NotesAtHKU

Profit maximization under perfect competition

Perfect compeition

Many unassociated buyers and sellers
Identical products

Price takers: $MR=P$

Cost expressions

Fixed cost (FC) are cost that do not vary with output, unlike variable cost (VC).

\begin{align*} TC & = TVC + FC \\ MC & = \frac{\Delta TC}{\Delta Q} \\ AVC & = \frac{TVC}{Q} \\ AC & = \frac{TC}{Q} \\ & = AVC + \frac{FC}{Q} \\ \end{align*}

Profit

\begin{align*} \text{Total Profit } \pi &= TR - TC \\ & = P \times Q - (FC + TVC)\\ & = P \times Q - AVC \times Q - FC\\ PS & = TR - TVC \\ \end{align*}

Economic profit includes the opportunity cost in total cost, whereas accounting profit does not.

Production runs and profit maximization conditions

Short run is the period over which some inputs are fixed
- Profit maximization at $P=\text{Intersect}(MC, MR)$
Long run is the period over which all inputs are variable (no fixed cost)
- $TR=TC \to$ no economic profit (can still have accounting profit)
- Profit maximization at $P=\text{Intersect}(MC, ATC=AVC)$

They are not defined by time, but by the ability to change inputs.

Explanation of short run profit maximization

$\because MR>MC\implies MP > 0 \implies$ firms can increase profit by increasing output
$\because MR<MC\implies MP < 0 \implies$ firms can increase profit by decreasing output
Note that $MR=MC$ does not represent the total profit, the total profit depends on $TR$ and $TC$ .

Number of firms

The equilibrium price is affected by the number of firms $n$ in the market.

Convert individual firm supply curve to market: $MC(q)\to MC(\frac{q}{n})$
Convert market demand curve to individual: $MB(q)\to MB(n\times q)$

To find the number of firms in the market:

Short run: $D=S\to Q, P=MC \to q, n=\frac{Q}{q}$
Long run: $P=\text{Intersect}(MC, ATC=AVC) \to MB(P) \to Q, n=\frac{Q}{q}$

Entry, exit and shutdown conditions

Entry is when new firms enter the market, exit is when existing firms leave the market, and shutdown is when a firm stops production temporarily.

Run	Factor	Result
Short run	$P < AVC \cup TR < TVC$	Shutdown at $P=\min\{AVC\} = MC$
Long run	$P > ATC$	Entry of new firms
Long run	$P < ATC$	Exit of existing firms
Long run	$P < AC \cup TR < TC$	Exit at $P=\min\{AC\} = MC$

Changing cost industries

Increasing cost (decreasing, constant) industries have greater (lower, constant) cost as production expands.

The following table summarizes their characteristics:

Type	When $D\uparrow$ , MC and AC:	Long run supply curve
Constant	Unchanged	Flat
Increasing	Upward shift	Upward sloping
Decreasing	Downward shift	Downward sloping

The following table summarizes changes in the following quantities when demand increases in long run / short run markets ( $C \uparrow, \downarrow$ ):

Stage	D	S	Q	P	$\pi$	n
Initial L	-	-	$Q_0$	$P_0$	$\pi_0=0$	$n_0$
$D\Uparrow$ S	$\Uparrow$	-	$\Uparrow$	$P_1 \Uparrow$	$\Uparrow$	-
$C\downarrow D\Uparrow$ L	-	$\Uparrow$	$\Uparrow$	$\Downarrow P_1 > P_2 > P_0$	$\pi_2=0$	$\Uparrow$
$C\uparrow D\Uparrow$ L	-	$\Uparrow$	$\Uparrow$	$\Uparrow$	$\pi_2=0$	$\uparrow$

Flip arrow directions for decreasing demand.

Monopoly

Market power

Ability to raise price > marginal cost without losing all customers

Sources of market power	Examples
Patents	-
Laws preventing entry	US Postal Service
Scale	Subways, electricity, highways
Innovation	iPod, eBay

Derivation of Marginal Revenue

Quantity: $n$

Discrete: $MR = TR(n) - TR(n-1)$

Continous: MR curve has twice the slope of D curve. (Coefficienct of $Q\times2$ )

Simple monopoly pricing

Maximize profit at $MR=MC$ .

