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numerical mathematical economics

Elements Of Numerical Mathematical Economics With Excel Static And Dynamic Optimization 1st Edition Giovanni Romeo - Solutions

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 the rate of advertising expenditure s(t). The sales response to advertising can be assumed
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 ln 2K K' e t=4 dt s:t: Kð0Þ ¼ 5 and KðTÞ ¼ 7
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 ½UðfðKðtÞÞ K0 ðtÞÞdt s:t: Kð0Þ ¼ 5 and KðTÞ ¼ 3 where the production function is:
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 lðp; p0 Þe rt dt s:t: pð0Þ ¼ 5% and pð1Þ ¼ Free where: lðp; p0 Þ ¼ p0 bj 2 þ a p0 j þ p
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 and yð1Þ ¼ 5
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 T free Set up in the problem the necessary transversality conditions.
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 Set up, to solve the problem, the problem the necessary transversality condition.
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 ¼ 8 Replicate the exercise in Excel and check the solution.
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 " 1 þ ðD2yÞ 2 ðD2tÞ 2 #1=2 D2t s:t: yð1Þ ¼ 7 yð3Þ ¼ 17 D1y þ D2y ¼ 10 Plot the solution
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 portfolio of funds that could meet this requirement.
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 regression: Fund Retrunst ¼ a þ bðBenchmark ReturnstÞ þ εt
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 relative returns of the fund versus those of the benchmark.
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 D4 which have demanded units 5, 3, 8, and 17, respectively. The matrix of transportation costs is
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 which have demanded units 7, 3, and 5, respectively. The matrix of transportation costs is as given
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 D3 which have demanded units 52, 68, and 80, respectively. The matrix of transportation costs is
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 D3 which have demanded units 100, 80, and 60,respectively. The matrix of transportation costs is as
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 vast applications in the management and industrial engineering area. An important topic is solving the
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 produced, the required resources expressed in hours are as follows in Table 4: The firm employs eight
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 þ 6x2 42 x1 0; x2 0 7. Using the Solver find the solution of the following linear problem
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 Optimization
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 the six ores available (in tons); 2500; 2200; 800; 3000; 1000, and 1600.Write a general linear
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 IV; and product C in departments I, III, and IV. Departments I and II must each be scheduled for at
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 process, profitability, and sales potential of each item. Identify the objective function, formalize
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 d. A1 -2 A2 7 B1 e. A1 9 A2 -2 B1 4252232 B2 B2 -3 14 B2 f. A1 2 1 A2 -1 3
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 and TCðBÞ ¼ 1 2 YB þ 10YAa. Setting up the maximum profit conditions, build a set of isoprofit
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. Visualize the reaction curves and calculate the market equilibrium quantity Y ¼ Y A þ Y B and the
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 100 1 2 ya. Set up the profit function and using the Solver find the optimal quantity y* to
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 quantity y* to produce and the equilibrium price p* for a maximum profit.b. Plot the relevant
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, together with its profit area.b. Plot the relevant revenue and cost functions and identify in the chart
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 72p: Repeat Exercise 10 in this new scenario. Then answer to the following point.a. Once you have
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 = 2y+50. where the variable y measures the quantity produced. The demand function of the market is
