ECON 331 - Homework

Posted 2020-11-15 wejava

ECON 331 - Homework #2
due October 10th at 4pm in WMC3633
Late homework will not be considered. Show detailed calculations and/or provide detailed
explanations to get full credit. Partial credit may be given.
• Exercise 1.
We consider a closed economy with three industries, energy, manufacturing and sector. The
production of 1 unit of output of:
- energy requires 0.5 units of manufacturing, 0.3 units of sector, and 0.2 unit of itself;
- manufacturing requires 0.3 units of energy, 0.5 units of sector, and 0.2 unit of itself;
- sector requires 0.2 units of energy, 0.3 units of manufacturing, and 0.5 unit of itself.
(a) Write the matrix equation associated with the closed model.
(b) If possible, solve the equation by using Cramer’s rule. Otherwise, solve by Gaussian
elimination. Justify your answer.
(c) Find the vector of relative prices when the price of energy is set at $100. Explain your
notations and what you are doing.
• Exercise 2.
We consider a small section of the (original) 81x81 Leontief economic model with four industries.
The production of 1 unit of output of:
- petroleum requires 0.2 units of transportation, 0.4 units of chemicals, and 0.1 unit of
itself;
- textiles requires 0.4 units of petroleum, 0.1 unit of textiles, 0.15 units of transportation,
and 0.3 units of chemicals;
- transportation requires 0.6 units of petroleum, 0.1 unit of itself, and 0.25 units of chemicals;
- chemicals requires 0.2 units of petroleum, 0.1 unit of textiles, 0.3 units of transportation,
and 0.2 units of itself.
Consider now an open economy. If the economy produces 900 million dollars of petroleum,
300 million dollars of textiles, 850 million dollars of transportation, and 800 million dollars of
chemicals, how much of this production is internally consumed by the economy? Justify your
answer.
1
• Exercise 3.
In econometrics, one is often interested in estimating the effect of changing one variable, say
x, on another variable, say y: for instance, a school district might be interested in the effect
of the reduction of the size of its elementary school classes on the performance of its students.
The linear regression model postulates a linear relationship between x and y. More generally,
the linear regression model can be written with matrices: y = Xb + u where y is a given n × 1
column vector and X is a given (n × 2)-matrix. The associated residual sum of squares writes:
(a) Define the matrices P = X(X′X)
−1X′ and M = In−P. Justify why they are well-defined.
What are their sizes?
(b) Show that M is symmetric.
(c) Show that M2 = M.
(d) Show that MX = 0.
(e) Show that when b = (X′X)
−1X′
y, we have RSS = y

 

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