Qdx f px

Chapter 2: Market forces Supply & Demand

This chapter includes four important elements:

1. A “change in quantity” demanded or supplied as a result of a change in the current price.

This is a movement along the demand or supply curve. This helps us understand the slope of demand or supply with respect to the price and then estimate their own price elasticities.

2. A “shift or change in the demand or supply” as a result of a change in a relevant “factor other” than the current price. This change represents a change in the entire demand or supply or a shift. Understanding the factors that shift the demand or supply help us specify and estimate a demand or supply equation and estimate the other factors’ elasticities

How do we distinguish a “change in quantity demanded or supplied” from a “change in demand or supply”? If the factor that changes is on any of the axes (such as the current price is on the vertical axis), then there is a “change in quantity demanded or supplied”. But if the change is in a factor that is not on any of the axes such as income or cost of production, then there is a “shift or change in demand or supply”.

The student should define the slope of direct demand or supply with respect to current price as “change in quantity over change in price”. Not the other way!

Example:

(Direct) demand: Qdx = 6,060 – 3Px. Slope of demand = ∆Q/∆P = -3

Inverse demand: Px = 2020 -1/3Qdx. Slope of inverse demand = ∆P/∆Q= -1/3

3. Consumer and producer surplus

What is the usefulness of calculating the consumer surplus for the manager? The manager can use it in price discrimination and in valuing full economic prices. What’s the usefulness of knowing the producer surplus? The producer can use it to bargain with the distributor over the surplus above minimum cost of producing the good accruing to the distributor.

4. Market equilibrium and disequilibrium (or price restrictions)

“Market equilibrium” means supply equals demand and there is no surplus or shortage. This helps determine equilibrium price and quantity.

“Market disequilibrium” means that supply and demand do not intersect or are not equal at any price in the market. In this case, we have either a surplus (quantity supplied exceeds quantity demanded) or a shortage (quantity demanded exceeds quantity supplied). This helps us determine the size of shortage or surplus.

When a government intervenes in the market and buys the surplus to set a price above the equilibrium price, then there is a “price floor” as is the case with agricultural products.

If the government issues a decree and sets the price below the equilibrium price then there is a “price ceiling or control” which leads to shortages. Some governments set a rent control for apartments.

THE SUPPLY FUNCTION

Supply function and Shifts in Market Supply

Supply Specification: The simple supply equation is defined as:

Qs = a + bP

and the slope with the respect to the price ∆Q/∆P is positive. That is the supply curve is upward sloping.

The general (direct) market supply equation can be expressed as a function.

QS = f (P; Production cost, Taxes, Expected Price).

where “P” is the current period price for this good and is different from the expected future price. Change in current price causes a change in quantity supplied or a movement along the curve. The other factors (after the semi colon) are the shifters which cause a change or a shift in the entire supply. Production cost includes the cost of labor, represented by the wage rate “ PW ”, capital cost represented by “PR”, the rental price of capital (equipment), and the price of raw materials “PM”. “ T” will represent taxes and Pex represents the expected future price for this good. Then the general (direct) supply function can be rewritten as QS= f (P; PW, PR, PM, T, Pex)

For example, Qs = 2000 + 3P –PW - 4PR – 2PM –T + - Pex.

where (direct) slope of supply with respect to (w. r. t.) current price P, ∆Q/∆P, is +3, and the slope w.r.t., PW, ∆Q/∆PW, is -1, and w. r. t. PR , ∆Q/∆PR , is -4, ∆Q/∆Pex > 0 or <0.

If PW = $20, PR = $40, PM = $10, T = $100 and Pex = $15, then after substitution for the constant shifters, the general (direct) supply equation collapses to the simple (direct) supply equation:

Qs = 1715 + 3P, which is generally written as Qs = a + bP.

All the “other variables” have been lumped with the intercept and the simple direct slope is ∆Q/∆P = +3. In the simple (direct) supply equation, all the variables after the “; ” are the factors that are held constant and are usually lumped together to form the intercept a . They are the “Shifters”. As indicated above, simple direct supply equation is given by:

Qs = a + bP. (Here, a is the horizontal intercept and b is the direct slope).

The simple “inverse” supply function is P = -a/b + (1/b)Qs

where +1/b is the inverse supply slope and -a/b is the vertical intercept.

A graph of the simple supply function is given by S1 below (P is placed on the vertical axis). Using the above example, P = -1715 / 3 + 1/3*Qs where 1/3 is the inverse slope.

