INTRODUCTION TO MATHS I

MJ

QUESTION ONE( 25 MARKS)

  1. a) An economy has three industries, government (G), Agriculture (A) and manufacturing (M). The input output table for three industries is given as:
Input to    
G A M Final demand Total output
Output From G 400 200 200 200 1000
A 200 400 100 300 1000
M 200 100 300 400 1000
  Primary

 inputs

 200 300 400

 

Calculate the total output required for each industry when the final demand changes to 300,350 and 450 for government, agriculture and manufacturing respectively                                                                      ( 15 marks)

  1. b) Evaluate by means of integration by substitution:    (5 marks)
  1. c) Given the following equation, where

Calculate

  1. i) The cross price elasticity of demand. What is the relationship between good x and good y. Explain. (3 marks)
  2. ii) The income elasticity of demand (2 marks)

 

QUESTION TWO ( 15 MARKS)

  1. a) The demand function of a product is given as:

 

The supply function is given as:

  1. i) Determine the equilibrium price and quantity         (6 marks)
  2. ii) Determine the producer surplus under market equilibrium     (6 marks)
  3. b) State and explain the types of returns to scale                     (3 marks)

 

QUESTION THREE ( 15 MARKS)

  1. a) Given the utility function and  and income of the consumer=130
  2. i) Write the consumer’s optimization problem         (2 marks)
  3. ii) Find the optimal level of x and y.         (7 marks)
  4. b) Prove Euler’s theorem using the following function     (6 marks)

 

QUESTION FOUR ( 15 MARKS)

  1. a) Let the national income model be given as
  2. i) Identify the endogenous variables           (1 mark)
  3. ii) Find the equilibrium national income           (4 mark)
  4. b) The demand and supply functions of 2 commodities are given as

Determine the equilibrium prices and quantities for the 2 goods using cramer’s rule     (6 marks)

  1. c) Prove young theorem using the following function ( 6 marks)

 

QUESTION FIVE ( 15 MARKS)

  1. a) Find the rank of the matrix

(5 Marks)

  1. b) A manufacturers marginal revenue function is

Find the change in the manufacturers total revenue if production increases from 5 to 10 units  (4 marks)

  1. c) Given the total revenue and total cost functions below, find the profit maximizing level of output

TR(Q)=1200Q-2Q2

TC(Q)=Q3-61.25Q2+1528.5Q+2000                                                        (6 marks)

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