Find D curve
Obtian MR curve ^^ (discrete vs continous)
Find Profit-maximizing Q* quantity @ MR=MC
Price is intersection of D curve and Q=Q*

More elastic demand $\to$ Lower price, smaller markup Less elastic demand $\to$ Higher price, higher markup

Markup

\text{Markup} = \frac{P - MC}{MC} = \frac{-1}{1+\eta_d} \implies \propto \frac{1}{\eta_d}

Where $\eta_d$ is the price elasticity of demand.

Inefficiency

Q* < Qe -> DWL is the efficiency loss

CS top part, PS bottom part including rectangle

Price $P*$ Intersect(q=q*, D) is the price that monopoly charges. (Monopoly pricing)

Revenue is $P*\times Q*$

Total marginal cost (TVC) is the trapesium under left triangle and Q=Q*.

Producer surplus = Profit if no fixed cost (Profit=TR-TVC-FC=PS-FC) (PS=TR-TVC)

Price control on monopoly

Price ceiling: New MR will be line from $I = \text{Intersection}(D, P)$ to yAxis and xAxis (L-shaped). MR is constant for $q < x(I)$ , where $x(I)$ is the maximum quantity that the market sells.

Special points:

$P_c = \text{Intersect}(D,MC) \implies$ Profit maximizing, no dwl, socially optimal
$P_c @ \text{Intersect}(D,AC) \implies$ Zero profit, DWL triangle under $x(I)$ and above MC.

Eliminate DWL by taxes or subsidies

We can use the formula which shifts the supply curve by $ts$ . For $I=\text{Intersect}(D,MC)$ :

MR(x(I))=y(I)-ts

Price discrimination

Prices are charged on different customers for the same goods for reasons unrelated to difference in costs, in order to maximize profits.

The following are three types of price discrimination:

First degree price discrimination

Perfect price discrimination is where the monopolist charges based on the buyers' maximum WTP according to the marginal benefit.

$MR = MB$ . All $CS$ is now extracted by the seller to be the $PS$ .

This does not happen in real life, as it is impossible to know the maximum WTP of each consumer. However, it is a useful theoretical concept.

2020 Fall Final Q67-69

A souvenir store has six customers each day. Their willingness to pay for the souvenir are listed in the following table.

Customer	WTP
1	92
2	79
3	64
4	51
5	36
6	22

The marginal cost is constant at $28.

(a) With perfect price discrimination, the store’s producer surplus will be ?

(b) Suppose only consumers with willingness to pay below $70 will clip coupons and use them. The store owner will optimally provide a coupon that takes ? off

First, add $MR, MC$ columns:

Customer	WTP	MR	MC	Indicator
1	92	92	28
2	79	66	28	< (b 1st min. MR > MC)
				< (b sep)
3	64	64	28
4	51	38	28	< (b 2nd min. MR > MC)
5	36	6	28	< (a)
6	22	-20	28

(a) $PS = 92+79+64+51+36-5\times28$

(b) From the above table, we see that the coupon threshold is between WTP of customer 2 and 3. Then, for each separated group, find the minimum $MR>MC$ (maximize profit). Coupon amount = $66-38$

(c) Because for each group, we set the price as $79$ and $51$ , $CS = (92-79) + (64-51)$ $\implies DWL = 182 - CS - PS$ .

(Note: The $CS$ full formula is $CS = (92-79) + (79-79) + (51-51) + (64-51)$ )

2023 Spring Final Q66-68

Doris runs a wedding photography business in Utopia. The marginal cost of providing each unit of wedding photography service is $274. Doris has to incur a fixed cost of$ 70 per month. On a typical month, Doris expects to see 10 customer couples with the following reservation prices for wedding photography service. (Each customer couple will purchase either zero or one unit of wedding photography service.)

Customer couple	Reservation price ($ per unit)
A	230
B	260
C	290
D	320
E	350
F	380
G	405
H	430
I	450
J	470

(a) If Doris has to charge the same price for all wedding photography services, to maximize profit, she will charge ? per unit and sell to ? customer couples, and consequently make a monthly profit of ?.

