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numerical mathematical economics

Questions and Answers of Numerical Mathematical Economics

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.
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
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
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:
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,
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
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
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
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
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
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
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
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
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
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
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
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:
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
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
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
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:
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
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
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
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
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
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
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
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
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
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
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
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,
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.
Consider the following Cobb-Douglas production function, which relates the level of inputs L and K, to output product, y: y ¼ 2L0:5 K0:5 : Plot the surface graph and the contour graph of the given
Consider the following univariate production function, which relates the level of input labor, L, to the output product, y: y ¼ L0:5 Plot its graph together with its numerical first derivative,
Repeat Exercise 3 for the following expression: Z 1 0 4e 4x dx
Repeat Exercise 3 for the following expression: Z 2 1 1 x dx
Calculate the following numerical definite integral and compare it versus the exact integral: Z 2 1 ln x dx
Plot the following function: y ¼ 5x2 Then, calculate and plot its numerical first derivative. Using the Excel trendline option find the approximate numerical expression of this derivative.
Using the techniques seen in the chapter, build the Excel Data Table such that you are able to draw the tangent along the points on the curve represented by the following univariate functions: y ¼

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