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, where, as we are operating in the short-run, the availability of capital quantity is given at 81 and
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 ¼ L1=8 K3=8 The unit prices of inputs are pL ¼ 4, pK ¼ 12. The profit function is therefore
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. Consider the following Leontief InputeOutput production function, with inputs represented by labor,
A firm has estimated the following CobbeDouglas production function: y ¼ 2 L1=3 K1=2 while total direct costs budget constraint is C ¼ £ 90, and unit prices of inputs are pL ¼ 4, pK ¼ 9.a. Find with Solver the optimal combination of inputs, subject to the given cost constraint.b. Plot a
Consider two consumers. The first consumer has the following Leontief utility function: Uðx1; x2Þ ¼ minx1 5 ; x2 5 (i.e., the two goods are perfect complements). While the second consumer has the following Linear utility function: Uðx1; x2Þ ¼ 2x1 þ x2 (i.e., the two goods are somehow
Suppose we have a nonlinear utility function described as follows: Uðx1; x2Þ ¼ x1x2 Then, the commodity prices are given as p1 ¼ £ 24, p2 ¼ £ 8, together with the budget constraint: 3x1 þ 1x2 ¼ £ 80:a. Find with Solver the optimal quantities the consumer has to choose to maximize the
Suppose we have CobbeDouglas utility function described as follows: Uðx1; x2Þ ¼ x0:6 1 x0:4 2 Then, the commodity prices are given as p1 ¼ £ 3, p2 ¼ £ 1, together with the budget constraint: 3x1 þ 1x2 ¼ £ 10: The neoclassical consumer problem can be then formalized as: max fx1;x2g x0:6 1
Given a commodity bundle (x) ¼ (x1, x2,., x5) and the vector of prices (p) ¼ (£ 20, £ 5, £ 2, £ 4, £ 10) under the cardinal approach, assuming the additive property of total utility function, solve the following consumer problem: max fxg UðxÞ ¼ max fxig X 5 i ¼ 1 UðxiÞ where UðxiÞ ¼
Inventory and optimal procurement policy (deterministic single item inventory model). A company employs a certain raw material whose consumption in the production cycle is constant over time and it wishes to find the optimal policy of procurement. We have the following information: - Consumption
Output Profit Maximization Problem. A printing company faces the following total cost function: CðqÞ ¼ 900 þ 400q q2 where q2 denotes the volume savings. The income due to the advertisement is instead as follows: IðqÞ ¼ 600q 6q2 : Sales is £ 600 per advertisement to which we subtract
Output Profit Maximization problem. A company produces monthly a certain number of products of one type, which are sold at a unit price of 800. The monthly fixed cost is 180,000 and the unit production variable cost is 50. The company also faces selling costs equal to half products sold (e.g., for
Output Profit Maximization problem. A company wishes to find the optimal program for a production of a product in order to maximize its profit (i.e., the objective function). The total cost function has been estimated as for C(q) = 30,000+ 600q -0.5q while the demand faced by the company for this
Output Profit Maximization problem. A company shows the following total cost function: CðqÞ ¼ 12; 000 þ 500q q2 The sale price depends on the quantities sold according to the following equation: p ¼ 700 1:5q Identify the objective function profit P ¼ Pq CðqÞ and find the optimal
Output cost minimization problem. A company has the following total cost function for the production of two items: Cðq1; q2Þ ¼ q2 1 þ q2 2 10q1 12q2 þ 151: Find the optimal quantities to produce to minimize the total cost function.
Steepest descent. Using the numerical steepest descent technique and the VBA macro shown in the chapter, find the critical points of the following functions: fðx1; x2Þ ¼ 32 8x1 16x2 þ x2 1 þ 4x2 2 fðx1Þ ¼ 3x4 1 4x3 1 The following set of problems shows the typical applications of the
Using the graphical approach in Excel solve the following nonlinear optimization constrained problem: min fx1;x2g x2 1 þ x2 2 s.t. x1x2 25 x1; x2 0
Using the graphical approach in Excel solve the following nonlinear optimization con- strained problem: min(x-4)+(x-4) s.t. 2x+3x2 6 -3x-2x22-12 X1,X2 20
Using the Solver find the solution to the following nonlinear constrained optimization problem: minx+4x-8x,-16x +32 5.t. x+x 5 X1, X220
Using the Solver find the solution to the following nonlinear constrained optimization problem: min x+x+x (4020) 5.t. x+x+2x3 = 10
Using the Solver find the solution to the following nonlinear constrained optimization problem: min 5x+2x2-x3 (4020) S.L x=3 xX1
