Mathematics For Economics 3rd Edition Michael Hoy, John Livernois - Solutions

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Excel data set WP.XLS contains annual data from 1857–1987 on X = wages and Y = the consumer price index in the UK.15 It is commonly thought that wage pressures are a prime cause of inflation. You wish to investigate this claim by carrying out a time series analysis on the data. In particular,(a)
Use the data on Y = consumption and X = personal income from INCOME.XLS.Assume (perhaps incorrectly) that Y and X are cointegrated.(a) Estimate an error correction model. Begin with a model containing a deterministic trend and p = q = 4 and then carry out statistical tests to find an appropriate
Use the data in ORANGE.XLS to check the previous example. In particular, does the ECM have enough lags of both ΔX and ΔY?
Use the data on Y = personal income and X = personal consumption in INCOME.XLS.(a) Use Dickey–Fuller tests to verify that Y and X have unit roots.(b) Run a regression of Y on X and save the errors.(c) Carry out a unit root test on the residuals using an AR(1) model.(d) Carry out a unit root test
Excel file LONGGDP.XLS contains annual data on real GDP per capita for four of the largest English-speaking countries (USA, UK, Canada and Australia) for the years 1870–1993.10 Investigate whether there are common movements or trends between GDP in these different countries. In particular, you
Use the data in ORANGE.XLS to complete the previous example. In particular:(a) Do a Dickey–Fuller test to verify that both orange price series have unit roots.(b) Do a sequential test to verify that the Dickey–Fuller test on the residuals was done correctly. That is, is an AR(1) model for the
using the data in COMPUTE1.XLS.That is, verify that Y and X are stationary and then test to find a suitable ADL(p,q) specification.(b) Calculate the long run multiplier for the model estimated in (a).
Data set COMPUTE1.XLS contains variables of the same form as COMPUTER.XLS, however for a company in a different industry.(a) Repeat the analysis of
Use the variables Y = percentage change in sales and X = percentage change in computer purchases in data set COMPUTER.XLS to decide whether the model estimated in the table above is a good one. In particular:(a) Establish whether Y and X really do not have unit roots as was assumed in the
In Exercise 9.7, we tested for unit roots in many series. We noted that if a time series has one unit root, then its difference will be stationary. Verify that this is true for the series having unit roots in Exercise 9.7. That is, discuss how you would test to see if the change in the series has a
In this chapter we have recommended a strategy according to which you begin with an AR(p) with deterministic trend model, choose lag length (p), decide whether the deterministic trend should be included or excluded, and then test for a unit root. Carry out this strategy using the following
The data in FIG98.XLS was used to create Figure 9.8.(a) Calculate the autocorrelation function for this trend stationary series.(b) In light of your answer to (a), discuss whether the autocorrelation function is a useful tool for testing for a unit root.
Use the data in files FIG95.XLS, FIG96.XLS and FIG97.XLS, which were used to create Figures 9.5–9.7, respectively.(a) Calculate OLS estimates of ρ and ϕ in the two variants of the AR(1) model.(b) Relate your results in (a) to the question of whether any of the series contain a unit root.(c)
Use the data in files FIG95.XLS, FIG96.XLS and FIG97.XLS, which were used to create Figures 9.5–9.7, respectively.(a) Calculate the autocorrelation function for each time series using a maximum lag of 4.(b) Relate your findings in (a) to your answers to Exercise 9.3. Focus in particular on the
Use the data on Y = personal consumption given in INCOME.XLS.(a) Calculate the autocorrelation function for Y and ΔY with a maximum lag of 4(P = 4).(b) Plot these autocorrelation functions in a bar chart.(c) Interpret the results you have obtained in (a) and (b).
The file INCOME.XLS contains data on personal income and consumption in the US.(a) For each of these two series individually create an XY-plot between the variable and the variable lagged one period.(b) For each of these variables, calculate r1.(c) First difference each of these variables and
The file INCOME.XLS contains data on the logs of personal income and personal consumption.(a) Calculate and interpret descriptive statistics for both personal income and change in personal income data. Do the same for personal consumption and its change.(b) Plot and interpret figures analogous to
Development economists are often interested in the effect of education spending on economic growth. However, they suspect that the positive effects of raising education levels may take five or as many as ten years to manifest themselves in higher growth rates. In light of these considerations, use
Use the data set, SAFETY.XLS, which contains T = 60 observations on Y = accident losses and X = hours spent in safety training. Suppose you believe that six months is the maximum time that safety training might affect accident losses and accordingly, you set qmax = 6. Using the strategy described
Use the data set, SAFETY.XLS, discussed in the previous example for this question.This data set contains T = 60 observations on Y = accident losses and X = hours spent in safety training.(a) Create the explanatory variables you would use in a distributed lag model with lag length equal to 4. (If
Excel file, WAGEDISC.XLS contains data on N = 100 employees in a particular occupation. Suppose interest centers on investigating the factors that explain salary differences in this occupation with a view to addressing the issue of sex discrimination in this occupation. The data set contains the
and the following four nondummy explanatory variables:
For this question use data set HPRICE.XLS, the five dummy variables, D1 to D5, listed in
For this question use data set HPRICE.XLS and the five dummy variables, D1 to D5, listed at the beginning of the chapter (the dummy variables for whether a house has a driveway, recreation room, basement, gas central heating and air conditioning).(a) With five dummy variables, how many classes of
For this question use Y = the price of a house and the dummy variables D1 = 1 if the house has a driveway (=0 otherwise) and D2 = 1 if the house has a rec. room (=0 otherwise) from the house price example (it can be obtained from HPRICE.XLS).Without using regression techniques, calculate the
Interpret the statistical information in the above example. Are all of the explanatory variables statistically significant?
