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numerical mathematical economics

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

Why does a wide band around fixed exchange rate parities (as with the 15 per cent band in use in the EMS between 1993 and the end of 1998) make life more difficult for currency speculators?
Why might a revaluation of a currency only temporarily reduce a Balance of Trade surplus?
How is the deflationary policy of a strong country transmitted through a fixed exchange rate system?
How useful do you think equilibrium models are in analysing a world that is never in equilibrium?
The model to which New Classical economists applied rational expectations is described in the text as ‘market clearing’ and ‘monetarist’. How are these descriptions related? What must have been the principal assumptions of the model?
How are the following arguments discussed in this chapter affected, if at all, by an assumption of endogenous money?(a) the expectations-augmented Phillips curve(b) policy irrelevance
The Chambers Twentieth Century Dictionary defined ‘hysteresis’ as:the retardation or lagging of an effect behind the cause of the effect: the influence of earlier treatment of a body on its subsequent reaction.How then can hysteresis occur in labour markets? How can the existence of hysteresis
Why does ‘the combination of the rational expectations hypothesis and the assumption of continuous market clearing’ imply that output and employment fluctuate randomly around their natural levels?
Why was the ‘shoe leather cost’ of inflation so called? What other costs are there of anticipated inflation?
Why is the natural rate of unemployment referred to as ‘natural’?
Imagine that a rate of inflation of 4 per cent p.a. becomes established in an economy with an inflation target of 2 per cent p.a. (say, because of a sustained rise in the prices of energy and commodities). Use the IS/PC/MR model to show how a policymaker brings inflation back to 2 per cent p.a. if
In its discussion of the transmission mechanism, the Bank of England says that the effect of a change in interest rates on expectations is the most difficult to predict.Why might this be?
How are interest rate changes likely to affect the distribution of income between:(a) rich and poor; (b) borrowers and lenders; (c) old and young?
What difference would it make to the strength of UK monetary policy if all mortgages were fixed-interest-rate loans:(a) in the short run; (b) in the long run?
We said in the text that a change in the official interest rate does not necessarily result in an instant and identical change in market rates. How might we explain this.
Focusing upon people’s wealth, explain how a change in interest rates might affect the level of spending
Go to the statistical section of the Bank of England’s website(www.bankofengland.co.uk/statistics/statistics.htm), look for the publication‘Bankstats’, find table G1.1 and look at the relationship between interbank rates and the Bank of England ‘base rate’ in the last few years. The
Explain what is meant by a repurchase agreement and work an example to show how, by changing the terms of a repo deal, the central bank can raise and lower short-term interest rates.
Why is the demand for reserves by commercial banks highly interest-inelastic?
Explain briefly the disadvantages of attempting to regulate monetary growth by non-price methods.
Why, in practice, are commercial banks unconstrained in their access to reserves?
Distinguish between ‘interest-endogeneity’ and ‘base-endogeneity’.
In the light of Keynes's view about the difference between risk and uncertainty, explain each of the following ideas from Keynes:(a) it might be rational (rather than simple money illusion) for workers to make labour supply decisions based on relative money wages rather than real wages;(b)
Given the various points mentioned in box 3.1 as influencing the velocity of money, would you expect the long-term velocity of money to have increased or decreased over the past fifty years?
Both the inventory-theoretic model of the transactions demand for money and Tobin's portfolio model are commonly called Neo-Keynesian models. Why?
Why was the speculative demand for money so controversial?
Why did the testing of the demand for money grow rapidly at the expense of theorizing about the demand for money after Friedman published his theory?
Both the precautionary and the speculative motives for holding money arise from the existence of uncertainty — uncertainty about what in each case?
Why is the Quantity Theory of Money not a theory of the demand for money?What is it a theory of?
Economics generally tells us that one must analyse the factors influencing both the demand for and supply of important variables. Why, then, might the demand for money not be of great importance from the point of view of economic policy?
The money supply curve in figure 2.1 is one component in the derivation of the LM curve. What effect does changing the slope of the money supply curve have on the LM curve?
Why, according to the flow of funds approach, does the choice of exchange rate regime make monetary control more, or less, difficult for the authorities?
What steps might the authorities take to offset the monetary effects of events in question 3?
In the flow of funds analysis, explain the effect of an increase in the government’s budget deficit, ceteris paribus.
Would it be sensible always to act as if the weather forecast for the following day were always correct if, on average, weather forecasts were correct:10 per cent of the time?50 per cent of the time?90 per cent of the time?Does this question provide a reasonable analogy with the notion of analysing
Provide examples of transactions in our modern monetary economy that take place through barter or through a combination of barter and money.
The words ‘exogenous’ and ‘endogenous’ are used widely in economics — not just referring to money. What precisely do they mean? Provide other examples of their use.
Distinguish between ‘monitoring’ and ‘targeting’ in the context of money supply measures.
Distinguish between:(a) means of payment and medium of exchange;(b) inside money and outside money.
There is a distinction made in the economics literature between ‘consumption’and ‘consumption expenditure’. This distinction implies different definitions of‘saving’. What are these different definitions and how do they relate to the discussion in the text about information and
What limits currently exist on the amount of credit that can be obtained by households? Is there any attempt by the monetary authorities to control the amount of credit available?
For a short time, in the early years of the British settlement of Australia (the 1790s), rum was used as a currency. What advantages and disadvantages would rum have as a commodity money?
The text refers to the importance of the interest rate decisions of central banks.How often and when are interest rate announcements made by:e the Bank of England Monetary Policy Committee?¢ the European Central Bank?¢ the Federal Reserve Board of the United States?
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 customer orders may likely fall in the future. To do this, run a Monte Carlo simulation of 1000
