MATHEMATICS FOR ECONOMISTS II

QUESTION ONE
a) Find the derivatives of y with respect to x:
i. y = (6?5)(?2−4)−(2?)(?6)(?2−4)2
ii. y = 3?12 + ?3 u = ?2 + 2?2 (4 mks)
b) Given the following Marginal Propensity to Consume (MPC) function, derive the corresponding Consumption function:
MPC = 0.7 + 0.1?−1/4
And C= 80 when Y= 0 (3mks)
c) Compute the elasticity of Q with respect to P and state whether the function is elastic, inelastic or unit elastic
Q = 2?2 (3 mks)
d) Discuss the limitations of Static Equilibrium Analysis (4 mks)
e) Compute the following integral:
∫(?2/3- 7? + 5?3?)dx (3 mks)
f) A national income model is represented by the following functions:
Y = C + I +G
C = a +bYd
T = tY
G = G0
I = I0
Derive the following multipliers:
i. Government Expenditure Multiplier
ii. Income Tax Rate Multiplier (6 mks)
g) Find the partial derivative of Z with respect to x and y
Z = (3?4+3?5− ?3)6 (2 mks)
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QUESTION TWO
a) Given Demand and Supply functions in a one-commodity market model as:
Qd = a – bP
Qs = -c + dP
i. Derive the equilibrium price and quantity
ii. Using comparative static partial derivatives, compute:
a. Effect of a shift of the demand function on equilibrium quantity (????). Use a diagram to show an increase in the parameter a
b. Effect of change in the slope of the supply function on equilibrium price (????). Use diagram to show increase in the parameter d
(8 marks)
b) Magothe has a coffee firm in Kiambu county having the following functions:
Q= 0.8P – 20
TFC = 180
AVC = 4 + 2Q
Find Magothe’s profit maximizing level of output and his profit (7 marks)
QUESTION THREE
a) Wijenje has the following maize production function
Q = 40K0.4L0.6
Where Q is the quantity of maize produced while K and L are units of inputs capital and labour respectively. Supposing that the prices of K and L are Ksh 20 and Ksh. 40 respectively, and that he has a total of Ksh. 5000 to spend on the two inputs:
i. Using Lagrangean optimization technique determine the values of λ, K and L at profit maximization level
ii. What will be Wijenje’s maximum profit
iii. Using bordered Hessian matrix, confirm that the critical values present a maximum (15 marks)
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QUESTION FOUR
a) Noellene is a price discriminating monopolist having the following functions for her milk production firm:
P1 = 32 – 2Q1
P2 = 22 – Q2
TC = 10 + 2Q + Q2
Determine the prices and quantities for the milk in the two different markets (9 marks)
b) The following demand and supply functions were extracted from a perfectly competitive market
P = 80 – 12Q demand function
P = 20 + 110Q supply function
Determine Producer Surplus and Consumer Surplus at equilibrium (6 marks)
QUESTION FIVE
a) What is the usefulness of the Lagrangean multiplier in mathematical optimization (3 marks)
b) Faith has a mango firm in Kitui in which she has an objective of:
Maximizing profit = 60x -2×2 – xy – 3y2 + 80y
Subject to x + y =12 as the constraint
i. Compute the values of x, y and λ at profit maximization point (10 marks)
ii. What will be Faith’s profit (2 marks)

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