Using the data set HPRICE.XLS, calculate and interpret descriptive statistics and a correlation matrix for the five dummy variables listed in the example above. How can you interpret the mean of a dummy variable?
Use data set HPRICE.XLS and let Y = house price be the dependent variable and consider the following potential explanatory variables:X1= the lot size of the property (in square feet)X2= the number of bedrooms X3= the number of bathrooms X4= the number of storeys (excluding the basement).Exercise
(a) Create XY-plots using the four explanatory variables in the house pricing example one at a time (plot Y and X1, then plot Y and X2, and so forth).(b) Perform simple regressions using the explanatory variables one at a time(regress Y on X1, then regress Y on X2 and so forth).(c) Comment on the
Use data sets FIG51.XLS, FIG52.XLS, FIG53.XLS and FIG54.XLS.(a) Test whether β = 0 using the confidence interval approach for each of the four data sets.(b) Test whether β = 0 using the P-value approach and the four data sets. Use the 5% level of significance.(c) Redo (a) and (b) for α.
In addition to Y = deforestation rate, data set FOREST.XLS also contains data on W =the percentage increase in cropland (labeled “Crop ch”) and Z = percentage change in pasture land (labeled “Pasture Ch”)
Using the table above (or running a regression yourself using data set FOREST.XLS)test the hypothesis that α = 0.
for this data set.
The file ELECTRIC.XLS contains data on Y = the costs of production (measured in millions of dollars) and X = output (measured in thousands of kilowatt hours) for 123 electric utility companies in the US. Repeat
The file ADVERT.XLS contains data on Y = annual sales and X = advertising expenditures (both measured in millions of dollars) for 84 companies in the US.(a) Run a regression of Y on X and obtain 95% confidence intervals for α and β.(b) Write a sentence explaining verbally what the 95% confidence
The data sets used to calculate Figures 5.1, 5.2, 5.3 and 5.4 are in FIG51.XLS, FIG52.XLS, FIG53.XLS and FIG54.XLS.
Use the data in the example related to costs of production in the electric utility industry(ELECTRIC.XLS), where Y = cost of production and X = output.(a) Run a regression of Y on X.(b) Take log transformations of both variables.(c) Run a regression of ln(Y) on ln(X) and interpret your results
Data set EX46.XLS contains two variables, labeled Y and X.(a) Make an XY-plot of these two variables. Does the relationship between Y and X appear to be linear?(b) Calculate the square root of variable X.(c) Make an XY-plot of the square root of X against Y. Does this relationship appear to be
for a definition of these variables). Does there seem to be a nonlinear relationship between any pair of variables? Repeat the exercise using the data in the advertising example (ADVERT.XLS).
Using the data in FOREST.XLS, examine different XY-plots involving the variables X, Y, W and Z (see
(a) Using the data in FOREST.XLS (see Exercise 4.1) run a regression of Y on X.What is the R 2?(b) Calculate the correlation between Y and X.(c) Discuss the relationship between your answers in (a) and (b).
(a) Using the data in FOREST.XLS (see Exercise 4.1) run a regression of Y on X and obtain the residuals. How would you interpret the residuals? Are there any outliers?(b) Repeat question (a) for the other variables, W and Z, in this data set.
The Excel data set FOREST.XLS contains data on Y = deforestation, X = population density, W = change in cropland and Z = change in pasture land.Exercise 4.2 Using the data in FOREST.XLS (see Exercise 4.1), run a regression of Y on X and compare the actual observations to the fitted values.
(a) Using the data in FOREST.XLS, calculate and interpret a correlation matrix involving deforestation, population density, change in pasture and change in cropland.(b) Repeat part (a) using the following variables in the data set HPRICE.XLS:house price, lot size, number of bedrooms, number of
The file EX34.XLS contains four variables: Y, X1, X2 and X3.(a) Calculate the correlation between Y and X1. Repeat for Y and X2 and for Y and X3.(b) Create an XY-plot involving Y and X1. Repeat for Y and X2 and for Y and X3.(c) Interpret your results for (a) and (b).
