Suppliers in a market equilibrium model expect the prevailing price to be the current price reflected in the market supply equation. If the current price is constant, then the prevailing price equal

(i) Derive the linear second-order differential equation implied by this model. (3 marks) Step 1: Rewrite the market supply equation with the expected price: q(t) = F + G(p + b(dp/dt) + c(d^2p/dt^2)) Step 2: Equate the market supply (q(t)) to the market demand (q*) and simplify the equation: F + G(p + b(dp/dt) + c(d^2p/dt^2)) = A + Bp Step 3: Substitute the linear price-adjustment equation into the equation from step 2: F + G(p + b(α(q* - q(t))) + c(d^2p/dt^2)) = A + Bp Step 4: Eliminate q* from the equation by replacing q* with A/B: F + G(p + b(α(A/B - q(t))) + c(d^2p/dt^2)) = A + Bp Step 5: Collect the terms and rearrange to form a second-order linear differential equation in the form L(p) = 0: c(d^2p/dt^2) + (Gbα - B)(dp/dt) + (GB - B)p = GB(A/B - F) - A This is the linear second-order differential equation implied by the model. (ii) Given α = 0.1, G = 25, B = -20, b = 0.5, and c = 0.1, solve the differential equation. (7 marks) Step 1: Plug the given values into the differential equation: 0.1(d^2p/dt^2) - (25)(0.1)(-20)(dp/dt) - (25)(-20)p = (25)(-20)(A/(-20) - F) - A Step 2: Simplify the equation 0.1(d^2p/dt^2) + 50(dp/dt) + 500p = 500A - 20AF - 500F Now, solving the differential equation requires a specific technique depending upon the nature of the coefficients and the given boundary conditions (initial conditions). With the provided parameters, solving the differential equation analytically might be challenging. You should consider using numerical methods (e.g., Euler's method, Runge-Kutta methods) or specialized software (like Mathematica, MATLAB) to find the solution for the price function p(t) given initial values for p(0) and dp/dt(0). Additionally, you would need the values for A and F.