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Examples of shifts in Supply: Suppose labor production cost decreases and also assume no changes in the other variables including the current price “P”. A reduction in production cost implies an increase in profit (the difference between total revenues and costs), which should increase quantity supplied. The increase in the quantity supplied while the current price is assumed constant implies a right shift (an increase) in the supply curve from S1 to S2 in the graph below. The sign for wage rate should be -.

In conclusion, a decrease in the wage rate (W) implies an increase in the quantity supplied QS, assuming P is constant, which means a rightward shift in supply curve, and vice versa for an increase in (W), which implies a shift in supply to the left.

The same logic applies to decreases or increases in PR and PM.

Changes in Expected Future Price (Pex): These changes are applied to the price expected to prevail in the next period. Their effect on quantity supplied in this period depends on the storability of the good in question.

( Storable Good; e.g., oil )

In the special case when the good is storable (e.g., oil, gold, … etc) then an increase in the expected price implies storing the good instead of producing more of it. Then at the current price, current quantity supplied decreases, representing a shift to the left in the supply curve.

In general, an increase in expected price should shift the supply curve to the right, which is the normal case. This is particularly true for non-storable goods. If the good is non-storable such as milk, then an increase in Pex leads to an increase in current QS because the production of non-storable goods does not take much time to bring them on stream and thus, the firms worry about maintaining their market shares. Therefore, the supply curve shifts to the right, assuming the other variables are constant.

Taxes: There are basically two types of taxes: Specific and ad valorem. The specific tax is a fixed amount of money per unit sold (e.g., 10 cents per pound), while ad valorem is proportional to the value or the price (e.g., 10% of the price) which may not be constant. An example of a specific tax is the excise tax, which is a constant $ tax on each unit sold and the tax revenue is collected from the supplier. In this case the (inverse) supply curve shifts up in a parallel fashion by the amount of the tax.

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Fig. 2.7 A per Unit (Excise) Specific Tax

If the tax is ad valorem, then if the price increases, the amount of the proportional tax increases with the price. Suppose the tax rate =20%. If P =$10 then the tax amount is $2. If P=$20, then the tax is $4. In this case the shift in supply is really an upward rotation .

Note that after tax, P1 = S1 = (1+t%)*S0 where supply S0 =P0 is expressed as an inverse supply equation:

P0 = -a/b+ (1/b)Qs0 which is S0, where P0 is price before tax.

Then S1 is

P1= (1+ t%)*P0 = (1+ t%)* [-a/b + (1/b)Qs0], where P1 is price after tax.

Solve for Qs1 as a function of P1.

Direct supply after tax: Qs1 = a + (b/(1+t%))P1= a + bP1 + t*bP1

(note that (1+ t%) = 1+ 20%) = 1.20 in the above example)

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Fig. 2-8 An Ad Valorem Tax (t=20%)

That is, inverse S0 rotates upward to S1.

Producer Surplus

The points on the supply curve measure the minimum amounts (or prices) the producers are willing to “accept” for producing the good because the supply curve is a cost curve. Those amounts are tantamount to minimum costs necessary to produce different levels of the good, and these costs are usually lower than the market price. Supply price is a minimum price.

Suppose the (direct) supply equation for TVs is given by:

Qs = 2000 + 3P –PW - 4PR = 2000 + 3P –2000 - 4*100 = -400+ 3P.

where PR is the rental price of monitors (a complement) per unit representing capital cost and PW is the price of an input like labor or the wage rate.

Suppose PR = $100 and PW = $2000.

Then the simple (direct) supply equation is: Qs = -400 + 3P (where -400 is horizontal intercept).

The inverse supply equation is: P = 400/3 + (1/3) Qs (400/3 is vertical intercept).

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Fig. 2.9: Producer surplus

In Fig 2.9 (Producers Surplus), the cost per unit to produce the first unit of the good is $400/3 (point C) and to produce the 800 units per unit is $400 (point B).In this figure, suppose the market price is $400 and this market price applies to all units. Upon substitution, the quantity is 800 units Then the sales revenues received by the producers are P*Q = $400 * 800 = $320,000. This is the area of rectangle [0 A B 800].

The area under the supply curve up to the point where the price line intersects the supply curve is the minimum cost associated with producing 800 units (an integral). Then

Producer Surplus = Revenues received – minimum amount necessary to produce the good or PS = TR –VC where VC is variable cost.

