Microeconomics

Economic Optimization
Total revenue
TR = f(Q)
TR = P * Q
MR = 𝛿TR/𝛿Q

Revenue maximization
TR = f(Q)
dTR/dQ = 0
when MR = 0, revenue is max

Average cost minimization
MC = dTC/dQ
AC = TC/Q
set MC = AC, solve for Q

Profit maximization
π = TR - TC
π(Q) = TR(Q) - TFC - TVC(Q)  
dπ/dQ = dTR/dQ - 0 - dTVC/dQ = 0    ; if ignore TVC
dπ/dQ = 0    -> mπ = 0
dTR/dQ = 0 -> mR = 0
So: mπ = mR

π(Q) = TR(Q) - TC(Q)
dπ/dQ = dTR/dQ - dTC/dQ = 0
marginal profit
Mπ = MR - MC = 0
Mπ = 0 , MR-MC = 0
So: MR = MC

Accounting profit πA = TR - TC(explicit)
Economic profit πE = TR - TC(explicit + implicit)
πE = πA - implicit cost(opportunity cost)

In perfect competition, business are earning normal profit, and economic profit is zero.

πE = 0
Normal profit = TC(explicit + implicit)

Demand Analysis
Market basket: combination of good and services that gives the same amount of utility or satisfaction

Indifference curve:
A curve of all market baskets that provides same utility to consumer
indifference curves do not intersect, convex to the origin, slopes downward

A->B->C, the utility value of one unit of x becomes smaller compared to y, then the amount of x to substitute is larger
(the more unit you consume, the less satisfaction per unit you get)

Budget constraints
B = PxX + PyY
slope of B  = dY/dX
PyY = B - PxX
Y = B/Py - Px/Py X
slope of B = dY/dX = -Px/Py
combination of products that can be purchased for a fixed amount

Income effect: increase in consumption after price cut

Substitution effect: changes in consumption as consumer substitute cheaper products for expensive ones

Effect: The change of relative prices is the substitution effect (steep line to dotted line) and the change of purchasing power is the income effect (dotted line to parallel solid line)

Engle Curves:
The effects of changing income on consumption

Optimal Consumption
for utility max, slope IC = slope B
-MUx/MUy = -Px/Py
MUx/Px = MUy/Py

one dollar you spend on goods X, give you the same satisfaction or not, compared to one dollar you spend on good Y

Marginal rate of substitution (MRS)
MRS = dY/dX, slope of an IC
MUx = dU/dX
MUy = dU/dY
MUx*dX = -MUy*dY
slope IC = dY/dX = -MUx/MUy = MRSxy

MRSxy = MUx/MUy

On the consumption side, for utility to remain constant, the quantity goods X has to be given up for one extra unit of goods Y.

Elasticity
Point elasticity
 e = %ΔY/%ΔX = X/Y * DY/DX
Arc elasticity
E = ((Y2-Y1)/((Y2+Y1)/2)) / ((X2-X1)/((X2+X1)/2))

Price elasticity of demand = %ΔQ/%ΔP
Point elasticity
 e = (P/Q) * (DQ/DP)
Arc elasticity
E = ((P2+P1) / (Q2+Q1) ) * ((Q2-Q1) / (P2-P1))

completely inelastic demand, ep = 0, see below

completely elastic demand, ep = infinite
(happen in perfect competition market, price taker only)

Price elasticity and price changes

elastic demand  |ep| > 1.0
  luxury goods  %ΔQ>%ΔP
  P↓ => Q↑ ->  P↓ => TR↑
  P↑=> Q↓ ->  P↑ => TR↓
unitary elasticity |ep| =1.0
inelastic demand |ep| < 1.0
  necessity goods  %ΔQ<%ΔP
  P↓ => Q↑ ->  P↓ => TR↓
  P↑=> Q↓ ->  P↑ => TR↑

Production Analysis
Marginal Product
MPx = dQ/dX

Isoquant

Different input combinations used to efficiently produce a output

Marginal rate of technical substitution
MRTSxy = MPx/MPy

On the production side, for output to remain constant, the quantity input X has to be reduced for one extra unit of input Y.

imperfect substitution

perfect substitution

perfect complement

Margin revenue product
MRPx = dTR/dX = dQ/dX * dTR/dQ
          = MPx * MRQ

Margin revenue product of Labor
PL = MPL * MRQ =  MRPL
PL is wage of labor

If MRP > MC, profit will increase
Economy efficiency MRP = MC

Budget line (isocost curve)
B = PxX + PyY
slope of B  = dY/dX
PyY = B - PxX
Y = B/Py - Px/Py X
slope of B = dY/dX = -Px/Py

