Equilibrium Interest Rate
We rewrite here the supply curve for credit which is a function of interest rate r:
We can also rewrite the demand curve for credit which is a function of interest rate r:
We can solve for equilibrium by trying out a vector of interest rate points, or using nonlinear solution methods.
Alternatively, although this is not a system of linear equations, we can approximate these equations using first order taylor approximation, then they become a system of linear equations. We can then using linsolve to find approximate equilibrium Q and r.
First Order Taylor Approximation
Here, we discussed the formula for First Order Taylor Approximation: Definition of Differentials. Using the formula we have from there: We approximate the demand and Supply curves. Now x is the interest rate, is the demand or supply at interest rate x we are interested in. a is the interest rate level where we solve for actual demand or supply. We approximate the by using information from . For the problem here, let us approximate around , this is 100 percent interest rate. Note the demand and supply curves are monotonic, and they are somewhat linear for segments of r values. If they are not monotonically increasing or decreasing, we should not use taylor approximation.
Approximate the Supply
% For Approximation, need to get the derivative with respect to R
SDiffR = diff(S, r)
SDiffR =
% Now evaluate S at r = 1 and evaluate S'(r) also at r = 1
SatRis1 = subs(S, r, 1)
SatRis1 =
SDiffRris1 = subs(SDiffR, r, 1)
SDiffRris1 =
% We now have an equation that approximates supply
SupplyApproximate = SatRis1 + SDiffRris1*(r-1)
SupplyApproximate =
Approximate the Demand
% For Approximation, need to get the derivative with respect to R
DDiffR = diff(D, r)
DDiffR =
% Now evaluate D at r = 1 and evaluate D'(r) also at r = 1
DDiffRris1 = subs(DDiffR, r, 1)
DDiffRris1 = % We now have an equation that approximates supply
DemandApproximate = DatRis1 + DDiffRris1*(r-1)
DemandApproximate = Solve approximate Demand and Supply using a System of Linear Equations
Now we have two linear equations with two unknowns, we can rearrange the terms. Note that only r and are unknowns, the other letters are parameters. Starting with the equations from above:
We can plug this into matlab and solve for it
COEFMAT = [1, -b/4;1, k*h];
OUTVEC = [a-(3*b)/4; h + k*h];
approximateSolution = linsolve(COEFMAT, OUTVEC);
QEquiApproximate = approximateSolution(1)
QEquiApproximate =
REquiApproximate = approximateSolution(2)
REquiApproximate =
Now we have approximate analytical equations for demand and supply. If our was close to true equilibrium rate, we would have a good approximation of how parameters of the model, the constants, impact the equilibrium interest rate and quantity demanded and supplied.