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.

Macroeconomics (Part 1)

Short Run

Closed-economy IS-LM model, which focuses only on the relationship between the interest rate and output. Assume price is fixed.

Equilibrium Output

The 45 degree line means production and income are equal. The demand curve is
ZZ =  C + I + G = c0 + c1(Y-T) + I + G = c0 -c1T + I + G +c1Y
ZZ = autonomous spending + c1Y

 autonomous spending is the intercept with vertical axis
 c1 is the propensity to consume

Equilibrium happens when demand = production.

Goods Market
Y = C + I + G  and S = Y - T - C = I + G - T
so  S + T = I + G
if G = T => S = I

Financial Market
demand for money, depends on income and interest rate
  Md = $Y L(i)              $Y+  , L(i)-
A rise in income leads to rise in demand for money
A rise in interest leads to decrease in demand for money

intersection of money supply and money demand is equilibrium interest rate.
real money supply = real money demand
  Ms/P = Y L(i)

CB control currency and reserves
  high powered money (monetary base) = currency + reserves

Let CU be currency, D demand deposit, M money supply, c is proportion of money in currency, H is monetary base, θ is reserve ratio
M = CU + D ;  CU = cM ; D = (1-c)M ;  R = θD
H = CU + R
M =  (1/(c + θ(1-c))) H

Q: Can CB closely control the money supply?
Ans: No, CB control H and θ, but cannot control c, so CB cannot closely control money supply

zero lower bound: nominal interest rate cannot go below zero
liquidity trap: when nominal interest rate  is zero, monetary policy cannot lower it further, implying monetary policy is not effective

Goods Market in open economy

ZZ: demand for domestic goods
AA: domestic demand for domestic goods
DD: domestic demand for goods

As income goes up, some additional domestic demand falls on imports. So AA is flatter than DD. Exports do not depend on domestic income, AA and ZZ have some slope.

As income (output) goes up, imports increase, exports do not change. So NX decrease.

Medium Run

Medium run: 1) price can change 2) include labor market

Labor market:
Wage setting: W = PeF(u,z)          ; z: catchall variable for all other variables
Price setting P = (1 + m) W          ; m: mark-up of price over wage

equilibrium in labor market: F(u,z) = 1/(1+m)

Original Phillips Curve
π_e = π^-  expected inflation is rather constant
         _
π = π+ (m + z) - αu_t

So, negative relationship between unemployment and inflation.

Original phillips curve implies there is no natural rate of unemployment

Modified phillips curve
from P = (1 + m) PeF(u,z)
let F(u,z) = 1 - αu+ z             ; α:  strength of unemployment on wage
P = (1 + m) (1 - αu+ z)

so  after maths transform:π_t - π_t-1 = (m + z) - αu_t

if  0 = (m + z) - αu_t  and so the natural rate of unemployment is

u_n = (m + z) /α

Phillips curve and the natural rate of unemployment
π_t - π_t-1 =  -α(u_t - (m + z)/α)
let un = (m + z)/α ; un : natural rate of unemployment
π_t - π_t-1 = - α(u_{t -} u_n )

Implications:
The natural rate of unemployment is the rate that keep inflation rate constant, or we call this rate NAIRU - non accelerating inflation rate of unemployment

Not sustainable to keep unemployment rate below u_n, because if you do that, inflation gonna rise

Short to Medium Run : The IS-LM-PC model

Modified phillips curve
π_t - π_t-1 =  (m + z) -  αu_t

π_t - π_t-1 = (α/L)(Y - Y_n )             ; L: labor force size

Y_n is potential output, so when output is above potential output, inflation rises

IS-LM in equilibrium in the short run, but output is above potential output, so inflation rises, or economy overheating

A rising inflationary pressure makes central bank raise the policy rate. The at medium run, Y and r are at natural levels and inflation rate may not change.

Zero lower bound:
Economy in recession, and output gap is negative (Y - Y_n < 0).
Central bank need to reduce real policy rate to restore Y to Y_n and make inflation stable. But, zero lower bound constraint make it impossible to achieve a negative real policy rate.

Central bank controls nominal rate only.

deflation and negative output gap feed on each other. Lower output leads to more deflation, more deflation lead to higher real interest rate and lower output.

Long Run

Production function: Y = F (K,N)
assumed to be constant returns to scale, if K and N are doubled, output will also double.

