Answers: Algebra Assignment

Part 1: Equations

2. Use Excel to create a table and draw a graph for each of the following equations.   You should draw the graphs for values of X from -1000 to +1000.  Increments of 100 would be adequate in the table

a. Y = 4X - 2

b. Z = 4X³ - 100

c. 8X -4W = 10

Equations and graphs can be thought of as alternative means for demonstrating a relationship. You may have enough of a background in analytic geometry to recognize the equation types and the graphs that would correspond to them. For example, you could know that the first equation is linear which means that the graph is a straight line. Now that you have some skills with excel, however, you could use excel to draw the graphs.  The table below contains all of the data for increments of 1000.  The formulas that were used were simply the equations, with the exception of the last equation that needed to be rearranged first. If X in the table below is in cell A1, then the formula for Y and Z were

cell B2 = 4*A2 - 2

cell C2 = 4*A2^3 - 100

To get the last equation into the correct form we need to isolate W on the left side of the = sign.  We do this as follows:

8X - 10 = 4W

8/4*X - 10/4 = W

2*x - 2.5 = W

cell D2 = 2*A2 - 2.5

Once you have these formulas in the first row, you should highlight the row and drag the bottom right corner of cell D2 down to fill in the table. 

The graphs are

and

and

3a. The demand for electricity (Qd) in any municipality is dependent upon the price of electricity (P), the population (POP), and median income (Y).

• a. Assume that the demand function is linear in all the variables and construct an equation for the demand function.
• b. describe the expected restrictions on the parameters in the equation based on your knowledge of economic theory
• c. Assume that a study of demand for electricity has found demand to be higher in Providence than Hartford but that in Hartford demand is more responsive to price changes. Draw on one diagram the two demand curves.

The key here is the linear assumption which means that each of the right-side variables is raised to the power of 1.

b. The expected restrictions pertain to the signs on the parameters if the equation is to be consistent with the economic theory. Based on traditional economic theory you would expect that an increase in the population would increase demand so that the coefficient of POP would need to be positive since it is equal to the change in Qd for a one unit change in POP. A value of 3 would mean that any increase in population would increase demand by 3. Using the same approach we would expect the coefficient of price (P) to be negative indicating that price and quantity demand are negatively related. An increase in the price (P) will reduce the amount demanded (Qd). Finally we should expect the income effect to be positive – an increase in income increases demand so the coefficient must be positive.

c. The graphs appear below.  The Hartford graph lies within the Providence graph since at each price demand in Hartford is less than demand in Providence.  The fact that demand is more responsive to price in Hartford shows up as a flatter slope for the Hartford graph. Hartford is the Blue and Providence is the Maroon line.

3b. You are to assume that the cost of public services, police, fire, education... (C), increases proportionally with the size of the population (P) and that the revenues from taxes (T)  are proportional to the taxable real estate base (B), the proportion being the real estate tax rate (t). You are to use this information to derive the equation for the tax rate and determine what will happen to the property tax if there is an increase in the population greater than the increase in the amount of taxable property?

The secret here is understanding what proportional means mathematically.  The equation that specifies a proportional relationship would be a linear equation with a zero intercept. In this case we would have the first two equations that describe the appropriate proportional relationships.  The third equation builds in the constraint that the budget must be balances and taxes equal costs. 

(1) C = a*P  

(2) T = t*B

(3) T = C

These two equations must be used to 'solve' for the relationship between t, P, and B.

a*P = t*B [substitute equations 1 and 2 into equation 3]

t = a*P/B  [isolate t on left hand side of equation]

If the population rises faster than the tax base [ P/B rises], then we will see the tax rate rise since the right-side is now larger.

4a. The balance of trade problems has focused attention on the demand for imports. A recent econometric study of import spending in the US has produced the following equation:

M = 6100 - 2.1*CPI + .25*Y

where Y = income

CPI = price level

• a. Are the signs on the coefficients what you would expect? Explain.
• b. How would the import demand equation for Canada differ if import demand were more sensitive to changes in income in Canada;
• c. Draw on one graph the curves relating import demand (M) to Income (Y) for the two countries.

a. What does our economic tell us to expect? The value of imports should increase as income increases – if people have more income to spend then they will spend some of it on goods that are bought from abroad (imports). We would therefore expect a positive coefficient which is what we have in the equation. As the price level rises (CPI increases) we should find that domestic goods are more expensive which should make imported goods more attractive. In this situation there should be a positive relationship between the two variables, but the coefficient is negative which is inconsistent with this theory.

4b. The personnel director for Electronics Associates developed the following estimated regression equation relating an employee's score on a job satisfaction test to his or her length of service and wage rate:

y = 14.4 - 8.69*Xl + 135*X2

where:

• Xl - length of service (years)
• X2 - wage rate (dollars)
• y - job satisfaction test score (higher scores indicate more job satisfaction)

• a. Interpret the coefficients in this estimated regression equation.

