mutual fund versus its benchmark.According to the model of TreynoreMazuy,1 we want to run the following regression: EF ¼
Table 2 gathers the data for Y ¼ demand of roses in a town and the following independent variables. X2 ¼ quantity of roses sold X3 ¼ average wholesale price of carnations X4 ¼ average weekly
Table 1 gives data on GNP and four definitions of the money stock for the United States for 1970e83.a. Regress GNP versus each definition of money. Which definition of money seems to be closely
Inventory and optimal procurement policy (deterministic single item inventory model). This is the classical application of finding the optimal economic order quantity (aka: EOQ). See also exercise 17
There are two items with cost of surplus c1, cost of shortage c2, and discrete probability distribution for each item request as follows: Item 1 c1 ¼ $2; c2 ¼ $130 f1ðrÞ ¼ 8 >< >: 0123 0:4 0:3
Consider the following case.1 There are certain rather expensive items (some costing over $ 100; 000 each) known as insurance spares which are generally procured at the time a new class of ship is
Consider the following continuous probability density function for a certain type of goods required in a month: fðrÞ ¼ 8 >>>>>>>>< >>>>>>>>: 0 r < 80 f2ðrÞ¼1 5 þ r 400 80 r < 100 f3ðrÞ ¼
Wagner-Whitin dynamic programming approach. Use the demand and manufacturing data gathered in Table 7 to set up the optimal production and inventory schedule over four periods. No shortages are
A company has to plan a production schedule over a four-period horizon, where the re- quirements by period are 20, 10, 40, and 30 units, respectively. Costs for holding a unit of inventory are h =3,
A company has to organize its aggregate production schedule for the next 3 months. Units may be produced on regular time or overtime. The relevant costs and capacities are shown in Table 5 below.
A company has two manufacturing plants (PA and PB) and three sales stores (I, II, and III). The shipping costs from production centers to the store centers, the manufacturing data with capacity
Build a dynamic monthly production schedule Pt over 6 months via a linear program using the data in Table 1 for demands, unit production costs, and costs for holding a unit of inventory. Shortages
Optimal advertising policy. Let us consider a company that wishes to set an optimal plan for its advertising expenditure over a period of 3 years, assuming the profit available for this expenditure
Mine ore extraction. Suppose we are granted operating a mine for 10 years, with initial available ore equal to 1000. We want to choose the optimal ore extraction u(t) such that we maximize the
Optimal consumption. Let C(t) ¼ u(t) be the consumption in period t. Solve the following consumption model for T ¼ 2 and T ¼ 4, with K(0) ¼ 5,000, discounting rate 5%, a ¼ 1, A ¼ 1: max fug XT
Use the Excel Solver and the Data Table to solve the following discrete dynamic problem. max fug X2 t ¼ 0 y2 ðtÞ þ u2 ðtÞ þ y2 ð3Þ s:t: yðt þ 1Þ ¼ yðtÞ þ uðtÞ and y(0) ¼ 1. When
Use the Excel Solver to solve the following discrete dynamic problem. Repeat the exercise by hand, using the fundamental recursive equation of dynamic programming 11.2-1. max fug X2 t ¼ 0 1 y2
Use the Excel Solver to solve the following discrete dynamic problem: max fug X 3 t ¼ 0 2 3 uðtÞyðtÞ þ lnðyðTÞÞ s:t: yðt þ 1Þ ¼ yðtÞð1 þ uðtÞyðtÞÞ and y(0) ¼ 1.
Use the Excel Solver to solve the following discrete dynamic problem: max fug X 3 t ¼ 0 1 þ yðtÞ u2 ðtÞ s:t: yðt þ 1Þ ¼ yðtÞ þ uðtÞ and y(0) ¼ 0.
Solve in Excel, as well as by hand, setting up the recursive equations, a multistage allocation problem as in Section 11.4, with the following data: x0 ¼ 10; N ¼ 4; gðyÞ ¼ 3y; hðx yÞ ¼
Reconsider the problem of Example 1 reformulated in a continuous framework as follows: y_ðtÞ ¼ uðtÞ so that the following performance measure is minimized: JðuÞ ¼ yð2Þ þ Z2 0 u2 dt: With
You are given a route map as represented in Fig. 3 with costs from node to node of the route. Solve it in Excel for the minimum cost route to choose, to reach destination h from current locationa.
Solve in Excel the directed graphs given in Figs. 1 and 2, finding the shortest path from node 1 to node 9.