= Area of the triangle ABC

= ½ * H * B

= ½ *($400 – 400/3)*800

= $106,668. (for the wholesaler A)

Graphically, PS is the area below the price line and above the supply curve. It is a powerful tool for managers. In the above figure, suppose that the 800 units are 800 pounds of meat supplied by the meat wholesaler to the retailer which is the producer of steak (the restaurant). In this case, the restaurant manager (the retailer) can bargain with the meat wholesaler over the producer surplus (a maximum of $106,668) to capture some of it in the form of a lower price. Thus, the retailer can use the PS against the wholesaler.

MARKET DEMAND FUNCTION:

Specification as a general (direct) function:

QDX = f(PX ; Income, Prices of related goods, Advertising, other variables) or

QDX = f(PX; M, PY, PZ, A,H)

where goods “X” and “Y” are substitutes, and “X” and “Z” are complements. The variables after the semi colon are the shifters of the demand curve.

The simple (direct) demand depends on current own price and assumes all the other variables are constant. QD = c - dP (add or subtract the shifters to or from horizontal intercept). Direct price slope ∆QD /∆P = - d.

When it comes to income (M), there are two types of goods: Normal and Inferior. In case of normal goods, an increase in income (holding the other variables constant), would lead to an increase in purchasing power, which manifests itself in an increase in demand. Demand curve shifts to the right, assuming no change in current price. ∆QDX /∆M > 0

In case of inferior goods, an increase in income leads to a reduction in quantity demanded at the same price. Thus, the demand curve shifts to the left. ∆QDX /∆M < 0

Related goods can be substitutes or complements.

· If goods X and Y (Tea and Coffee) are substitutes , then an increase in PY (of coffee) would lead to a decrease in quantity demanded of Y(coffee). On the other hand, it is assumed that there is no change in PX (tea) then people switch from coffee to tea, which means quantity demanded of tea (QX) increases at the same price of tea PX. This implies a shift in demand for tea (X). Relation between PY and QX is ∆QDX /∆PY > 0.

· If the two goods X and Z (Printers and Computers) are complementary , then an increase in price of computers (PZ) would lead to a leftward shift in demand for printers (DX). Relation between PZ and QX is negative, ∆QDX /∆PZ < 0 .

Advertising (A) also shifts the demand curve. An increase in advertising shifts the demand curve to the right. There are two types of advertising: informative advertising which provides information about the existence or quality of a product, and persuasive advertising which alters the underlying taste of the consumer “You must buy it” or “The only thing you should buy”. ∆QDX /∆A > 0 .

Consumer Expectations (changes in expected prices, expected income etc): Demand for durable goods (e.g., cars) is affected by changes in expected prices. However, demand for perishable products (e.g. milk, eggs) is not affected much by expectation of higher prices.

Other factors (H) are special factors related to certain products such as “Health Scares” related to cigarettes” or the birth of a baby related to diapers.

The general linear (direct) demand equations can be written as

QDX = b0 - b1PX + b2PY - b3PZ + b4M + b5A.

The own direct slope with respect to PX is: ∆QDX /∆PX = -b1 < 0 (simple demand has a negative slope). (What’s the “indirect” demand slope for the simple demand?) Is it -1/b1?

The sign for income (M) depends on whether good X is normal or inferior.

For example, if the slope with respect to M is ∆ QDX /∆ M = +b4 (then the good is normal). If ∆ QDX /∆ M is negative, the good is inferior.

The slope with respect to PY is: ∆QDX /∆PY = +b2 >0 (positive means X and Y are substitutes). If ∆QDX /∆PZ = -b3 < 0 then X and Z are complements. ∆QDX /∆Z = b5

Demonstration Problem 2-1

A firm’s manager was given the estimate of the direct demand function or equation for his/her firm’s product X:

Qdx = 12,000 – 3Px + 4Py – 1M + 2Ax

Please answer the following questions:

1. What type of goods are X and Y (with respect to the price of Y)?

2. What type of good is X (with respect to income)? Normal? Inferior? Why?

3. How does advertising affect this firm’ product?

4. Let Py = $400, M =$1,000 and A = $100. Derive the simple inverse demand and calculate the inverse slope (hint: plug the numbers in the equation and solve for Px ).

Is it: PX =12,800/3 -1/3QD?

Consumer Surplus (area below the demand curve above the price line)

Points on the demand curve signals the maximum amount a consumer is willing to pay per unit for a certain amount of a product. This maximum amount falls as more of a product is consumed and it is also different from the market price. Demand price is a maximum price.

