Question

2. Consider the following model of Supply and Demand. where P is the price of the good, Q is quantity demanded and Qs is quantity supplied. G) What condition should o satisfy in order for the second equation to be a reasonable supply function. (ii) What condition should ß and satisfy in order for this system to have a unique equilibrium. uming a unique equilibrium exists express the system in matrix form and use matrix algebra to find the equilibrium values of Q and P. (iv) Using calculus derive an expression for how the equilibrium price changes when the supply curve shifts.

Answer 1

Solution:

Denoting alpha by a, beta by b, gamma by g and delta by h, simply for the ease of writing.

According to the question, then, demand function is: Qd = a - bP

And supply function is: Qs = g + hP

i) We are required to find a condition on h, such that Qs is a reasonable supply function. Law of supply states that there exists a positive relation between the price, P and quantity supplied, Qs, (i.e.,, the supply curve is upward sloping) meaning that more quantity is supplied at a higher price.

Thus, for the given supply function, since we see a 'plus' sign before hP, for law of supply to satisfy (or for this function to be a reasonable one) we require h to be positive, or h > 0.

ii) At equilibrium, Qs = Qd. So, the two equations, demand and supply functions, given, take the following form:

Q = a - bP

Q = g + hP

Now, we have two linear system of equations, in two unknowns, Q and P (rest are simply parameters or constants).

This means that there can exist either unique solution, or infinitely many solutions or no solution at all. In order for existence of no solution, the two lines (supply and demand curves) must be parallel, and in case of infinitely many solutions, the two must be coincident. Thus, in both these cases, the two curves must have same slope. In case of unique solution however, the lines must intersect once, implying the two slopes must be different.

Hence, for given system to have a unique equilibrium, we simply want the two lines to have different slopes.

Slope of a demand curve = = -(1/b)

Denoting alpha by a, beta by b, gamma by g and delta by h, simply for the ease of writing. According to the question, then, demand function is: Qd = a - bP And supply function is: Qs = g + hP i) We are required to find a condition on h, such that Qs is a reasonable supply function. Law of supply states that there exists a positive relation between the price, P and quantity supplied, Qs, (i.e., the supply curve is upward sloping) meaning that more quantity is supplied at a higher price. Thus, for the given supply function, since we see a 'plus' sign before hP, for law of supply to satisfy (or for this function to be a reasonable one) we require h to be positive, or h > 0. ii) At equilibrium, Qs = Qd. So, the two equations, demand and supply functions, given, take the following form: Q = a - bP Q = g + hP Now, we have two linear system of equations, in two unknowns, Q and P (rest are simply parameters or constants). This means that there can exist either unique solution, or infinitely many solutions or no solution at all. In order for existence of no solution, the two lines (supply and demand curves) must be parallel, and in case of infinitely many solutions, the two must be coincident. Thus, in both these cases, the two curves must have same slope. In case of unique solution however, the lines must intersect once, implying the two slopes must be different. Hence, for given system to have a unique equilibrium, we simply want the two lines to have different slopes. Slope of a demand curve = partial differential P/partial differential Q d = -(1/b) Slope of a supply curve = partial differential P/ partial differential Q s = 1/h

Slope of a supply curve = = 1/h

So, for unique solution, -(1/b) should not equal (1/h)

or h unequal to -b (or -h is not equal to b).

iii) In (ii), we obtained following equations:

Q = a - bP

Q = g + hP

With small modification of adding 'bP' on both sides of first equation and subtracting 'hP' on both sides of second equation, we have

Q + bP = a

Q - hP = g

In matrix notation, this can be expressed as:

Let's denote A = , X = , and B = $egin{bmatrix} a\g end{bmatrix}$

So, we have, AX = B

On pre-multiplying bot sides by A^-1 (i.e., A inverse), we have A^-1AX = A^-1B

IX = A^-1B or simply X = A^-1B, I is an identity matrix of dimension 2*2. We have to solve for X matrix using matrix algebra.

Using basics of matrix formulations, we must know the following:

A^-1 = adj(A)/|A| (adj of a matrix refers to/stands for adjoint or adjugate of a matrix)

|A| (that is determinant of A) = (1)*(-h) - (b)*(1) = - h - b.

From (ii), we have already established that -h is not equal to b, |A| is not equal to 0, and so, A^-1 exists.

adj(A) =

So, A^-1 = [-1/(h+b)]*

And, = [-1/(h+b)]* $egin{bmatrix} a\g end{bmatrix}$

= [-1/(h+b)]* (using matrix multiplication)

So, Q = (ha+bg)/(h+b) and P = (a-g)/(h+b) is the unique equilibrium solution.

iv) Shifting the supply curve: Supply curve shifts if value of g changes. That is if g decreases, supply curve shifts down, and if g increases supply curve shifts up.

With equilibrium price, P = (a - g)/(h + b), we have to find how P changes with change in g.

Change in P = * Change in g

= -1/(h+b)

So, if supply curve shifts up (down) by amount , equilibrium price decreases (increases) by .