Consider the system of two linear differential equations: y_ 1ðtÞ ¼ y2ðtÞ y_ 2ðtÞ¼ y2ðtÞ þ uðtÞ which needs to be controlled to minimize: JðuÞ ¼ 1 2 ZT 0 y2 1ðtÞ þ u2 ðtÞ dt:
Consider the system of two linear differential equations: y_ 1ðtÞ ¼ y2ðtÞ y_ 2ðtÞ¼ y2ðtÞ þ uðtÞwhich needs to be controlled to minimize: JðuÞ ¼ 1 2 ZT 0 y2 1ðtÞ þ y2 2ðtÞ þ u2
The following second-order differential system: y_ 1ðtÞ ¼ y2ðtÞ y_ 2ðtÞ ¼ 2y1ðtÞ y2ðtÞ þ uðtÞ needs to be controlled to minimize the performance measure: JðuÞ¼½y1ðTÞ 1 2 þ
Find the optimal control for the system: y_ 1ðtÞ ¼ ayðtÞ þ uðtÞ to minimize: JðuÞ ¼ 1 2 Hy2 ðTÞ þ ZT 0 1 4 u2 ðtÞdta. Solve the problem with H ¼ 5, T ¼ 15, a ¼ 0.2, and y(0) ¼
Consider the system of two linear differential equations: y_ 1ðtÞ ¼ y2ðtÞ y_ 2ðtÞ ¼ 2y1ðtÞ y2ðtÞ þ uðtÞ which needs to be controlled to minimize: JðuÞ ¼ 1 2 ZT 0 y2 1ðtÞ þ 1
Solve the following LQR problem, which slightly changes Example 1 in Section 10.8. min fug JðuÞ ¼ 1 2 ½y1ðTÞ 5 2 þ 1 2 ½y2ðTÞ 2 2 þ 1 2 Z2 0 1 2 u2 ðtÞdt s:t: yð0Þ ¼ 0 and
Bang-bang optimal maintenance and replacement model. 10 Consider a machine whose resale value gradually declines over time, and its output is assumed to be proportional to its resale value. It is
Bang-bang optimal advertising policy control problem. The model presented here attempts to determine the best advertising expenditure for a firm, which produces a single product and sells it into a
Bang-bang optimal investment control problem. Solve the following optimal control investment model: max fIðtÞg JðIðtÞÞ ¼ Z 1 0 KðtÞ 1 2 IðtÞ dt s:t: K_ðtÞ ¼ IðtÞ dKðtÞ
Optimal consumption. Solve the following optimal control consumption model: max fcðtÞg JðcðtÞÞ ¼ Z 5 0 bt lnðcðtÞÞdt s:t: K_ðtÞ¼ cðtÞ Kð0Þ ¼ 5; Kð5Þ ¼ 0 and where b ¼ 1
Solve in Excel the following problem, using the numerical technique of the steepest descent, with the VBA code proposed in the chapter: min fug JðuÞ ¼ Z 1 0 u2 dt y_ ¼ y þ u with initial
Solve in Excel the following problem, using the numerical technique of the steepest descent, with the VBA program proposed in the chapter: min fug JðuÞ ¼ Z 1 0 1 2 u2 dt þ y2 ð1Þ y_ ¼ y þ u
Solve the following bang-bang control problem with a horizontal terminal line: max fug JðuÞ ¼ Z 4 0 1dt s:t: y_ ¼ y þ u yð0Þ ¼ 5; yðTÞ ¼ 11; T free and. u˛U ¼ ½1; 1: The exact
Solve the following bang-bang control problem: max fug JðuÞ ¼ Z 4 0 3ydt s:t: y_ ¼ y þ u yð0Þ ¼ 5; yð4Þ 300 and. u˛U ¼½0; 2:In this problem, terminal time T is fixed, but the terminal
Solve the following bang-bang control problem: max fug JðuÞ ¼ Z 4 0 3ydt s:t: y_ ¼ y þ u yð0Þ ¼ 5; yð4Þ free and u˛U ¼½0; 2:
Maximize the following performance measure: JðuÞ ¼ Z 1 0 u2 dt s:t: y_ðtÞ ¼ y þ u and y(0) ¼ 1, y(1) ¼ 0.
