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ACTIVITY ONE
Learning Objectives
LO1. Students to consolidate basic meaning of bivariate functions
LO2. Students to learn how to confidently use bivariate functions in economics
Students are given the bivariate function : Y = X1 X2
They are put into small groups and asked to produce lists of all meaningful economic relationships connecting the variables utility, income, tax rate, consumption, disposable income, output, labour , capital, government revenue, interest rate, investment, apples, oranges, etc. which can be described by the function:
A complete answer involves specifying what the labels Y, X1, and X2 mean and any restrictions on and are.
Task 1
Construct one example where and are both positive
Task 2
Construct one example where is positive but is negative
Task 3
Construct one example where and both take on specific numerical values.
ACTIVITY TWO
Learning Objectives
LO1. Students to consolidate basic meaning of bivariate functions
LO2. Students to learn how to confidently use bivariate functions
Task One
Students are given the bivariate function : Y = X12 X23 and asked to complete the final column and comment on their ansers.
X1X2Y(i)56?(ii)57?(iii)34?(iv)35?
BiVariate Functions  ANSWERS
ACTIVITY ONE
There are a large number of solutions and, of course, the purpose is not to create a particular solution but rather to practise the confident use of bivariate functions as used in economics.
Possible solutions include
Task 1: Y=utility, X1 = oranges, X2 =apples
Task 2: Y= Consumption, X1 = income, X2 =interest rate
Task 3: Y= Government revenue, X1 = income, X2 =tax rate, and =1
ACTIVITY TWO
Task One
5400
8575
576
1125
Comparing (i) with (ii) , note that Y increases by (8575 5400) = 3175 due to a unit increase in (X2) whereas comparing (iii) with (iv) Y increases by only (1125576)= 549 due to a unit increase in (X2)
Message: The absolute change in Y, when only one independent variable changes, depends on the starting level of both independent variables.
Partial Differentiation  ACTIVITIES
ACTIVITY ONE
Learning Objectives
LO1. Students to consolidate basic meaning of partial derivative functions
LO2. Students to learn how to confidently calculate partial derivative functions in economics
Students are given the bivariate function : Y = X12 X23 which reduces to the univariate function Y=8 X12 when X2= 2.
Students are divided into two groups. Students in Group A find the ordinary derivative of Y=8 X12 ( answer is 16 X1) and evaluate it for X1 = 2, 3, 4 etc. Students in Group B find H1 , the partial derivative of Y= X12 X23 w.r.t. X1 ( answer is 2X1 X23) and evaluate it for (X1,X2) = (2,2), (3,2), (4,2) etc
They then compare notes by reporting their answers and filling up the following table:
Value of X1Group A
Ordinary derivative of Y=8 X12Group B
H1 ,Partial derivative of Y= X12 X23 w.r.t. X1 (evaluated when X2=2)234
The numerical results in Column 2 and 3 should be the same.
Task 1
Create a table similar to the above when X1 is constant at 3 but X2 is varying.
Task 2
Draw a rough sketch of H1against X1 holding X2 constant at X2 =2
Task 3
Draw a rough sketch of H1against X2 holding X1 constant at X1 =4
ACTIVITY TWO
Learning Objectives
LO1. Students to consolidate the idea that the partial derivative (like the ordinary derivative) is a function (not a number)
Task One
Students are given the bivariate function: Y = H(X 1, X2) = X12 X23 and asked to complete the following table and comment on their answers
X1X2H1H2(i)56??(ii)57??(iii)34??(iv)35??
Partial Differentiation  ANSWERS
ACTIVITY ONE
Task One
In the tables, columns 2 and 3 should produce the same answers. Students should appreciate why they are equivalent. Group A is reducing the bivariate function to a univariate function first and then differentiating whilst Group B is differentiating first and then setting a specific value for the constant X2.
Task Two
Task Three
ACTIVITY TWO
Task One
2160, 2700
3430, 3234
576, 432
750, 675
Comparing (i) with (ii) and (iii) with (iv) note that both H1 and H2 numerically increase due to a unit increase in (X2)
Message: The value of both partial derivatives (as only one independent variable changes) depends on the starting level of both independent variables.
