# 4 calculation questions about Mathematical Economics

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There have 4 questions about mathematical Economics, please finish it on time.

Have to finish it in 12 hours.

Econ 3302 Assignment 1 Due in class on October 2nd (Tuesday) in the class The assignment can be done in groups of 2 or 3 students. Individual submissions are also welcome. In answering the questions below, make sure that you show all your steps clearly. Question 1 (6 points) A. Give an example of each of the following. (3 points). 1. y is function of x, but x is not a function of y. 2. y is not a function of x but x is a function of y. 3. y is a function of x and x is a function of y. B. In each of the following cases, determine the real roots (if any) (3 points) (i) 2x2 -4x + 6 =0 (ii) x2 -2x + 1 =0 (iii) 2x2 -5x + 2=0 Question 2 (9 points) Consider an economy with two sectors - food (f) and manufacturing (m). Each sector uses labour (L) to produce its output, and labour is mobile across both sectors, so that in aggregate labour market equilibrium, workers earn the same wage (w) in both sectors. The supply of labour in the economy is exogenously given. In addition, food requires land (T) and manufacturing requires capital (K). These factors are not mobile at all, and their quantities are exogenously given. This model can be expressed in terms of the following equations. 1. Lf = a - bw + eT0 2. Lm = c - dw + gK0 3. Lf +Lm = L0 a > 0, b > 0, e > 0 c > 0, d > 0, g > 0 Equation (1) describes the demand for labour in food production (Lf), where w is the wage and T is the supply of land. Similarly, equation (2) depicts the demand for labour in manufacturing production (Lm), with K standing for the supply of capital. Equation (3) is the equilibrium condition for the economy's labour market: the total demand for labour should equal the exogenously given supply of labour (L0) A. Solve for the equilibrium wage using the equilibrium condition. (3 points) B. Determine the impact of an increase in the stock of capital by 2 units on the equilibrium wage. (2 points) Then find the impact on the equilibrium wage of an increase in the stock of land by 2 units. (2 points) Would the equilibrium wage increase, decrease, or remain unchanged if the quantity of capital increased by 2 units and, simultaneously, the quantity of land decreased by 2 units. Explain your answer. (2 points) Question 3 (7 points) A. Simplify each of the following expressions. (2 points) (i) (x0.5 + x3/4x-1/4)/x0.25) (ii) x-2x3/2/x1/3 (iii) (x1/4y-2)-3 (iv) (64x9)1/3 (16y)-2 B. Consider the consumer demand function for blueberries: Q = P-aRb Xc where a, b and c are positive constants, Q = quantity of blueberries demanded, P =price of blueberries, R= price of raspberries, and X= consumer income. 1. Suppose the initial values of P, R and X are P0, R0 and X0. Now both P and R rise by 20 percent (X remaining unchanged). Show how demand Q would change as a result? Specifically, would demand rise by more than 20 percent, less than 20 percent, or by 20 percent? Show your work. (3 points) 2. Economic theory tells us that if P, R and X all rise simultaneously by the same percentage, demand Q would not change? In the demand function given above, what condition would ensure this? Explain. (2 points) Question 4 (20 points) A. The population of Nova Scotia (NS) was estimated to be 920 thousand in 2011, and to be growing at the rate of 2.4% annually. The population of New Brunswick (NB) is 730 thousand in 2011 and growing at the rate of 3.7% annually. When would the population of the two provinces (NB and NS) be the same? Write down the function and show how you solve it. (5) B. The natural growth of a species of tree as a function of its age can be expressed as follows: π(π‘) = 40 + 44π‘ + 4.4π‘ 2 β 0.02π‘ 3 where time, denoted by t, is measured in years, and volume of timber, denoted by π(π‘), is measured in feet cube. Answer the following questions. 1. Find the age at which the volume of timber produced by a tree stops growing. (3) 2. Write the expression for Rate of Growth of timber produced by a tree. Donβt solve it. Express it with generic notations, i.e. f(x) or fβ(x) etc. (5) 3. Suppose interest rate and price of timber are constant and denoted by π and π respectively. If the value of a tree is defined as ππππ’π(π‘) β‘ π π(π‘) at any t, express the Present Value of a tree as a function of its age. (3) 4. Denote the present value as PV(t). Now express the changes in present value as a function of t. In other words, find an expression that shows the marginal change in its present value. (4) Hints: β’ β’ Rate of growth in X = dX/X (dX is the change in X, Rate of Growth or % change in X is dX/X) Future Value = Present Value x ert

Faithe
School: Purdue University

Due in class on October 5th (Thursday) in the class

Question 1 (6 points)
A. Give an example of each of the following. You can answer this by drawing rough graphs
and givingan explanation in one line in each case (3points).
1. y is function of x, but x is not a function of y.
y=x2 +6; for every x value, y has only one value.
X is not a function since; x=β(y-6) ,an equation that yields more than 1 values for x; thus,
x is not a function of y

2. y is not a function of x but x is a function of y.

x=y^2+4; for every y value, x has only one value.
The y is not a function since of x ; y=β(x-4) ,an equation that yields more than 1 values for
y; thus, y is not a function of x

3. y is a function of x and x is a function of y

.= 2y=x+1; For every value of y there is only one value of x and vice versa.

B. In each of the following cases, determine the real roots (if

any) (3 points)
(i) 2x2 -4x +6=0
(ii) x2 -2x +1=0
(iii) 2x2 -5x +2=0

And

b β4ac
2

1. b2 β4ac < 0 There are
no real roots.
2. b2 β4ac = 0 There is
one real root.
3. b2 β4ac > 0 There are
two real roots.

(i) 2x2 -4x +6=0
16-4*2*6=-48

(ii) x2 -2x +1=0
4-4*1=0

(iii) 2x2 -5x +2=0
25-4*2*2=9

0 Real root
1 real root
2 Real roots

Question 2 (9 points)
Consider an economy with two sectors - food and manufacturing. Each sector uses labour to
produce its output, and labor is mobile across both sectors, so that in aggregate labour market
equilibrium, workers earn the same wage ...

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Anonymous
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