The coefficients tell us about the nature of the relationship between the level of job satisfaction and the number of years worked and the wage rate. Based on the equation, an increase in the wage rate (X2) increase job satisfaction because of the positive coefficient.  The level of job satisfaction decreases with the number of years worked, however, because of the negative coefficient on the variable X1.

• b. Develop an estimate of the job satisfaction test score for an employee that has 4 years of service and makes $6.50 per hour.

Here all you need to do is plug in the values into the equation.  y = 14.4 - 8.69Xl + 13 5X2 to get y = 14.4 - 8.69*4 + 13 5*6.5 = 857.14

5. Below you will find a number of behavioral equations.

(1) C = a + b*YD

(2) I = d + e * r

(3) QD = a + b*P + d*K

(4) R = a + b*YD + e*U

(5) U = a + b*p

(6) S = a + b*U + c*p

(7) w = a + b*U + c*YD

Where

• C = consumption
• YD = disposable income
• I = investment
• r = interest rate
• P = price
• QD = demand
• K = income
• R= retail spending
• p = inflation rate
• U = unemployment
• S = share of vote won by incumbent
• w = growth rate in wages

1. You should identify the parameters in each equation and specify the signs that you expect them to have.  For example, in the demand equation you would expect a negative relationship between price and demand and that would show up in a negative coefficient.

(1) C = a + b*YD Consumption spending should increase as disposable income increases so b should be positive coefficient,  Also, if you expect that the increase in consumption would be less than the increase in disposable income, then it would be a fraction.

(2) I = d + e * r   Investment spending, business spending on new machines and factories, should be higher when interest rates are lower because this lowers the cost of funds.  You would expect that the coefficient of r would be negative.

(3) QD = a + b*P + d*K     You would expect that demand will decrease as prices rise and increase as income rises.  This would be reflected in a negative coefficient on price and a positive coefficient on income.

(4) R = a + b*YD + e*U   Retail sales would be expected to be larger when the economy is doing well.  You would expect that retail sales will increase when disposable income increases and they should decrease when unemployment rises.  The coefficient of disposable income should be positive and the coefficient of unemployment should be negative

(5) U = a + b*p  One of the central issues in macroeconomics is the relationship between the unemployment rate and the inflation rate.  It even has a name - the Phillips curve.  The expectation is that it should be a negative relationship and thus the coefficient should be negative.

(6) S = a + b*U + c*YD   Economics does tend to influence votes and people tend to reward the incumbents with reelection. You would expect the share of the vote going to the incumbent to increase in good times, so it should decrease when unemployment rises and increase when disposable income increases.  The coefficient of U should be negative and the coefficient of YD should be positive.

(7) w = a + b*U + c*p   Wages should increase more in good times and when prices are rising faster.  You would expect that the coefficient of U is negative and the coefficient of p is positive. 

1. How will the following behavioral changes show up in the equations?  More specifically, you are to determine how the following changes will be reflected in changes in the parameters.  For example, if demand became more responsive to price changes, then the coefficient of price in the demand equation would increase in size (in absolute value).  If the coefficient was -2, an increase in sensitivity would increase it to -3.

1. An increase in the sensitivity of investment spending to interest rates.
2. An increase in the sensitivity of retail spending to unemployment.
3. A decrease in investment spending.
4. A decrease in the sensitivity of unemployment to the inflation rate.
5. An increase in the income elasticity of demand.
6. A decrease in sensitivity of election results to inflation.
7. A decrease in the sensitivity of wage growth to unemployment.

The key to this question is recognizing that when we talk about sensitivity that we are talking about slope of a line and the coefficients / parameters in the above equations.  In the simple equation Y = a + bX, b will be interpreted as the measure of the sensitivity of Y to changes in X.  a = DY / DX. 

• An increase in the sensitivity of investment spending to interest rates.

We are talking about equation 2 and the parameter e.   Based on your economics background, you will expect  that e should be negative since economic theory tells you to expect that an increase in interest rates will reduce investment spending.   An increase in the sensitivity of investment to interest rates means that the absolute value of the coefficient rises.

• b. An increase in the sensitivity of money demand to interest rate changes.

We are talking about equation 4 and the parameter e.   Based on your economics background, you will expect  that e should be negative since economic theory tells you to expect that an increase in interest rates will reduce money demand as people want to reduce their holdings of cash which earns no interest.    An increase in the sensitivity of money demand to interest rates means that the absolute value of the coefficient rises.

• c. A decrease in investment spending.

We are talking about equation 2 and the parameter d.   A decrease in the level of investment spending means that the absolute value of the coefficient falls.

• d. A decrease in the sensitivity of unemployment to the inflation rate.

We are talking about equation 4 and the parameter e.   Based on your economics background, you will expect  that e should be negative since economic theory tells you to expect that an increase in interest rates will reduce money demand as people want to reduce their holdings of cash which earns no interest.    A decrease in the sensitivity of money demand to interest rates means that the absolute value of the coefficient falls.