People with a university education tend to hold higher paying jobs than those with fewer educational qualifications. This could be due to the fact that a university education provides important skills that employers value highly. Alternatively, it could be the case that smart people tend to go to
(a) Using the data in HPRICE.XLS, calculate and interpret the mean, standard deviation, minimum and maximum of Y = house price (labeled “sale price” in HPRICE.XLS), X = lot size and Z = number of bedrooms (labeled “#bedroom”).
(a) Using the data in FOREST.XLS, calculate and interpret the mean, standard deviation, minimum and maximum of deforestation and population density.(b) Verify that the correlation between these two variables is 0.66.
Construct and interpret descriptive statistics for the pasture change and cropland change variables in FOREST.XLS.
The file FOREST.XLS contains data on both the percentage increase in cropland from 1980 to 1990 and on the percentage increase in permanent pasture over the same period.Construct and interpret XY-plots of these two variables (one at a time) against deforestation. Does there seem to be a positive
(a) Recreate the histogram in Figure 2.2.(b) Create histograms using different class intervals. For instance, begin by letting your software package choose default values and see what you get, then try values of your own.
(a) Recreate Figure 2.1.(b) File INCOME.XLS contains data on the natural logarithm of personal income and consumption in the US from 1954Q1 to 1994Q2.6 Make one time series graph that contains both of these variables.(c) Transform the logged personal income data to growth rates. Note that the
4. How do we proceed if the sign of the comparative-statics effect cannot be determined unambiguously?
3. In a model based on an optimization problem, how may the second-order conditions help us in the comparative-statics analysis?
2. What are the sufficient conditions under which we can carry out the comparative-statics analysis of a general economic model?
1. (a) What does comparative-statics analysis do? (b) How does it do it?
4. Short-run cost minimization. In the model of the cost-minimizing firm in this section, suppose that the firm has a fixed amount of capital K = K avail- able. Solve its cost-minimization problem, derive its cost functions, discuss their properties, and compare them with the long-run functions
3. In the example on farmer's-land allocation in this section, solve the problem in terms of outputs rather than land allocations. Draw the level curves of the net-revenue function in (y, y2)-space, and illustrate the solution as a point of tangency. Show what would happen diagrammatically if the
2. (a) max y=2x+4x2-x-0.5x-2x1x2 subject to 2x1 + x2 = 10 (b) max y=x1x2 subject to x + x = 16 (c) max y=x+x2 subject to (x2/25) + (x/9) = 1 (d) min y=2x1 + x2 subject to (0.2x0.5 +0.8x20.5)-2=1
1. Solve the following constrained maximization and minimization problems, illustrating your answer with diagrams in each case: (a) max y=2x+3x2 subject to 2x + 5x = 10 (b) max y=x25x0.75 subject to 2x+5x=10 (c) min y=2x+4x2 subject to x0.25x0.75 -10 (d) max y(x+2)(x2+1) subject to x1 + x2 = 21
5. A monopolist supplies two markets, one at home, the other abroad. The de- mand functions are 91 = 10-P, 92=5-0.5p2 where q denotes home sales and q2 foreign sales. The firm's total-cost func- tion is C = 0.5(91 +92) (a) Find its profit-maximizing output and prices (no arbitrage between the
4. A company owns an inventory of 100 units of a good. It must sell the entire inventory over the next three periods. The profit function for sales within any one period is (x)=50x, -0.5x, 1=1,2,3 It wishes to maximize the present value of profit V = (x1)+x(x2) + (x3) where 0.8 is its discount
3. A firm is considering bidding for the franchise to sell cola and hot dogs at a baseball stadium. It estimates the demand functions for cola and hot dogs respectively as Dc 20-4pc - PH DH 15-pc-5PH.where Dc is demand for cola in thousands (of cans), Dy is demand for hot dogs in thousands, pc is
2. Find the maxima or minima of each of the functions in question I subject to the constraints 0x10 1 x2 20
1. Find the stationary values of the following functions and use the second-order conditions to determine which give maxima, minima, or saddle points: (a) y = 0.5x+2x (b) y=x+x2-x-x+x1x2 (c) y=10x+2x2 -0.5x-2x+5x1x2 (d) y=2x1 x2-x-x (e) y=x+x-4x1x2 (f) y xx2+2/x1 + 4/x2 (g) y = 2x-4x2
8. State and explain necessary conditions for a point (x.....x) to yield a local minimum of the function f subject to the interval constraints a xb. i=1,..., n.
7. State and explain necessary conditions for a point (xx) to yield a local maximum of the function f subject to the interval constraints a xbi i = 1,..., n.