Using the Solver find the solution to the following nonlinear constrained optimization problem: max fx1;x2;x3g x2 1 þ x2 2 þ x2 3 s.t. x2 1 x1x2 þ x2 2 x2 3 ¼ 1 x2 1 þ x2 2 ¼ 1
Using the Solver find the solution to the following nonlinear constrained optimization problem: min fx1;x2;x3g x2 1 2x1x2 þ x2 2 þ 5x2 3 s.t. x1 þ x2 þ 2x3 ¼ 10:
Construct in Excel the circle contour lines for various levels of f for the following function: fðx1; x2Þ ¼ x2 1 þ x2 2
Bivariate functions nonlinear optimization. Find the critical points of the following functions ad assess whether they are minima, maxima, or saddle points, studying the sign definiteness of the Hessian matrix. Plot the surface charts using the Excel Data Table. Enumerate also the partial
Using the technique of the graphical tangent, via the Excel Data Table shown in the Example 3 find, using the Solver, the exact minima or maxima of the following univar- iate functions and visualize the tangent equation on the chart at the critical points: y= x+1 y= (4x+15x)e (x-1)(x-6)
Enumerate in Excel the following univariate functions and study the concavity, con- vexity. Find the critical points and assess whether they are maxima or minima. Find then the exact minima or maxima of the function using the Solver. Using the numerical second derivative also find the inflexion
Unemployment versus Inflation and Long-Run Phillips relation: advanced ODEs modeling. The original formulation of the Phillips curve is as follows: w ¼ fðUÞ ðf 'ðUÞ < 0Þ (1) where w is the rate of growth of money wage W and U is the rate of unemployment. Later, economists have used the
The following system presents complex roots. Solve it using the Euler method, and see if the steady state is a focus or a center. y_ðtÞ ¼ 1 1 1 1 yðtÞ
Solve the following Walrasian model using the Euler method: p_ N_ ¼ kb ks g 0 p N þ ka gc : with k ¼ 5; a ¼ 2; b ¼ 0:65; g ¼ 1; s ¼ 0:50; c ¼ 2 Study the discriminant of: l1; l2 ¼ kb 2 1 2
The following system presents complex roots. Solve it using the Euler method, and see if the steady state is a focus or a center. y_ðtÞ ¼ 0 1 1 0 yðtÞ
Solve the system of two differential equations proposed in Section 4.6 which models the saddle point situation in the tourism battle between two regions using the Euler method shown in Section 4.5 and plot the phase diagram.y_ðtÞ ¼ 1 3 5 3 yðtÞ
Solve in Excel using the direct method the following complete system of linear ODEs and plot the phase diagram: y_ðtÞ ¼ 0 1 1=4 0 yðtÞ þ 2 1=2
Solve in Excel using the direct method the following homogenous system of linear ODEs and plot the phase diagram: y_ðtÞ ¼ 4 1 4 4 yðtÞ
The exponential cardinal utility function assumes the following differential equation: dUðxÞ dx ¼ 1 k UðxÞ þ 1: Solve the differential equation using the Euler method and plot the total utility function together with the marginal utility. Set the risk aversion factor k ¼ 1.
Capital growth. Let the following be the function of the aggregate output of economy: Y ¼ ða þ akÞt 1=2 :Then, let the capital accumulation equal to saving (scaled with s marginal propen- sity to save) as follows: K(t) = sY. We can condense the above equations into the following nonautonomous
Given the following expression of force of interest. dðtÞ ¼ 1 þ 0:05t calculate the resulting form of the capital accumulation function y(t).
Using the Euler method solve the following nonautonomous first-order differential equation: y_ðtÞ ¼ 2ty þ 2t yð0Þ¼ 2
Using the Euler method solve the following nonautonomous first-order differential equation: y_ðtÞ ¼ 2ty þ t yð0Þ ¼ 0
Using the Euler method, solve the following nonautonomous first-order differential equation: y_ðtÞ ¼ t y yð0Þ ¼ 0:5
Using the Euler method, solve the following first-order differential equation: y_ðtÞ ¼ y yð0Þ ¼ 1
Consider the following economic model: CðtÞ ¼ C0 þ bYaðt 1Þ YðtÞ ¼ CðtÞ þ I which can be condensed in the following nonlinear difference equation: YðtÞ ¼ C0 þ bYaðt 1Þ þ I: Construct the step-chart and phase diagram with: a ¼ 0:5 C0 ¼ 5 b ¼ 0:5 I ¼ 5
Consider the following nonlinear first-order difference equation: yðtÞ ¼ y2 ðt 1Þ þ 3 16: Construct the step-chart, phase diagram and examine the global stability.
Consider the following nonlinear first-order difference equation with initial condition: yðtÞ ¼ yaðt 1Þ construct the step-chart and phase diagram with: a ¼ 0:5 a ¼ 0:5 a ¼ 2 a ¼ 2 and analyze the convergence or divergence of the system.