Unconstrained Optimisation  ACTIVITIES
ACTIVITY ONE
Learning Objectives
LO1. Students to consolidate writing down an objective function which is to be maximised.
LO2. Students to learn how to obtain the FOC
LO3. Students learn how to solve the FOC
Task One
Students are given the following problem. A multiproduct firm has cost function given by TC=ax2 +bxy +cy2 + K where x and y are the quantities produced of its two outputs. The prices of these are fixed in world markets at p and q respectively. How much of each output should the firm produce?
Students should now be split into four groups and each group be given the following tasks. The groups should not be in contact, i.e. each group works in isolation from all others.
Group A: Write down the optimisation problem in standard form, making the notation clear.
Group B: Given the function = px + qy [ax2 +bxy +cy2 + K], find the partial derivative functions x and y.
Group C: Given the problem and the partial derivatives x = p 2ax by and y = q bx 2cy
What are the FOC?
Group D: Find the solution to the equations p 2ax by = 0 and q bx 2cy = 0
ACTIVITY TWO
LO1. Students learn how to operate SOC to check for a maximum
Task One:
Continuing with the above problem students are now divided again into three groups. Each group works in isolation as before and is assigned the following tasks.
Group A: Calculate the second partial derivative functions: xx, xy and yy ,yx
Group B: Given the answers to the second partial calculations, find the value of and xx
Group C: Using economic intuition and the fact that = 4ac b2 and xx = 2a, discuss whether the SOC are satisfied.
Unconstrained Optimisation  ANSWERS
ACTIVITY ONE
Task One
Group A : Choose x and y to Maximise = px + qy [ax2 +bxy +cy2 + K]
Group B: x = p 2ax by and y = q bx 2cy
Group C: p 2ax by = 0 and q bx 2cy = 0
Group D: x* = [2cpbq]/[4ac b2] and y*= [2aq bp]/[ 4ac b2]
Then the sequence of answers is put together by the Lecturer and the whole group discusses the question Are the values of x* and y* found by this procedure the firms optimal output levels? [Answer: We cannot be sure because the Second Order Conditions have not been checked] .
ACTIVITY TWO
Task One
Group A: 2a, b, 2c,  b. Note that xy = yx, both being equal to b.
Group B: = 4ac b2 and xx = 2a
Group C : It is immediately obvious that part of the SOC is satisfied because xx is clearly negative. But what about ? Economic intuition tells us that if the price of y, i.e. q goes up, it would be utterly perverse for the firm to sell less y. Similarly for x. In other words, we expect supply curves to be upward sloping!
From the solutions to FOC we note that x* = [2cpbq]/[4ac b2] and y*= [2aq bp]/[ 4ac b2].
The impact of p on x* and of q on y* will be positive only if 2c/[4ac b2] and 2a/[ 4ac b2] is strictly positive. But 2c and 2a are clearly positive. Therefore in order for upward sloping supply curves, we require [4ac b2] to be strictly positive. But [4ac b2] is precisely the value of . Hence, > 0 and the SOC are satisfied.
Constrained Optimisation  ACTIVITIES
ACTIVITY ONE
Learning Objectives
LO1: Students to learn how to obtain the FOC
LO2: Students learn how to solve the FOC
LO3: Students learn how to interpret the FOC
Task One
Students are given the following problem:
(a) Maximise H= p1X1 + p2X2 s.t . G= X12 + X22 d" r2 and (X1,X2)e"0
(b)Using your answer to (a) write down the FOC and interpret
(c) Using your answer to (c) solve these three equations
ACTIVITY TWO
Learning Objectives
LO1. Students consolidate solving and interpretation of FOC.
This is an individual activity. All students are given the following problem:
A small country is seeking to maximise its export revenue by selling its coffee and coconuts on the world market where they fetch $ 6 and $ 8 per ton respectively. In producing coffee and coconuts, the country is constrained by its production set which is given by:
X12 + X22d" 100
Where X1 denotes coffee production in tons and X2 denotes coconut production in tons.
