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ECO2IQA Introduction To Quantitative Analysis

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Question:

A consumer has a demand function for a certain product given by Qd = 100 − 10P. What is the price elasticity of demand when P = 2 a. We can not find the point elasticity because the point is not given to us. The inverse supply function for StaubAir vacuum cleaner is P = 150+2Qs. If the equilibrium price in the market is $250 per unit, then the producer’s surplus is 
 
Question 1: Supply and Demand 
Suppose the demand and supply functions for wine are given by Qd = k − 4P and Qs = 1 + 3P respectively, where k is a constant parameter. Find the market equilibrium price and quantity in terms of k. In each of the following cases, sketch the demand and supply functions and indicate the equilibrium price and quantity:
a. k = 3
b. k = 4
c. k = 2.
What happens when k 

Question 2: Migration and Matrices
In applied economics, we are often interested in migration of workers from one geographical region to another. Suppose we divide a hypothetical country Loonyland into three regions: R1, R2, R3. We can find the proportion of the workers of those three regions who stay put or migrate to another region. For instance, p12 is the proportion of the workers in region 2 that move to region 1 while p33 is the proportion of workers in region 3 who do not migrate to any of the two other regions.

We can denote these transition proportions by pij , i, j = 1, 2, 3 and can state these proportions of workers that stay put or migrate to another region in terms of the following transition matrix P Suppose the number of workers in these regions (in millions) at a point in time t is denoted by (xti) and presented in the vector xt regional migration over n periods in time can be determined using the following equation:

xt = Pxt−1, t = 1, 2, ...n. (1)
1. (1 Mark) Using equation (1) only, write the expressions for x1 and x2 and simplify them to express x2 in terms of x0 and P.
2.  Using the transition matrix P and the initial endowment of workers x0, find x1.
3. Find the number of workers in each region after migration in period 1 if

4. Find the number of workers in each region after migration in period 2.
5. What would be the number of workers in each region after an influx of international migrants to each region in Loonyland by 5, 000, 000 after period 1.
Question 3: National Income Model 
Consider the following three-sector national income determination model:
C = 20 + 0.85 (Y − T)
T = 25 + 0.25Y
I = 155
G = 100

1.  Determine the exogenous and endogenous variables in the system.
2.  Solve the model presented in the above system of equations using the determinant and the inverse matrix for the equilibrium values of unknown variables.
3. Verify your solution in part (2) above by solving these simultaneous equations
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