A first-order system is described by the following equation of motion in the state variable: y_ ¼ y þ u with initial condition y(0) ¼ 4 and y(T) free. Find in Excel the optimal control
Find the extremal in the following constrained problem, with two differential equation constraints: min J ¼ Z 1 0 1 2 y2 1 þ y2 2 þ y2 3 dt s:t: y0 1 ¼ y2 y1 y0 2 ¼ 2y1 3y2 þ y3
Find the extremal in the following isoperimetric constrained problem: min J ¼ Z 1 0 1 2 y2 1 þ y2 2 þ 2y' 1y' 2 dt s:t: Z 1 0 y2 2dt ¼ 8 ðIsoperimetric constraintÞ Exe and y1ð0Þ ¼ 0;
Find the extremal for the following constrained CoV problem, involving two state variables and a constraint represented by a differential equation: J ¼ Z 1 0 y1 y2 2 dt s:t: dy1 dt ¼ y2 and
Optimal adjustment of labor demand.7 Consider a firm that has decided to raise the labor input from L0 to an undetermined optimal level L T after encountering a wage shock reduction at t ¼ 0. The
Optimal rate of advertising expenditures. Let us consider the case of a company that produces a good having a seasonal demand pattern. Rate of sales depends not only on the time of year t but also on
Optimal consumption Ramsey model. Solve the Ramsey problem, finding the optimal capital path K*(t), such that the following functional is maximized: UðKÞ ¼ Z 1 0 ½UðCðtÞÞe t=4 dt ¼ Z 1 0
Optimal consumption Ramsey model. Solve the Ramsey problem, finding the optimal capital path K*(t) such that the following functional is maximized: UðKÞ ¼ Z 1 0 ½UðCðtÞÞdt ¼ Z 1 0
Unemployment and inflation. Repeat the problem of finding the optimal path of inflation rate in Section 9.6, transformed now in the following problem with free terminal value: min JðpÞ ¼ Z 1 0
Find the extremal of the following functional in Excel using the Solver, and compare the numerical solution to the exact solution: J ¼ Z 1 0 y þ yy0 þ y0 þ 1 2 ðy0 Þ 2 dt s:t: yð0Þ ¼ 0
Find the extremal of the following functional in Excel using the Solver, and compare the numerical solution to the exact solution: J ¼ Z 2 0 y2 þ t 2 y0 dt s:t: yð0Þ ¼ 0 and yð2Þ ¼ 2
Find the extremal of the following functional in Excel using the Solver, and compare the numerical solution to the exact solution: J ¼ ZT 0 h ty þ ðy0 Þ 2 i dt s:t: yð0Þ ¼ 1; yðTÞ ¼ 10 and
Find the extremal for the following CoV problem, involving two independent functions: J ¼ Z p 4 0 y2 1 þ y0 1y0 2 þ y2 2 dt s:t: y1ð0Þ ¼ 1; y2ð0Þ ¼ 3=2 and y1 p 4 ¼ 2; y2 p 4 ¼ free
Solve the following CoV problem in Excel using the Solver, and compare the numerical solution to the exact solution: min J ¼ Z 1 0 h 10ty þ ðy0 Þ 2 i dt s:t: yð0Þ ¼ 1 and yð1Þ ¼ 2
Solve the following CoV problem in Excel using the Solver, and compare the numerical solution to the exact solution: min J ¼ Z 5 1 h 3t þ ðy0 Þ 1=2 i dt s:t: yð1Þ ¼ 3 and yð5Þ ¼ 7
Solve by hand, using the Lagrange multipliers technique, on a discrete basis and over three stages only, the following CoV problem: min J ¼ Z2 0 h 12ty þ ðy0 Þ 2 i dt s:t: yð0Þ ¼ 0 and yð2Þ
Solve using the contour lines in Excel the following CoV discrete form of the shortest distance problem: J ¼ X2 i ¼ 1 " 1 þ ðDiyÞ 2 ðDitÞ 2 #1=2 Dit ¼ " 1 þ ðD1yÞ 2 ðD1tÞ 2 #1=2 D1t þ "
Another restriction is that the resulting asset allocation should have no more than 30% invested in the Emerging Markets area. Is the allocation you have chosen satisfying this constraint?
The insurance company restricts then to invest in a portfolio of funds which could possibly show no more than 1.50% Tracking Error Volatility versus the benchmark. Identify on the MVF the optimal
On the basis of the Tracking Error Volatility and Alpha of the funds, run the optimization program and build a Minimum Variance Frontier (MVF).
The alpha of each of the 16 funds. The alpha of a fund is a financial indicator of over/ under performance of the mutual fund versus the benchmark, and it is measured via the following linear
Funds monthly relative returns and relative Tracking Error Volatility (TEV) for each fund versus the benchmark given by the insurance company in the 3 years period. The TEV is the volatility of the
Matrix of correlation for all these 16 funds.