Task One
Find the optimum production values X1* and X2* of coffee and coconuts and the export revenue earned. At what rate would the export revenue increase if the production set constraint was slightly relaxed.
Task Two
Re solve the problem if the production constraint was changed to X12 + X22d" 121. By how much does export revenue change? How could you obtain this answer without resolving the whole problem? If t !"*./0Dr
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ACTIVITY THREE
Learning Objectives
LO1. Students learn how to operate convexity conditions to check for a maximum
Task One
Students are asked to draw shapes of convex and non convex sets
Task Two
Students are asked to note common economic problems in which the relevant constraints do indeed form a convex set.
Task Three
Students are asked to note how to identify convex superior sets that arise commonly in economics.
Task Four
Students should check whether the convexity conditions are satisfied for the problem above.
Constrained Optimisation  ANSWERS
ACTIVITY ONE
Task One
(a) Answer: L = p1X1 + p2X2  [ X12 + X22  r2]
It is very important that students write the Lagrangean correctly, particularly noting the NEGATIVE sign between the objective and the constraint. They also need to note that in this problem r2 is c.
(b) Answer:
L1 = p1  2 X1=0 (1)
L 2= p2  2 X2=0 (2)
[ X12 + X22  r2]=0
(c) Answer:
To solve these three equations we start with 1) and 2).
Rewrite (1) and (2) as:
p1 = 2 X1 (4)
p2 = 2 X2 (5)
Divide 4) by 5) to get:
p1/ p2= X1/ X2 (6)
From 4) and 5) since both p1 and p2 are strictly positive, it follows that (X1*,X2*) and * are strictly positive.
Since *>0, it follows that:
X12 + X22  r2=0. (7)
Now solve 6 and 7 to get:
X1* = r p1/" [p12 + p22 ] and X2* = r p2/" [p12 + p22 ]
To find the value of *, substitute X1* = r " p12/[p12 + p22 ] in 4) to get:
* = {" [p12 + p22 ]}/2r
Thus the full solution is:
X1* = r p1/" [p12 + p22 ] and X2* = r p2/" [p12 + p22 ] and * = {" [p12 + p22 ]}/2r
As a final step substitute (X1*,X2*) from above into H to obtain:
H* = r p12/" [p12 + p22 ] + r p22/" [p12 + p22 ]
ACTIVITY TWO
TASK ONE
This is just a numerical version of the problem analysed in Activity 1.
L=6X1 + 8X2 [X12 + X22 100]
FOC are:
(1): L1 = 6  2 X1 = 0
(2): L2 = 8  2 X2 = 0
(3): [X12 + X22 100] = 0
From (1) and (2), = X1/ X2 (A)
From (A) both X1 and X2 are strictly positive. Hence from (1) so is >0 hence from (3)
X12 + X22 100 = 0 (B)
Substitute (A) into (B) to get:
(X1* = 6 , X2* = 8 ) and R* = 100. Substiv(hirST_×ėŗ
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The rate at which the export revenue increase if the production set constraint was slightly relaxed = * = . R* = 100
TASK TWO
If the constraint changes to X12 + X22d" 121, then one can resolve the whole problem to get:
(X1** = 6.6 , X2** = 8.8 ) and R ** = 110. Hence dR* = 110100= 10.
Much easier to solve by using the information contained in the Lagrange Multiplier, i.e. dR*/d(r2)= *
Since * = , and d(r2)=121100=21, then dR *= X 21 = 10.5 which is approximately same as by long method.
ACTIVITY THREE
Task One
Convex sets are triangles, circles , semi circles, quarter circles, rectangles squares etc. The best example of a nonconvex set is a doughnut.
Task Two
Consumer theory ( budget set is a triangle, hence convex), Producer theory ( production function with nonincreasing returns to scale)
Task Three
Indifference curve shapes or isoprofit lines which are linear.
Task Four
Yes they are since H= p1X1 + p2X2 is linear and hence the Superior set S [
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