Y/N = F(K/N, 1)         K - capital, N - worker

If only one factor of production is doubled, the output less than double (decreasing returns to scale)

In the interactions between output and capital, there is a steady state, as shown by equation below
  sf(K*/N) = 𝛿(K*/N) or s(Y*/N) = 𝛿(K*/N)
where s: saving , 𝛿: depreciation

It is interpreted as, additional machines invested = machines that are depreciated.

If savings > depreciation, Yt/N will rise as depicted in diagram below.

The golden rule level of consumption is the saving rate that yields the highest level of consumption in steady state.

We have endogenous growth mode, where a steady growth is achieved without technological progress. The model indicates that in the long run, growth rate depends on savings rate and rate of spending on education.

To include technological progress effect on rate of growth, let
 Y = F(K, N, A)  where A: tech, K: capital, N: labor
 Y = F(K, AN)   another form
 Y/AN = F(K/AN, 1)
or Y/AN = f(K/AN)  output per effective worker is a function of capital per effective worker
Let I = S = sY
I/AN = sY/AN

The gA is growth rate of technology progress
The gN is growth rate of workers

The above graph says output per effective worker (Y/AN) and capital per effective worker (K/AN) converge in the long run.

I = 𝛿K + (gA + gN)K  = (𝛿 + gA +gN)K
I/AN = (𝛿 + gA +gN)K/AN

At steady state :
Y/AN, K/AN  the LR growth is 0
Y/N, K/N       the LR growth is gA
Y, K               the LR growth is gA + gN

Expectations

IS relationship
Y = C(Y-T) + I(Y, r+x) + G
A(Y, T, r, x) = C(Y-T) + I(Y, r+x)            A:aggregate private spending
Y = A(Y, T, r, x) + G
adding expected values of future variables
Y = A(Y, T, r, Ye, Te, re) + G

The effects of expansionary monetary policy and expectation

Without expectation, monetary expansion would lead to a fall in policy rate from r to r" and output rises from YA to YB. WIth expectation, IS also shifts right and output rises from YA to Yc.

Budget deficit reduction

SR: A -> B, Y0->Y1  lower spending and lower output
MR: r↓ , Y1 -> Y0  due to lower interest rate, unchanged output, higher saving and higher private investment (Because output unchanged, gov spending is reduced, so private investment is increased.)
LR: I↑(Y, r↓-x)  higher investment leads to higher capital and higher output

Expected policy change will lead to future consumer response, but not present consumer response (false)

Exchange Rate

real FX rate = nominal FX rate (domestic Price / foreign Price)
 ε = EP/P*  ; E - nominal exchange rate, P - domestic price, P* - foreign price
if P < P* , meaning domestic goods is cheaper, therefore  ε < E

In the short run, price level is fixed, a country with fixed exchange rate regime, cannot use monetary policy and cannot adjust exchange rate

In medium run, price can change, so a country can adjust its real exchange rate through price changes, instead of adjusting its nominal exchange rate.

Uncovered Interest rate parity
   (1 + i) = (1 + i*) E / E*

the E* is expected future exchange rate
the i* is foreign interest rate
the i is domestic interest rate

Approximate formula
   i = i* - ( E* - E ) / E

Real effective exchange rate (REER) : weighted average of a country's currency relative to an index or basket of other major currencies, adjusted for inflation

nominal exchange rate and real exchange rate, in SR, tend to move in the same direction, in middle run, will they move in the same direction? (uncertain)

IS-LM analysis

1) Use the IS-LM framework to determine SR equilibrium of tax cut. Explain its effect to Y, r, C, I, S
Ans: The effects of a cut in taxes, the IS curve shift to the right, Y increase, r does not change.
C ↑ as T goes down
S + T = I + G   =>  I = S if T = G

2) Use the IS-LM-PC framework to evaluate a medium run equilibrium of tax cut. Explain its effect to Y, r, C, I, S
Ans: The effects of a cut in taxes, the IS curve shift to the right, Y increase, the change in inflation is positive. so inflation goes up. It is rising inflationary pressure. CB raise its policy rate, r ↑. Y goes back to original output. S is uncertain. I is uncertain. C will increase.

3) If tax cut partly involves reduction in import tariff for energy efficient machines such that many firms are now able to buy these machines and significantly reduce their energy costs. Use IS-LM-PC framework to evaluate the simultaneous effects (medium run equilibrium) of tax cut and reduction in energy cost on Y, r, C, I and S in comparison with 2.
Ans: The tax cut shifts IS curve to the right, economy under inflationary pressure, the reduction in energy price shift PC curve to the right (cost reduced, AS increase, PC shift to the right wrt to output). Change in inflation is zero. Y ↑, r same, C ↑, I and S uncertain. See drawing below.