Lets us look at the demand for water. Suppose at zero the consumer is willing to pay $5 to have the first liter of water (see Fig. 2-5a).

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Fig. 2-5: Consumer Surplus

In discrete terms , after this consumer consumes the first liter he/she is willing to pay $4 for the second liter. Once this consumer has enjoyed 2 liters, it is willing to pay $3 per liter and so on.

For the continuous, case, what is the total value (benefit) of 2 liters of water? (Area under the demand curve to the horizontal axis = area of rectangle + area of triangle).

Max total benefits = $3 x (2 liters) + ½*($5 - $3)*(2 liters) = $6 + $2 = $8.

In the market, the consumer does not pay different prices for different units. Here, the market price after buying the second liter is $3. Total consumer expenses are $3x2 units = $6.

Consumer Surplus = Total Maximum Willingness to pay - Total expenses = $8 - $6=$2.

This concept is useful in disciplines that emphasize price discrimination where producers try to capture CS from consumers. You can also calculate CS directly by calculating the area of the shaded triangle above. CS = ½*H*B = ½*($5 - $3)*(2 liters) = $2.

Market Equilibrium

Market Equilibrium: Supply intersects demand. It includes the equilibrium quantity “Qe” (or Q*) and equilibrium price Pe (or P*). After the equilibrium, there is no shortage or surplus. Quantity supplied equals quantity demanded as shown in the graph below.

(Fig. 2-10: Market Equilibrium)

Demonstration Problem 2-4:

Simple Direct Market Demand: QD = 6 - 0.5*P.

Simple Direct Market Supply: QS = 4 + 2*P.

Market Equilibrium: QD = QS .

6 – 0.5Pe = 4 + 2Pe

0.5Pe + 2Pe = 6 - 4

2.5Pe = 2

Solve for Pe. Then

Pe = 2 / 2.5 = $ 0.8 (equilibrium price also called P* ).

Plug Pe in either supply or demand equation to determine the equilibrium quantity:

Qe = 4 + 2Pe = 4 + 2(0.8) = 5.6 units (equilibrium quantity)

Thus, market equilibrium = (Pe; Qe) = ($0.8; 5.6 units).

The graph of this market equilibrium is given by

(Fig. 2-10: Market Equilibrium)

Free Market Mechanism: The tendency of the market price to change as a result of market forces in order to clear the market (i.e., to equate QS and QD).

· If P1 > Pe then QS > QD (Surplus).

Then there would be a downward pressure on “P”, and once “, until QS = QD at Pe.

· If P2 < Pe, then QD > QS (shortage) and there would be an upward pressure on the price, shrinking the shortage, until QS = QD at Pe.

PRICE RESTRICTIONS AND MARKET (DIS)EQUILIBRIUM

There are two types of market disequilibrium: Price controls (or ceiling) and Price floor (or support). Disequilibrium means supply does not equal demand.

Price Control or Ceiling (PC) : Government’s intervention (e.g., rent control) prevents market price from moving up to clear the market and achieve equilibrium. Thus PC < Pe ; where PC is the ceiling price. That is, price ceiling is below equilibrium price.

http://daphne.palomar.edu/llee/101Chapter08.pdf

(Figure 2-11a: Price ceiling)

Price controls such as rent controls lead to shortages because the controlled price is too low.

Total shortages = QCD – QCS.

If these are apartments, then the total shortage can be divided relative to equilibrium into two parts:

Qe - QCS = # of existing apartments that are taken off the market relative to Qe.

QCD – Qe = # of new apartments which are sought by new renters relative to Qe

Demonstration Problem 2-5 (apartments)

Demand: QD = 100 – 5P (where Q is in 10,000 units and P is in $100, and the zeros can be ignored).

Supply: QS = 50 + 5P

a. Calculate market equilibrium

QD = QS

100 – 5Pe = 50 + 5Pe → Pe = $5 (one hundred) and Qe = 75 (0,000) units

b. Assume ceiling PC = $1 (one hundred). Calculate the total shortage.

QCD = 100 - 5 PC = 95(0,000 ) units (total quantity demanded at price ceiling).

QCS = 50 + 5 PC = 55 (0,000) units.