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 determine the unit price (Table 2).a. Run a Monte Carlo simulation for the possible total sales
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 has in stock only two items, what is the probability that he will not be able to meet the customer
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 probability will be 0.95 that he will have enough items to meet the customer demand for a time
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 cars will add to the queue in a 30-second interval. Plot the Poisson distribution.
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 and assume the time is measured in minutes. Therefore, 30 calls per hour is 0.5 calls per minute, so
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 the following.a. What is the probability that exactly 3 people out of the 10 interviewed will
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 for x ¼ # of Heads from 0 to 5. Do this using the random number generator tool in Excel. TABLE 1 Flip
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 sample space U of this experiment.
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 the global F-test, test simultaneously the hypothesis that all the slope coefficients are jointly
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 and plot the regression.
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 prediction interval at 95%.c. Use the built-in Excel Forecast Sheet to make a sale prediction for 2019.d.
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 a þ bEB þ gEB2 to assess whether the fund shows a market timing ability or not. Remark: this
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 family disposable income in town X5 ¼ the trend variable taking values of 1; 2; and so on; for the
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 related to nominal GNP?b. Run now a multiple regression of GNP versus all the money data supplies.c.
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 in Ch. 5. We have the following data: C ¼ 15 unit variable cost of purchase p ¼ 35 unit price of
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 0:2 0:1 Item 2 c1 ¼ $5; c2 ¼ $60 f2ðrÞ ¼ 8 >< >: 012 0:2 0:5 0:3 What is the optimal inventory
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 under construction. These spares are bought even though it is known that it is very unlikely that any
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 1:5 r 400 100 r 120 0 r > 120 Costs of inventory are as follows: c1 ¼ cost of surplus ¼ 200
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 permitted (i.e., when the demand cannot be met, being temporarily out of stock). Month D, Demand A, Fixed
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, h = 2, and h3h4 = 1. No shortages are allowed. The produc- tion costs are given in Table 6. These
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. TABLE 5 MANUFACTURING DATA Capacity Production Costs Period Regular Time Over Time Regular Time Over
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 constraints, and unit costs of production, as well as the demands data for the store with sales price and
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 are not allowed.TABLE 1 Month Demand Unit Production Costs (E) Unit Inventory Costs (E) 1 1,300 98 5.0
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 is 50,000 £ at the beginning of the period. The company knows that reinvesting in advertising will
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 present value of cash flows from the mine over the 10 years. The model can be therefore formalized as
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 t ¼ 0 bt ln½uðtÞ s:t: Kðt þ 1Þ ¼ AKaðtÞ uðtÞ 0 for t ¼ 0; 1; 2; /;ðT 1Þ and
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 you use the Data Table, assume u(t) can only take the discrete values 1.0, 0.6, 0.20, 0.4, 1.0.
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 ðtÞ þ 2u2 ðtÞ yðt þ 1Þ ¼ yðtÞ uðtÞ and y(0) ¼ 5. (Hint: to solve the problem by hand
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 2ðx yÞ; a ¼ 0:50; b ¼ 0:70
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 initial condition y(0) ¼ 1.5 solve the problem in Excel as if it were a CoV problem for y(T) free
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. You can only travel one way as indicated by the arrows. Plot in a chart the optimal route.
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: Solve the problem setting yð0Þ ¼½ 5 2 , T ¼ 2, and y(T) free.
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 ðtÞ dt: Solve the problem setting yð0Þ ¼½ 4 2 , T ¼ 2, and y(T) free.
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 ZT 0 n ½y1ðtÞ 1 2 þ 0:0025u2 ðtÞ o dt: Solve the problem with T ¼ 5 and yð0Þ ¼½ 0 0 ,
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 5.0.b. Solve the problem with H ¼ 5, T ¼ 15, a ¼ þ 0.2, and y(0) ¼ 5.0.
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 2 y2 2ðtÞ þ 1 4 u2 ðtÞ dt:Setting up the Hamiltonian Maximum Principle conditions, it turns
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 yð2Þ free y_ 1ðtÞ ¼ y2ðtÞ y_ 2ðtÞ¼ y2ðtÞ þ uðtÞ: Set up in Excel the system of
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 possible, somehow, to slow down the rate of decline of the resale value, applying preventive
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 market which can absorb no more than M dollars of the product per unit of time, that is, M is the
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 Kð0Þ ¼ 1; Kð1Þ free and I(t)˛[0, 1] where d is the capital depreciation rate, set equal to 15% and
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 1þ0:05 is the discount factor and K(t) is the capital available to the consumer. We assume there is no
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 condition y(0) ¼ 4 and y(1) ¼ 0. (Hint: in this case, as y(1) ¼ 0, namely the state variable is
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 with initial condition y(0) ¼ 4 and y(1) free. (Hint: as y(1) is free to vary we need to integrate
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 solution is for T* ¼ ln[2]y0.6931. In such a problem, with a free terminal time but a fixed endpoint, the
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 state is free to vary, only subject to y(T)ymim, and the problem is called to have a truncated
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 (unconstrained) that minimizes the following performance measure: JðuÞ ¼ Z 1 0 1 2 u2 dt þ y2 ð1Þ
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 y1ð0Þ ¼ 5; y2ð0Þ ¼ 5; y1ð1Þ ¼ free; y2ð1Þ ¼ freea. Find the necessary conditions to solve the
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; y2ð0Þ ¼ 0; y1ð1Þ ¼ free; y2ð1Þ ¼ free
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 boundary conditions as follows: y1ð0Þ ¼ 0; y2ð0Þ ¼ 0; y1ð1Þ ¼ free; y2ð1Þ ¼ freea. Set
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 adjustment of labor input will imply a cost that varies with L’(t), the rate of change of LðtÞ.

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