• e. An increase in the income elasticity of demand.

We are talking about equation 3 and the parameter d.   Based on your economics background, you will expect  that b should be positive since economic theory tells you to expect that an increase in income will increase demand.    An increase in the sensitivity of demand to income means that the absolute value of the coefficient rises.

• f. A decrease in demand for money.

We are talking about equation 4 and the parameter a.   A decrease in the level of money demand means that the absolute value of the coefficient falls.

Extra Questions

6. The cost (c) of truck transportation depends upon the gas and maintenance of the vehicle and the salary of the driver. The cost of operating a truck (excluding the cost of the driver) is dependent on the speed (s) of the truck. This fixed cost is $5 and the cost increases by $.50 for every one mile/hour increase in the speed. The driver's salary is $6 per hour.

• a. Assuming the trip is 300 miles, you are to derive the function relating the total operating cost to the speed of the trip. Hint: To do this, begin by developing two separate cost functions - one for the driver's expense, and one for the gas and maintenance.

The problem here is to develop the equation that specifies the relationship between the speed at which one drives and the total cost.   The two components of cost are the driver's cost and the operating cost.  The cost per hour is $6 and the driver's cost (Cd) equals the number of hours on the road (H) times the cost per hour [equation 1].   The number of hours depends upon the speed (S) and the distance (D) [equation 2].  Combining these two equations we can get an equation for Qd specified in terms of speed [equation 3].  The operating cost (Co) is $5 plus $.5 times the speed (S) [equation 4].

• (1) Cd = 6*H
• (2) H = D/S
• (3) Cd = 6*D/S
• (4) Co = 5 + .5*S

Adding equations 3 and 4 we get the equation for total cost (C) which is the solution to the problem once we replace D with 300 [equation 5].

• (5) C = 6*D/S + 5 + .5*S

• b. Assuming the legal limits to speed are 40-70 mph, what speed should the driver drive if he/she is concerned with minimizing cost?

There are a few ways to do this.  One would be trial and error - plug in various values for S and calculate C. A second would be to use calculus and take the derivative of the equation and solve for the value of S where the derivative is zero.  The third option would be to set up a spread sheet similar to what appears below. In cell b2 you will put the equation =6*300/a2 + 5 + .5*a2. 

7. The three equations below are segments of the estimated demand equations for motor vehicles, furniture, and food.

lnQ1 = .718lnY + .534lnY-1 + .355lnY-2 + .178lnY-3 - .398lnP

R2 = .56

lnQ2 = .156lnY + .116lnY-1 + ..077lnY-2 + .039lnY-3 -.596lnP

R2 = .89

lnQ3 = .531lnY + .398lnY-1 + .265lnY-2 + .133lnY-3 - .334lnP

R2 = .76

where:

Q = quantity demanded Y = income

P = price -1 = lagged one period

• a. Identify the equations and briefly describe the basis for your selection. (Ex. Do you believe that the first equation would most likely be the food, furniture, or automobile equation?)

This is the problem that causes the most problems, but this is only because of the fact that it is based on a set of nonlinear equations and it includes a time dimension.  Or maybe it is because there is no answer that is perfectly correct.  What we do know is that the coefficients in a log linear equation (which we have here) should be interpreted as elasticities. Economic theory suggest would indicate that the income and price elasticity of demand for food would be the smallest of the three, while it would be difficult in advance to rank the other two by elasticities.  If we assume that price elasticity of demand increases as the size of the expenditure increases, then motor vehicle demand has a higher price elasticity than appliance demand. 

The problem is that if we rank them based on price elasticity, food would be the last equation.  Ranked on income elasticity, however, food would be the second equation [the income elasticity would be the sum of the lagged and current income terms].  Bottom line - it is not clear which equation is the food equation, but if we rank them on the price elasticity then the ranking will be Q1 (appliances), Q2 Motor vehicles) and Q3 (food).

• b. Use your answer to part a to describe the differential behavior of industry sales over the business cycle.

The business cycle refers to the cyclical movements of income (GDP).  If we follow the ranking in part a then we would look to the income elasticities to determine the movement over the business cycle - the higher the elasticity the more cyclical demand would be.  based on this we would expect that in the year that income dropped one percent, demand for appliances (Q1) would fall .718 percent, demand for motor vehicles would fall .156 percent and demand for food would drop .531 percent. 

• c. If income falls 2 percent in 1991, 1.5 percent in 1992, and rises 1 percent in 1993, how much will demand have changed for Q2 from its present level if no other changes take place?

These changes need to be plugged into the second equation with the 1991 figure being the Y-2.  The answer would be percentage change in Q2  = .156*1 - .116*1.5 - .077*2.  =-.17.  Demand should be down by .17 percent.