The following system of difference equations gives the Samuelson model: YðtÞ ¼ CðtÞ þ IðtÞ þ G0 CðtÞ ¼ gYðt 1Þ IðtÞ ¼ a½CðtÞ Cðt 1Þwhich can be condensed into the following second order difference equation: YðtÞ ¼ gð1 þ aÞYðt 1Þ agYðt 2Þ þ G0: Solve in
Solve the following second-order difference equation and set up in Excel the step-chart: yðtÞ¼ 3yðt 1Þ 2yðt 2Þ
Solve the following first-order difference equations and set up in Excel the step-charts: yðtÞ¼ 1 3 yðt 1Þ yðtÞ ¼ 1 3 yðt 1Þ þ 6 yðtÞ¼ 1 4 yðt 1Þ þ 5
The IS-LM set of equations is given below: C ¼ a þ bð1 tÞY I ¼ e lR G ¼ G L ¼ kY hR M ¼ M The economy in equilibrium will satisfy the following: Y ¼ C þ I þ G C ¼ a þ bð1 tÞY I ¼ e lR M ¼ kY hR With four endogenous variables Y, C, I, R and four exogenous variables G,a,
The same set of equations is given below, except for the government spending, that you will use as instrumental variable: C ¼ 15 þ 0:8ðY TÞ T ¼ 25 þ 0:25Y I ¼ 65 R G ¼ G0 L ¼ 5Y 50R M ¼ 1; 500: Solve for a target level of government spending, such that the desired level of M ¼
The IS-LM model gives the equilibrium conditions in the good and money markets. The goods market (IS) has been econometrically estimated as follows: C ¼ 15 þ 0:8ðY TÞ T ¼ 25 þ 0:25Y I ¼ 65 R G ¼ 94 where G ¼ 94 is the private consumption, C the tax revenues, T, the private
Solve in Excel the Leontief InputeOutput problem, determining the equilibrium quantities of the unknown x1x3 ¼ 0, such that: x ¼ Ax þ d using the following matrix information: A ¼ 2 6 4 0:30 0:50 0:30 0:20 0:20 0:30 0:40 0:20 0:30 3 7 5 d ¼ 2 6 4 20 10 40 3 7 5 The following problems relate
Resorting to the bordered matrix, determine the sign of the following constrained quadratic form: QðxÞ ¼ 3ðx2Þ 2 þ ðx3Þ 2 þ 4x1x2 þ 2x2x3 s.t. x1 x3 ¼ 0
Using the model shown in Section 3.7 solve the following problem. Suppose you, as economist, gathered the following macroeconomic annual data for a country Variable Actual Level (bln) Y* NX* 1,700 50 850 z 279 -TXM 200 714 536 486 t 42%The economic policy makers desire to know how they may change
Set up an Excel chart such that the following system of linear inequalities is satisfied: 8 >< >: x1 þ 3x2 6 4x1 2x2 4 2x1 x2 2
Solve the following linear system using the Excel Solver and the inverse: 1 2 2 3 x1 x2 ¼ 3 1 : Resorting to the RouchéeCapelli theorem, analyze whether the system admits only one solution or not.
Solve the following linear system using the Excel Solver, the inverse, and the Cramer rule: 2 6 4 214 321 133 3 7 5 2 6 4 x1 x2 x3 3 7 5 ¼ 2 6 4 16 10 16 3 7 5: Resorting to the RouchéeCapelli theorem, analyze whether the system admits only one solution or not.
Solve the following linear system using the Excel Solver, the inverse, and the Cramer rule: 2 6 4 11 1 2 3 1 7 1 2 3 7 5 2 6 4 x1 x2 x3 3 7 5 ¼ 2 6 4 9 16 57 3 7 5: Resorting to the RouchéeCapelli theorem, analyze whether the system admits only one solution or not.
You work as a business consultant for a company. Using the econometric techniques, you have estimated the firm’s total cost function as C(q) ¼ q3 5q2 þ 14q þ 75. Enumerate now the marginal cost function C0 (q) and obtain the analytical equation form using the Excel trendline.
Using the following demand function: q ¼ 10 2p1=2 calculate the Consumer Surplus at p0 ¼ 4 and q0 ¼ 6.
The Consumer Surplus is defined as the difference between the maximum price the consumers are willing to pay and the price they actually pay. It is the net gain of the buyers. In mathematical terms, we can derive the consumer surplus as follows. Let p ¼ D1 (q) be the inverse demand function and
Suppose a firm starts at t ¼ 0 with a capital stock of K(0) ¼ £ 500 while the investment law detected in the past is equal to I(t) ¼ 6t 2 and this will apply over the next 10 years as well. Enumerate the planned level of capital stock. Compare the numerical integral versus the exact integral.

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