Question

math modeling

9. The following simple model is intended to represent the dynamics of supply and demand. Let P denote the selling price of a certain product and Q the quantity of this product being produced. The supply curve Q = f(P) tells how much should be produced at a given price to maximize profit. The demand curve P = g(Q) tells what price buyers should pay given a certain level of production in order to maximize their utility. (a) Select a specific product and make an educated guess as to the form of the supply curve Q = f(P) and the demand curve P = g(Q). (b) Use the results of part (a) to determine the equilibrium levels of P and Q. (c) Formulate a dynamic model based on the assumption that P will tend toward the level dictated by the demand curve, while Q will tend to the level given by the supply curve. (d) According to your model, is the (P; Q) equilibrium stable? Does it matter whether you assume a discrete–time or continuous–time model? (Economists usually assume a discrete–time model in order to represent the effect of a time delay.) (e) Perform a sensitivity analysis for the assumptions you made in part (a). Consider the question of stability.

Answer 1

Answer (a) Let the specific product be a book.

The supply curve of book is

Qs= 5 + 3 P

Here Qs is the quantity supplied of the book and P is the Price. Plus sign shows that supply curve is upward sloping which means quantity supplied of book increases with increase in price.

The demand curve of the book is

Qd = 25 - P

Here Qd shows the quantity demanded of the book and P is the price. the negative sign shows that demand curve is downward sloping. This means quantity demanded of book decreases with increase in price.

Answer (b) Equilibrium will take place where quantity demanded of book is equal to quantity supplied of book i.e. Qs= Qd

5+3P = 25 - P

This implies 4 P = 20

P = 5

Putting the value of P in Qs we get value of equilibrium quantity.

Qs = 5+ 3P

Qs = 5+3 (5)

Qs = 20

So at equilibrium Price of book = 5 and quantity of book = 20