6. Explain what is meant by the saddle point of a function.
5. Explain why these second-order conditions cannot be framed in terms of the second-order partials fii only.
4. State and explain sufficient conditions for a point (xx) to yield a local minimum of a function f over its domain R".
3. State and explain sufficient conditions for a point (x.....x) to yield a local maximum of a function f over its domain R".
2. What types of points make up the set of stationary points?
1. State and explain necessary conditions for a point (x.....x) to yield a stationary value of a function f over its domain R".
10. Do parts (a), (d), and (e) of question 10 for the general CES production function y =A[8x + (1-8)x2] defined on x1 > 0 and x2 > 0. A>0,0 < -1
9. Consider the following specific CES production function defined on x >0. X2>0: y= f(x1, x2)= [0.3x +0.7x]-1/2 (a) Find an expression for the MRTS, and show that isoquants are strictly convex to the origin. (b) Use the determinant condition in theorem 11.12 to show that fis quasiconcave. (c) Show
8. Illustrate Euler's theorem for the Cobb-Douglas production function f(x1, x2, x3)=Axxx A>0. a > Ovi defined on R. and a = 1.
7. Use a Venn diagram to show the following relationships between: (a) The set of functions that is quasiconcave and the set that is concave. (b) The set of functions that is quasiconvex and the set that is convex.
6. Show that the function y = Axxx. 7. ++ A > 0, a; > 0 Vi defined on R is quasiconcave. Show that it is concave only if we add the restriction that a ++3 1.
5. Use theorem 11.9 to show that the function 1/4 1/2 y=xx y=x x defined on R is strictly concave.
4. For each of the following functions find the total differential, and use this to draw a representative level curve. (a) y=x1 + x2 for x1, x2 0 1/2 1/3 (b) y=xx for x1, x2 >0 (c) y=xx for x1, x2 > 0 In which cases are the level curves strictly convex to the origin? Explain by finding the second
3. Compute all the first- and second-order derivatives of the function f(x1, x2, x3)=ax1+xx 4. and show that Young's theorem applies.
2. Compute all the second-order partial derivatives for the function given in question 1. Determine the signs of these and provide an economic interpreta- tion.
1. Find the marginal-product functions for the production function f(x1, x2) = Axx for A>0.0 0.
6. Show that using the formula given by equation (11.11) to find the elasticity of substitution, , for the Cobb-Douglas production function yields the same result as was found in example 11.39.
5. For the Cobb-Douglas production function f(x1, x2) = Axx, A>0,a. > 0 defined on R+, show that Euler's theorem applies so that f1x1+f2x2 = kf (x1, x2) where k is the degree of homogeneity of f.
4. Show that the production function y = xxis homogeneous of degree 7/6.
3. Show that: 4. 1/4 1/3 1/4 (a) y=xxx satisfies the conditions of quasiconcavity and strict concavity on R+ 1/2 1/3 1/4 (b) y=xxx satisfies the conditions of quasiconcavity but not strict concavity on R 1/2 2/3
2. Show that the following functions defined on Rare quasiconvex. Which are also convex? (a) f(x1,x2)=x+x (b) f(x1, x2)=3x+5x (e) f(x1, x2)=2x+3x2-xx
1. Show that the following functions defined on Rare quasiconcave. Which are also concave? (a) f(x1, x2)=x"x2" 1/2 1/4 (b) f(x1, x2)=x 1/3 2/3 (c) f(x1, x2)=xx
10. A = [24] Find the matrix of the the quadratic forms below. Assume that x is in R. (a) 8x+7x-3x-6x1x2+4x1x3 - 2x2x3 - (b) 4x1x2+6x1x38x2x3
9. Obtain the orthogonal decomposition of
8. Find the eigenvalues of (b) B= [2] 3 6 8 A 0 1 6 00 2
7. Find the eigenvalues of (a) A =
6. Let V be the set of points inside and on the unit circle in xy-plane defined by such that x+y2 Showby means of a specific example that V is not a vector space.
5. Provide two different bases for R such that the vectors of the bases are not orthogonal to each other and are not of unit length.
4. Find the rank of the following matrices: [1 0 2 [10 1 A 0 1 0 10 2 B=01 0 2
3. (a) If A is a 9 x 7 matrix, what is the largest possible rank of A? (b) If A is a 7 x 9 matrix, what is the largest possible rank of A? Explain.
2. If A is a 7 x 9 matrix with three linearly independent rows, what is the rank of A?
1. Establish whether the following pairs of vectors are linearly independent: (a) y = [8] w= (b) y = (c) y = == [i] W W = -[4--0--0
12. What does it mean to say that a matrix is indefinite?

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Mathematics For Economics 3rd Edition Michael Hoy, John Livernois - Solutions

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