A company owns three production centers P1, P2, and P3, whose production capacity is of units 10, 15, and 8, respectively. These products have to be shipped in four depositary centers D1, D2, D3, and
A company owns three production centers P1, P2, and P3, whose production capacity is of units 5, 7, and 3, respectively. These products have to be shipped in three depositary centers D1, D2, and D3
A company owns three production centers P1, P2, and P3, whose production capacity is of units 61, 49, and 90, respectively. These products have to be shipped in three depositary centers D1, D2, and
A company owns three production centers P1, P2, and P3, whose production capacity is of units 50, 80, and 110, respectively. These products have to be shipped in three depositary centers D1, D2, and
Implement a linear binary program for the investment opportunities shown in Table 5, with investment budget constraint equal to £ 1000.Transportation problems modeling. The linear programming has
A firm produces an item of two types: for the high-end market and for the mass market. Both products require the same type of raw materials but in different proportions. For each ton of output
Using the Solver find the solution of the following linear problem and represent it geometrically in a chart: max fx1;x2g 4x1 þ 6x2 s.t. 3x1 þ 2x2 6 x1 þ 5x2 10 x1 0; x2 0
Using the Solver find the solution of the following linear problem and represent it geometrically in a chart: max fx1;x2g 8x1 þ 2x2 s.t. 4x1 þ x2 16 2x1 þ 3x2 12 x1 0; x2 0
Using the Solver find the solution of the following linear problem and represent it geometrically in a chart: max fx1;x2g 4x1 þ 7x2 s.t. 2x1 þ 5x2 10 6x1 þ 3x2 18 x1 0; x2 0
Using the Solver find the solution of the following linear problem and represent it geometrically in a chart: max fx1;x2g 4x1 þ 2x2 s.t. 3x1 þ 4x2 24 700x1 þ 225x2 3; 150 1:8x1 þ 2x2 18 7x1
Using the Solver find the solution of the following linear problem and represent it geometrically in a chart:s.t. min 9x1 +12x2 {x1:32) 3x1 + x2 12 x1 + x2 9 x1+2x2 8 x10, x2 Free II. Static
Consider again the blending production problem of Example 4 shown in the chapter and suppose that the customer order is for 4000 tons of alloy and that the producer has only the following amounts of
A plant can manufacture three products A, B, and C. The plant has four departments, I, II, III, and IV. Product A must be processed in departments I and II; product B in departments I, II, III, and
Three products A, B, and C are produced in a plant, and for a given period there is the problem of deciding how much of each produc t to produce. Table 1 below contains information on the production
Solve the zero-sum games with respect to Player A with the graphical method, considering the following (2$2) pay-off matrices. B1 B2 a. A1 1 0 A2 0 0.5 B1 B2 b. A1 3 6 A2 4 5 B1 B2 c. A1 1 2 A2 6 B1
Cournot Duopoly Model with costs. Suppose the following linear demand curve in the market: Y ¼ 100 2p and equal linear total cost function for each duopolist as follows: TCðAÞ ¼ 1 2 YA þ 10YA
Cournot Duopoly Model with no costs. Suppose the following linear demand curve in the market: pðYÞ ¼ 20a. Build a set of isoprofit curves for each duopolist and visualize them in a chart.b.
Chamberlin monopolistic firm. Consider a firm in a monopolistic market that has the following total cost function: TC ¼ 0:03y3 0:1y2 þ 50y þ 100 while the short-run demand of the firm is: p ¼
A monopolist has the following total cost function: TC ¼ 6 þ y2 while its downward sloping demand is as follows: p ¼ 40 ya. Set up the profit function, and using the Solver calculate the optimal
A perfect competitive firm has the following short-run total cost function: TC ¼ 2y2 þ 98 while the market clearing price is p* ¼ 40.a. Calculate the optimal quantity to produce for the firm,
There are 30 firms in a perfect competitive market that faces total costs according to the following total cost function: TC ¼ 5y2 while the demand function of the market is as follows: y ¼ 300
Perfect Competition. On a natural resource perfect competition market, we have 100 firms that face the following short-run total cost function (see the constant term representing the fixed costs): TC
Input profit optimization problem. A firm sells a product at a unit price of £ 100, while the production function to produce this type of item is represented by the following CobbeDouglas function,
Input profit optimization problem. A firm sells a product at a unit price of £ 8, while the production function to produce this type of item is represented by the following CobbeDouglas function: q
Consider the following Leontief InputeOutput production function: yðx1; x2Þ ¼ minx1 5 ; x2 5a. Plot the surface chart using the Excel Data Table.b. Construct the chart for a set of isoquants. 7.

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