Question

Suppose the following table represents the market demand and supply: Price per apple (P) Quantity demanded ( $2 $4.5 $8 Quantity supplied (Q) 10 35 70 64 16 a (10 points) Calculate the linear demand function: Q-a -bP. Draw the linear demand in a graph with price in the vertical axis and quantity demanded in the horizontal axis. Label all points including the intercept terms. Calculate the slope of the linear demand function What is the economic meaning of the slope? b (10 points) Calculate the linear supply function: -cdP. Draw the linear supply in a graph with price in the vertical axis and quantity demanded in the horizontal axis. Label all points including the intercept terms. Calculate the slope of the linear supply function. What is the economic meaning of the slope? c (10 points) Solve for the equilibrium price and quantities. Draw both the supply and demand functions in the same graph and indicate the market equilibrium. On the same graph show the effect of an increase in consumers' wealth. What is the implication for equilibrium price and quantity?

Answer 1

Solution:

We are given the following table:

Price per apple (P)	Quantity demanded (Qd)	Quantity supplied (Qs)
$2	64	10
$4.5	44	35
$8	16	70

We find the equation for the linear curves using the two point formulation of equation of a line: if the two points are (x1, y1) and (x2, y2), equation of a line becomes: y - y1 = (y2-y1/(x2-x1))*(x - x1).

In this case, price P is represented by the vertical axis (or y-axis) and quantity demanded or supplied is represented by horizontal axis (or x-axis). Thus, we have the following:

a) Let the two points on demand curve be: (64, 2) and (16, 8)

So, forming linear demand function: P - 2 = (8-2/(16-64))*(Qd - 64)

P - 2 = -(1/8)*(Qd - 64)

8P - 16 = - Qd + 64

So, Qd = 80 - 8P

Price (in $s) 104 (0,10) Demand curve: 80 8P (16,8) 6 5 (44,4.5) 4 (64,2) 2 (80,0) -10 0 10 20 30 4050 60 70 80 90 100 Quantity (in units)

Slope of the linear demand curve (we already calculated) = (8-2/(16-64)) = -1/8 = -0.125

Slope of linear demand curve can also be calculated as $\partial P/\partial Qd$ = 1/($\partial Qd/\partial P$) . This slope measures the steepness of a demand curve, stating what is the relation between quantity demanded and price. Since, it is negative it means as price increases, quantity demanded falls, keeping other determinants of demand unchanged.

b) Let the two points on demand curve be: (10, 2) and (70, 8)

So, forming linear demand function: P - 2 = (8-2/(70-10))*(Qs - 10)

P - 2 = (1/10)*(Qs - 10)

10P - 20 = Qs - 10

So, Qs = 10P - 10

Price (in $s) 1 10 curve: Qs 10P-10 70,8) 6 5 (35,4.5) 4 2 9(10,2) (0,4) 10, -10 0 10 20 30 4050 60 70 80 90 100 Quantity (in units)

Slope of the linear supply curve (we already calculated) = (8-2/(70-10)) = 1/10 = 0.10

Slope of linear demand curve can also be calculated as $\partial P/\partial Qs$ = 1/($\partial Qs/\partial P$) . This slope measures the steepness of a supply curve, stating what is the relation between quantity supplied and price. Since, it is positive, it means as price increases, quantity supplied increases (positive relation between price and quantity supplied), keeping other determinants of supply unchanged.

c) Equilibrium occurs where the quantity demanded, Qd, equals the quantity supplied, Qs:

Qd = Qs

80 - 8P* = 10P* - 10

18P* = 90

P* = 90/18 = 5

Q* = 10*5 - 10 = 40 units

So, equilibrium price is $5 and equilibrium quantity is 40 units.

Implication of equilibrium price and quantity is that at this point, what all consumers demand at this particular price, sellers are willing to supply at that same price. Due to such adjustment and no discrepancy between what is demanded and what is supplied, equilibrium in the market occurs.

When the consumers' wealth increase, since, consumers are the buyers, and impact demand side of the market, this change in their wealth will affect only the demand curve and not the supply curve. Furthermore, since wealth is a determinant of demand other than price, it will be a (parallel) shift in the demand curve, and not a movement along demand curve. Lastly, such increase in wealth will result in consumers' demanding more at a particular price (since, they are capable of buying more now with real wealth increased), there will be a rightward shift in the demand curve. Following is the graph indicating such change in wealth (notice, that now the equilibrium price and quantity have increased as now more is demanded, sellers can charge a bit higher price):