Matrix Functions in Business - Assignment Example

Mathematics Assignment 2 Name University Mathematics for Business Assignment Question One The markets for beef and milk denoted by i and ii respectively in a typical period t have demand functions summarised as shown in the matrix function below. The matrix equation is transformed into a scalar equation by introducing an identity matrix. Given the fact that the two products have similar determining factors in a market, the identity matrix as given in this case is a square matrix. In this regard, we use a 2x2 identity matrix. The identity matrix is 1 0 0 1 The scalar equation for the first matrix becomes Pt= (a2+b2)½+b11q1t+b12q 2t, and Pt2= (a2+b2)½+ b21q1t+b22q 2t, b) The coefficient of b11 is one. It is the element of the identity matrix that determines the inverse and can also give the roots of the matrix. Using a negative or the inverse of the matrix gives a function with a negative sign. Since the prices of any commodity cannot be stated in negatives, the coefficient of b11 has to be positive. c) The coefficient of b12 is zero. This is the second in the first column of the identity matrix. Using the inverse affects the ultimate pricing. Additionally, other quantities can change while the identity matrix cannot. d) The coefficient of b11 is one. This is because the identity of a square matrix with two elements is 1. Using a negative or the inverse of the matrix gives a function with a negative sign. Since the prices of any commodity cannot be stated in negatives, the coefficient of b11 has to be positive. e) The coefficient of b12 is zero. This is because the identity matrix has two elements namely one and zero. Using the inverse affects the ultimate pricing. Additionally, other quantities can change while the identity matrix cannot. f) b12 and b21 should be the same. When using the matrix equation, it is clear that the square matrix is in the form of a 2x2 matrix. The fact that the matrix is used for the two equations linking the meat and milk prices, an identity matrix is applied. In this case, the inverse is not used but the identity in the first order using one and zero in both diagonals. Considering the diagonal similarity, b11 and b22 are the same. Therefore, b12 and b21 should be the same. Question Two Farmers’ supply function is given by Q=c+ Dp, where p, a and q are vectors while D is a square matrix. Determine the two component scalar equations from the combined supply equation. From the equation provided, the vector quantity c1and c2 are assumed to be equal. The assumption is based on the fact that the other factors that influence production are constant. Given this and using a square matrix to replace d11, d12, d21 and d22, the equation is transformed into the following. Q1t= (c1+c2)½+d11pit+d12+p2t and Q2t= (c1+c2)½+d21pit+d22p2t b) Interpret the coefficient of d11 The coefficient of d11 takes a positive sign. The coefficient is an element of the identity square matrix. The coefficient does not change but is different if the inverse of the matrix is used. c) Interpret the coefficient of d12 The coefficient of d12 is zero. This is also an element of the identity matrix. Using the square matrix, the element enables the calculation of the combined function. d) Explain whether d11 is likely to be positive or negative The elements of an identity matrix can either be negative or positive. However, this is dependent on the identity being inverse. Given this scenario, the element d11 is positive. This drawn from the fact that the square matrix used in this case is a unit and the values are zero and one. e) Explain whether d12 is likely to be positive or negative Whereas d11 takes a positive sign, the element d12 does not have a sign. This is because the element is assigned zero in the identity matrix used. In this regard, the identity matrix does not assign the element a sign unlike in the case were inverse of a square matrix. Question Three The farmers’ predicted prices for period t is equated to the real prices in period t-1. These are associated in to the supply function given by qt=c+Dpt-1 where p-1 gives the p1which is the predicted price. Given that the annual produce equals the planned supply, price adjustments and market forces influence the stability of the price. From the above, the recurrence relationship governing the behaviour of the market is determined in the following way. The general equation is given by Xt=Fxt-1+b The value of the xt is attained based on the fact that the variable b rrepr5esents the unpredictable changes that are beyond the farmers’ ability. Assuming increasing rates in the order of 1+r over time t, this forms the coefficient of F in the general function. Therefore, in the first equation, c replaces the F and (1+r) replaces D. Consequently, the function can be rewritten in the following form Qt=atx0 The x0 represents the seed value which decreases with time through the formula (t-1). The seed value for the farmers’ case is p0. Q=Dtpt-1 Therefore, the equation is Q=DtP0 Replacing D with (t-1). The result is Q= (1+r)t.P0 Question Four The convergence to steady state equilibrium is possible if the various variables satisfy the models. To begin with, the F noted in the equation Xt=Fxt-1+b must be a stable matrix. This implies that F1 must be converging to zero. Additionally, xt must converge toward x→x1. In a scenario where x1 is in steady state, A must be equal to the difference in the values of I and F. In essence A=I-F. Question Five a) Assuming that the efficient markets hypothesis applies, determine the supply of beef and milk to the market in the next year? Using Matlab, q represents milk while p represents milk. The supply is 790.7362 units for milk and 445.2896 units of beef. b) What will be the prices at which this supply clears in the beef and milk markets next year? The selling price at which the supply will clear for milk is 5 while the price for beef is 35 per unit. Image one: The market relationship governing market stability c) Is the relationship governing the evolution of short-run market prices stable? Explain. The relationship is stable. Considering the price fluctuations and the clearing price, it is indicative of the equilibrium between supply and the variables affecting consumer demand in this case the price. The image above reveals the price and quantity variation generated using Matlab. The image indicates that the various variables affecting the market prices and supply are effective in the specific market. d) Determine the long run equilibrium prices of beef and milk. The long run equilibrium prices are as follows. For milk the price is $ 7 and beef 35 is per each unit of the product. e) Determine the long-run equilibrium quantities of beef and milk. The equilibrium quantities supplied in the market corresponds with the equilibrium prices, for milk, the quantity is 950 and beef is 420. Read More