Total shortage = QCD – QCS = 95 – 55 = 40 (0,000) apartments.

c. If the average apartment has three persons, then

# of displaced residents = (3)*( Qe - QCS ) =(3)*(75-55) = 60 (0,000) persons

# of new residents = (3)*( QCD – Qe ) = (3)*(95-75) = 60 (0,000) persons

How Do Businesses Deal with Losses Created by Price Ceilings?

Price ceilings provide a gain for buyers and a loss for sellers. Sellers would like to avoid the loss if they can.

1. One way to do so is called a black market. In this case, the sellers illegally raise the price and hope to get away with it. So, for example, tickets to popular events are sold by scalpers at high prices. (In California, ticket scalping is not illegal if it is not conducted at the place the event takes place.) While there are many other examples, black markets are not smart; it is just too easy to be caught. It is also not smart because of the existence of gray markets.

2. A gray market is a way of getting around the price ceiling without actually doing anything illegal. There are two forms of gray market. (a) One form of gray market involves charging for goods or services that were formerly provided free . If the rent cannot be raised on the apartment, there is nothing preventing the landlord from charging for the parking space, charging for use of the elevator, charging for gardening and cleaning services, forcing the tenants to pay for electricity and water, and so forth. In New York, a rent-controlled apartment near Central Park might rent for $300 to $400 per month; in a free market, the rent would probably be $2,000 per month. To get in, one needs the key. This has been known to cost $1,000. This is not a refundable deposit; this is a charge to have the key. It is obviously worth it to be able to rent the apartment for $300 to $400 per month. A Berkeley apartment owner converted his apartment into a church. To be able to live there, one had to pay church dues of $1,200 per year in addition to the rent. Gasoline stations would commonly charge for washing the windows, checking the tires, and so forth. The price of oil used in oil changes would be raised. (Those having oil changes at the station were favored in access to gasoline during the years of the price ceiling. In these years, Americans had the cleanest engines in history.) Some gas station owners ran the line to the gasoline pump through the car wash. One San Diego station forced people to have a $7 car wash to get to the gasoline pump. ($7 in these years is the equivalent of about $20 today.). This practice was later declared illegal. (b) The second form of gray market is to provide less service for the same price .

Welfare Impact of Price Ceiling

Since there is a shortage, there should be an allocation mechanism to allocate the good among the consumers. The most common mechanism is (first come, first served). In times of severe shortage, consumers must spend some time to wait in line or search for the good or apartment. Suppose the demand is for gasoline and the consumer wants to buy 10 gallons. Moreover, assume this consumer must wait for two hours in line to get the gasoline and that this consumer’s time is worth $5 an hour. This means the consumer is spending $1 per gallon in terms of waiting time to purchase gasoline (non-pecuniary price), in addition to the price ceiling per gallon (pecuniary price). This is called the full economic price.

Example:

Full economic price can be depicted graphically as:

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( Figure 2-11b: Full Economic Price and Welfare Impact of Price Ceiling)

Example : Suppose the maximum price the consumers are “willing to pay” per unit is PF = $11 (called full economic price and is assumed) and the pecuniary price ceiling per unit is PC = $5.

Thus ( PF – PC ) = (11-6) = $6 is called the non pecuniary price per unit the consumers are willing to pay by waiting in line (the implicit price per unit for waiting in line). Full econ price per unit is PF = PC + (PF - PC)

Full economic price = pecuniary dollar price + non pecuniary price. Note that PF is greater than the equilibrium price Pe.

Example: Calculating Full Economic Price Using Equations. In the apartment example above: The supply equation under the ceiling is

QSC = 50 + 5 PC = 55 units (by plugging PC = $1 in this supply equation) (STEP 1)

Next, set the demand equation under ceiling equal to 55 units and change PC to PF in this equation:

QDC = 100 – 5PF = 55 units (STEP 2)

Then solve this equation for full economic price, PF = (100 - 55)/5 = $9. (STEP 3)

Compare this full economic price to:

Equilibrium Pe = $5 and to ceiling price PC = $1.

The non-pecuniary price of the good is: (STEP 4)

PF – PC = $9 - $1 = $8. This is the value of your time waiting in line or searching per unit.

Non busy consumers with very low opportunity cost of waiting time may benefit from the price ceiling, while those with high value for opportunity cost of waiting time may be hurt by the relatively low price ceiling. If a politician’s constituents have a relatively low opportunity cost of time, that politician naturally will attempt to invoke a price ceiling.

Another mechanism to allocate the good that is in short supply is to sell the good to the regular customers (e.g., gas stations during crises sell to their regular customers).