Eco5020F
Advanced Microeconomics
Term 1: Producer and Consumer Theory
Open book
80 marks
Duration 4 hours
Question 1
40 marks
For this candidate indirect utility function
𝑣 ∗ (𝑝1 𝑝2 , 𝑦) = 0.022 𝑦 6 𝑝1−4 𝑝2−2
1.1 Determine the function’s legitimacy as indirect utility function by verifying the usual properties
1.1.2 Homogeneity of degree zero in 𝐩, 𝑦
4
1.1.3 Strictly increasing in 𝑦
2
1.1.4 Decreasing in 𝐩
3
1.1.1 Continuity
3
1.2 Solve for this function’s Marshallian demands
7
1.3 List thee properties that you expect the demand functions to obey.
9
1.4 Invert the indirect utility function to find the expenditure function
3
1.5 What are the two ways in which the Hicksian demand functions can be derived from the
information above?
4
Question 2
15 marks
Battese and Coelli (1992) fitted the following OLS model estimated with with a maximum likelihood
routine for a pooled sample of n = 129 small-scale rice farms in India:
ln(𝑠𝑎𝑙𝑒𝑠) = 𝑎0 + 𝑎1 ln(𝑙𝑎𝑛𝑑)
+ 𝑎2 ln(% 𝑖𝑟𝑟𝑖𝑔𝑎𝑡𝑒𝑑 𝑙𝑎𝑛𝑑) + 𝑎3 ln(𝑙𝑎𝑏𝑜𝑢𝑟) + 𝑎4 ln(𝑏𝑢𝑙𝑙𝑜𝑐𝑘𝑠) + 𝑎5 ln(𝑐𝑜𝑠𝑡) + 𝜀




Sales are in Indian rupees
Land is in hectares
Irrigation captures farm quality. Quality is proxied by the percentage land that is irrigated
Labour is in hours


As the main source of draught power, bullocks, serves a proxy for capital. This variable is
measured in hours
Costs, in rupees, include fertiliser, manure, pesticides and other purchased inputs
The estimation results were:
Coefficient
Constant
Land
% irrigated land
Labour
Bullocks
Cost
Std error
3.71
0.62
0.80
0.74
-0.45
0.079
t-statistic
0.66
0.15
0.27
0.14
0.16
0.048
5.62
4.13
2.96
5.29
-2.81
1.66
Log likelihood statistic
-50.806
Note that a t-statistic >|2| is significant
2.1 Comment on the plausibility of the estimate
7
2.2 What are the function’s global returns to scale?
4
2.3 How would you go about finding the profit maximising level of output for this technology? Give
reasons. Hint: It has to do with first and second order conditions for maximisation and returns to
scale.
4
Question 3
25 marks
3.1 Use the Lagrange method to derive a cost function for the following CES technology
1⁄
3
𝑦 = (𝑥1
1⁄
3
+ 𝑥1 3 )
ℒ(𝑥1 , 𝑥2 , 𝜆) = 𝑤1 𝑥1 + 𝑤2 𝑥2 + 𝜆 (𝑦 −
3.2 Show that your answer is concave in 𝑤
15
1⁄
(𝑥1 3
+
1⁄ 3
𝑥2 3 ) )
10