EXERCISE 1 -

Q1. Graph the functions

(a) y = -x² + 5x - 2            

(b) y = x² + 5x - 2

with the set of values -5 ≤ x ≤ 5 constituting the domain. It is well known that the sign of the coefficient of the x² term determines whether the graph of a quadratic function will have a "hill" or a "valley." On the basis of the present problem, which sign is associated with the hill? Supply an intuitive explanation for this.

Q2. Graph the function y = 36/x, assuming that x and y can take positive values only. Next, suppose that both variables can take negative values as well; how must the graph be modified to reflect this change in assumption?

Q3. Condense the following expressions:

(a) x⁴ x x¹⁵          

(b) x^a x x^b x x^c

(c) x³ x y³ x z³

Q4. Find:

(a) x³/x^-3

(b) (x^1/2 x x^1/3)/x^2/3

Q5. Show that x^m/n = (n√x)^m. Specify the rules applied in each step.

Q6. Prove Rule VI and Rule VII.

EXERCISE 2 -

Q1. Given the market model:

Q_d = Q_s

Q_d = 21 - 3P

Q_s = -4 + 8P

find P* and Q* by (a) elimination of variables and (Use fractions rather than decimals.)

Q2. Let the demand and supply functions be as follows:

(a) Q_d = 51 - 3P

     Q_s = 6P - 10

(b) Q_d = 30 - 2P

      Q_s = -P + 5P

Find P* and Q* by elimination of variables. (Use fractions rather than decimals.)

Q3. According to (3.5), for Q* to be positive, it is necessary that the expression (ad - bc) have the same algebraic sign as (b + d). Verify that this condition is indeed satisfied in the models of Probs. 1 and 2.

Q4. If (b + d) = 0 in the linear market model, can an equilibrium solution be found by using (3.4) and (3.5)? Why or why not?

Q5. If (b + d) = 0 in the linear market model, what can you conclude regarding the positions of the demand and supply curves? What can you conclude, then, regarding the equilibrium solution?  

EXERCISE 3 -

Q1. Find the zeros of the following functiuons graphically.

(a) f(x) = x² - 8x + 15

(b) g(x) = 2x² - 4x - 16

Q2. Solve Prob. 1 by the quadratic formula.

Q3. (a) Find a cubic equation with roots 6, -1, and 3.

(b) Find a quartic equation with roots 1, 2, 3, and 5.

Q4. For each of the following polynomial equations, determine if x= 1 is a root.

(a) x³ - 2x² + 3x - 2 = 0

(b) 2x³ - ½x² + x - 2 = 0

(c) 3x⁴  - x² + 2x - 4 = 0

Q5. Find the rational roots, if any, of the following:

(a) x³ - 4x² + x + 6 = 0

(b) 8x³ + 6x² - 3x - 1 = 0 

(c) x³ + ¾x² - 3/8x - 1/8 = 0

(d) x⁴ - 6x³ + 7¾x² - 3/2x - 2 = 0

Q6. Find the equilibrium solution for each of the following models:

(a) Q_d = Q_s

      Q_d = 3 - P²

      Q_s = 6P - 4

(b) Q_d = Q_s

      Q_d = 8 - P²

      Q_s = P² - 2

Q7. The market- equilibrium condition, Q_d = Q_s, is often expressed in an equivalent alternative form, Q_d - Q_s = 0, which has the economic interpretation "excess demand is zero." Does (3.7) represent this latter version of the equilibrium condition? If not, supply an appropriate economic interpretation for (3.7).

EXERCISE 4 -

Q1. Given the following model:

Y = C + l₀ + G₀

C= a + b(Y - T)    (a > 0,    0 < b < 1)              [T: taxes]

T = d + tY            (d > 0,   0 < t < 1)               [T: income tax rate]

(a) How many endogenous variables are there?

(b) Find Y*, T*, and C*.

Q2. Let the national-income model be:

Y = C + l₀ + G

C= a + b(Y - T₀)  (a > 0, 0 < b< 1)

G = gY                (0 < g <1)

(a) Identify the endogenous variables.

(b) Give the economic meaning of the parameter g.

(c) Find the equilibrium national income.

(d) What restriction on the parameters is needed for a solution to exist?

Q3. Find Y* and C* from the following:

Y = C + l₀ + G₀

C = 25 + 6Y^1/2

l₀ = 16

G₀ = 14