Supply & Demand Analysis Lecture Notes PDF

Supply & Demand Analysis Sensitivity: Internal  Function A function, f, is a rule which assigns to each incoming number, x, a uniquely defined outgoing number, y. A function can be thought of as a ‘blac...

Supply & Demand Analysis Sensitivity: Internal  Function A function, f, is a rule which assigns to each incoming number, x, a uniquely defined outgoing number, y. A function can be thought of as a ‘black box’ that performs a dedicated arithmetic calculation. We can write: y = 2x + 3 or f(x) = 2x + 3 Sensitivity: Internal  Independent & Dependent Variables The incoming and outgoing variables are referred to as the independent and dependent variables respectively. The value of y clearly depends on the actual value of x that is fed into the function. E.g. in microeconomics the quantity demanded, Q of a good depends on the market price, P. We could express this as Q = f(P) Q = f(P) – this is the Demand Function P = g(Q) The two functions, f and g, are said to be inverse functions; that is, f is the inverse of g and equivalently, g is the inverse of f. Sensitivity: Internal  We hypothesise that the function is linear so that: P = aQ + b a and b are called parameters Sensitivity: Internal  Models are based on economic laws and help to explain and predict the behaviour of real-world situations. The closer the model comes to reality, the more complicated the mathematics is likely to be. What is the difference between a model and an equation? Sensitivity: Internal  Graph of typical linear demand function Sketch a graph of the demand function: P = -2Q + 50 So, need to determine the value of P when Q = 0 and the value of Q when P = 0. If Q = 0, then P = -2(0) + 50; P = 50 If P = 0, then 0 = -2Q +50; 2Q = 50; Q = 25 Sensitivity: Internal Sensitivity: Internal  Elementary theory shows that demand usually falls as the price of a good rises. Therefore slope of the line is negative. P is a decreasing function of Q a < 0 and from the graph b > 0 Sensitivity: Internal  Task P = -3Q + 75 1. Is this a demand or supply curve? 2. Where does the curve intersect and x and y axes? 3. What is the value of: a) P when Q = 23 b) Q when P = 18 Sensitivity: Internal  Model so far is crude. Why? Other factors to consider? Q = f (P, Y, Ps, Pc, T) Sensitivity: Internal  Questions? Sensitivity: Internal  Endogenous variables are variables that are allowed to vary and are determined within the model (e.g. P and Q). Exogenous variables are constant and are determined outside the model (examples?). What is the difference between normal, inferior and superior goods? Sensitivity: Internal  Supply Function The supply function is the relation between the Q of the good that producers plan to bring to the market and the price, P, of the good. P in this case is said to be an increasing function of Q. P = aQ + b With slope a > 0 and the intercept b > 0 Sensitivity: Internal Sensitivity: Internal  But….. Sensitivity: Internal Sensitivity: Internal  Equilibrium Sensitivity: Internal  Example The demand & supply functions of a good are given by: P = -2QD + 50 P = 0.5QS + 25 Where P, QD and QS denote the price, quantity demanded, and quantity supplied respectively. a) Determine the equilibrium price and quantity. Sensitivity: Internal  a) P = -2QD + 50 P = 0.5QS +25 Set both lines equal to one another: -2QD + 50 = 0.5QS + 25 -2.5Q = -25 Q = 10 P = -2Q + 50 P = -2(10) + 50 P = 30 Point of intersection = (10, 30) Sensitivity: Internal  Determine the equilibrium price and quantity for: P = -4QD + 20 P = 2QS + 2 -4Q + 20 = 2Q + 2 -4Q - 2Q = 2 – 20 -6Q = -18 Q=3 P = -4(3) + 20 P = -12 + 20 P=8 Sensitivity: Internal  Example The demand & supply functions of a good are given by: P = -2QD + 50 P = 0.5QS + 25 Where P, QD and QS denote the price, quantity demanded, and quantity supplied respectively. Determine the effect on the market equilibrium if the government decides to impose a fixed tax of £5 on each good. Sensitivity: Internal Sensitivity: Internal  b) Determine the effect on the market equilibrium if the government decides to impose a fixed tax of £5 on each good. Is it the Demand or Supply curve that will be affected? It’ll be the supply: P = 0.5QS + 25 If the government imposes a fixed tax of £5 per good then the money that the firm actually receives from the sale of each good is the amount, P, that the consumer pays, less the tax, 5: that is, P-5. P – 5 = 0.5Q + 25 P = 0.5Q + 30 Now, need to find the new market price and quantity. Sensitivity: Internal  P = -2Q + 50 (original demand curve) P = 0.5Q +30 (new supply curve) -2Q + 50 = 0.5Q +30 -2.5Q = 30 – 50 Q=8 P = -2(8) +50 P = 34 Graphically, the tax shifted the supply curve upwards by 5 units. A P of £34 means that not all of the tax is passed on to the consumer. Sensitivity: Internal  Additional tax question for revision … The supply and demand equations of a good are given by the following formulas: P = 3Q + 24 P = -Q + 160 Find the equilibrium price and quantity if the government imposes a fixed tax of £16 on each good. Which is the Demand curve and which is the Supply curve? Sensitivity: Internal  P = 3Qs + 24 (original supply curve) P = -Qd + 160 (original demand curve) P – 16 = 3Qs + 24 P = 3Qs + 40 (new supply curve with the tax) P = 3Q + 40 P = -Q + 160 3Q + 40 = -Q + 160 3Q + Q = 160 – 40 4Q = 120 Q = 30 P = - (30) + 160 (substitute Q in to original demand curve) P = 130 Sensitivity: Internal  Answer Equilibrium Quantity = 30 Equilibrium Price = £130 Sensitivity: Internal  Solve … QD1 = 10 – 2P1 + P2 QD2 = 5 + 2P1 – 2P2 QS1 = -3 + 2P1 QS2 = -2 + 3P2 Determine the equilibrium price and quantity for this two-commodity model Sensitivity: Internal  In equilibrium, we know that Qs is equal to Qd for each good so, QD1 = 10 – 2P1 + P2 QS1 = -3 + 2P1 10 – 2P1 + P2 = -3 + 2P1 10 – 4P1 + P2 = -3 – 4P1 + P2 = -13 (1) Do the same for the second good: QS2 = -2 + 3P2 QD2 = 5 + 2P1 – 2P2 5 + 2P1 – 2P2 = -2 + 3P2 5 + 2P1 – 5P2 = - 2 2P1 – 5P2 = - 7 (2) Sensitivity: Internal  We have therefore shown that the equilibrium prices P1 and P2, satisfy the simultaneous linear equations – 4P1 + P2 = -13 (1) 2P1 – 5P2 = - 7 (2) – 4P1 + P2 = -13 (1) 4P1 – 10P2 = - 14 (2) *2 -9P2 = -27 P2 = 3 – 4P1 + P2 = -13 (1) (substitute in the value for P2 ) – 4P1 + 3 = -13 – 4P1 = -16; P1 = 4 Sensitivity: Internal  Equilibrium quantities can be deduced by substituting these values back into the original supply equations: P2 = 3; P1 = 4 QD1 = 10 – 2P1 + P2 QD1 = 10 – 2(4) + 3 QD1 = 5 QD2 = 5 + 2P1 – 2P2 QD2 = 5 + 2(4) – 2(3) QD2 = 7 Sensitivity: Internal  Questions? Sensitivity: Internal