(b) Suppose Doris does not know each customer couple’s reservation price but she knows that all customer couples with a reservation price above $333 never use discount coupons. Those with reservation prices below$ 333 use them whenever they are available. If Doris makes coupons available in wedding magazines, those customer couples who clip and present them get to pay a discounted price for wedding photography service. Others pay the regular list price. To maximize profit, Doris should set the list price at ? per unit and the discount price at ? per unit

(c) Suppose instead Doris knows each customer couple’s reservation price and can practice perfect price discrimination. Doris will provide wedding photography service to ? customer couples in total, and consequently make a monthly profit of ? in total.

First, add number of buyers at price and set price columns:

Customer	Price	No. buyers	Profit with set price @ price	Profit	Indicator
A	230	10	-510	775
B	260	9	-196	819
C	290	8	58	833	(c)
D	320	7	252	817
				...	(b sep)
E	350	6	386	...
F	380	5	460	...	(a)
G	405	4	454	...
H	430	3	398	...
I	450	2	282	...
J	470	1	126	...

Note:

Profit with set price is calculated as $\text{No. buyers}\times(\text{Price} - MC) - FC$ .
Profit is calculated as $SUM(\text{Price}) - MC \times \text{No. buyers} - FC$ Sum range from each row to end (count is same as no. buyers!)

(a) Maximum profit is at $P=380$ , $Q=5$ , and profit of $460$ (Row F).

(b) Find separated groups by threshold $333$ . Then, discounted price is at $320$ , and listed price unchanged ( $380$ ).

Second degree price discrimination

Discrimination by Hurdles is where the monopolist charges different prices based on the quantity consumed.

With multiple $MR$ curves, part of the $CS$ is extracted by the seller.

Examples:

Buy one, get one half the price

Third degree price discrimination

Market segmentation is where the monopolist charges different prices based on the elasticity of different groups.

With multiple $MR,MB$ curve pairs, part of the $CS$ is extracted by the seller.

Conditions required:

Ability to identify different groups with different elasticities
Ability to prevent resale between groups

Examples:

Movie tickets on weekdays vs weekends
Business vs economy flight tickets

Note that arbitrage makes it more difficult implement such discrimination.

Recall that market demand curve is the horizontal aggregation of individual demand curves.

Arbitrage

Arbitrage is the practice of buying a product in one market and selling it in another market at a higher price.

To prevent arbitratge, the transportation cost between the two markets must be higher than the difference of the prices.

Anti-competitive behaviors

Other forms of price discrimination includes:

Tying: a base good is sold at a lower price, and a complementary good is sold at a higher price.

Examples:

Printer and ink
Game console and games

Bundling: requiring products to be purchased toghether in a package.

Examples:

Microsoft Office

Example application

Given $MC=2, FC=3$ , and bundling cost $bc=1$ . The following table gives the WTP of two people P1 and P2 for goods A and B:

Person \ Goods	A	B	WTP
P1	9	5	14
P2	4	6	10

Min. cost	4	5

To find the maximized profit:

\begin{align*} \text{Buy 1A 1B} & = 9 + 6 - 2\times2 - 3 & = 8\\ \text{Buy 1A 2B} & = 9 + 5\times2 - 2\times3 - 3 & = 10\\ \text{Buy 2A 1B} & = 4\times2 + 6 - 2\times3 - 3 & = 5\\ \text{Buy 2A 2B} & = 4\times2 + 5\times2 - 2\times4 - 3 & = 7\\ \text{Buy 1 bundle} & = 14 - 2\times2 - 3 - 1 & = 6\\ \text{Buy 2 bundles} & = 10\times2 - 2\times4 - 3 - 1 & = 8\\ \end{align*}

Therefore, the seller should sell separately and make a profit of 10.

Profit maximization

Profit maximization under perfect competition

Perfect compeition

Cost expressions

Profit

Production runs and profit maximization conditions

Number of firms

Entry, exit and shutdown conditions

Changing cost industries

Monopoly

Market power

Derivation of Marginal Revenue

Simple monopoly pricing

Markup

Inefficiency

Price control on monopoly

Eliminate DWL by taxes or subsidies

Price discrimination

Price discrimination

First degree price discrimination

Second degree price discrimination

Third degree price discrimination

Arbitrage

Anti-competitive behaviors

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