Expansion path
for utility max, slope Isoquant = slope B
-MPx/MPy = -Px/Py
MPx/Px = MPy/Py

Output elasticity  = %ΔQ/%ΔX
Point elasticity
 e = (X/Q) * (DQ/DX)
Arc elasticity
E = ((X2+X1) / (Q2+Q1) ) * ((Q2-Q1) / (X2-X1))

Degree of operating leverage (DOL)  = %Δπ/%ΔQ
 e = (Q/π) * (Dπ/DQ)
DOL = (P-AVC)/(P-AC)

Cost Analysis
TC = TFC + TVC
AC = AFC + AVC
MC = dTC/dQ

short run cost curves

Profit contribution π_c
πc = P - AVC  per unit
πc = PQ - AVC*Q  total
      = TR - TVC
P > AVC => πc +
P < AVC => πc  -

Learning curve

learning rate  (AC1 - AC2) / AC1 *100

Breakeven analysis
total  revenue = total cost
  P * Q = TFC + AVC * Q
  QBE = TFC / (P - AVC)

Cost elasticity  = %ΔTC/%ΔQ
Point elasticity
 e = (Q/TC) * (DTC/DQ)
Arc elasticity
E = ((Q2+Q1) / (TC2+TC1) ) * ((TC2-TC1) / (Q2-Q1))

Price Theory
Competitive markets
P = MR = MC

Imperfectly competitively markets
TR = PQ
MR = dTR/dQ = d(PQ)/dQ
       = PdQ/dQ + QdP/dQ
       = P(1 + Q/P * dP/dQ)
MR = P(1 + 1/ep)
max π: MR = MC
P* = MC/(1 + 1/ep)   -> P* is profit max price per unit

MOC = (P-MC)/MC => P=MC(1+MOC)  ; markup on cost
Optimal markup on cost = -1/(ep+1)

MOP = (P-MC)/P)  ; markup on price
Optimal markup on price = -1/ep

Competitive Market
- essential identical products
- large number of buyers and sellers
- free entry and exit
- opportunity for normal profits in the LR (LR: P = MC = AC)

SR: MC is the supply curve.
P=MC=MR (point A)
π= TR - TC  = rectangle CBQ1

LR: MC is the supply curve.
P=MC=AC
π=0

Monopoly
- product has no substitutes
- only one seller
- restricted entry and exit
- opportunity for economic profits in the LR (LR: P > AC)
- pricing power

For SR and LR , P > Cost, monopoly equilibrium Q is at MR = MC. MC is the supply curve.

For deadweight loss, it occur because P > Cost and Qm < Qc.

social benefits of monopoly
- economies of scale
- invention and innovation

Monopolistic Competition
- differentiated products
- many buyers and sellers
- free entry and exit
- opportunity for normal profits in the LR (SR: P > MC, LR: P = AC)

when P=AC,  there are normal profit , but zero economic profit

Qn: compare perfect competition and monopolistic competition, why perfect competition is more efficient in the LR?
- perfect competitive market P= AC at the lowest AC
- monopolistic competition P = AC , but that AC is not the lowest AC
- so perfect competition is more efficient than monopolistic competition

Oligopoly
- identical or differentiated products
- interdependent price output decisions
- few competitors
- opportunity for economic profits in the LR (P > MC, P = AR > AC)
- restricted entry and exit
- output setting models (cournot, stackelberg)
- price setting models (sweezy, bertrand)

Cournot model
- firms make simultaneous and independent output decisions
- duopoly, two firms
- find output reaction curve

Stackelberg model
- sequential output settings, big firm set output levels, smaller firm follows
- price leadership

DL = DT -Sf , DT is total demand, Sf is follower supply curve
Price leader faces demand curve DL, as a monopolist, max profit where MRL = MCL, at Q1 and P1. Followers supply output of Q3 - A1.

Bertrand model
- firms make simultaneous and independent price decisions
- find price reaction curve

Sweezy model
- kinked demand curve

This model faces a kinked demand curve, indicating that competitors will react to price reductions by cutting their own prices and causing the segment of the D curve below the kinked to be relatively inelastic. Price increases are not followed, causing the portion of the D curve above the kink to be relatively elastic.