ECON 4020 Test 1

Log-preferences

lnc + θlnℓ → max c,ℓ
s.t. c + Bℓ = F

c = 1/(1+θ) * F
ℓ = θ/(1+θ) * F/B

Perfect Complements

min{c, θℓ} → max c,ℓ
s.t. c + Bℓ = F

Set c = θℓ
Plug into BC: θℓ + Bℓ = F
ℓ = F/(θ+B)
c = θℓ = θF/(θ+B)

Perfect Substitutes

c + θℓ → max c,ℓ
s.t. c + Bℓ = F

Substitute constraint out: c = F - Bℓ
F - Bℓ + θℓ → max ℓ
F + (θ-B)ℓ → max ℓ
If θ>B, set ℓ as LARGE as possible.
If θ<B, set ℓ as _^small as possible.

Rigid Preferences

ⓒ is exogenously fixed constant
c + Bℓ = F

ⓒ + Bℓ = F
ℓ = (F - ⓒ) / B
c = ⓒ

One-sided preferences

lnc → max c,ℓ
c + Bℓ = F

c = F
ℓ = 0

Product / Value Added Approach

GDP = Sum of value-added goods and services across all production units in an economy

Value Added

FinalValue - IntermediateValue

Expenditure Approach

GDP = C + I + G + NX
GDP = Consumption + Investment + Government + Net Exports

Income Approach

GDP = Sum of all income received by economic agents contributing to production

Nominal GDP Formula

GDP = P1*Q1

Real GDP Formula

Base Year 1:
RGDP¹₁ = P1*Q1
RGDP¹₂ = P1*Q2

Base Year 2:
RGDP²₁ = P2*Q1
RGDP²₂ = P2*Q2

Percent Increase

(New - Old) / Old * 100%

Growth Rate

Year 1: g₁ = RGDP¹₂/RGDP¹₁
Year 2: g₂ = RGDP²₂/RGDP²₁

Average Growth Rate

g = sqrt(g₁g₂)

Implicit GDP Price Deflator: ID

ID = NominalGDP / rGDP * 100%

Consumer Price Index (CPI)

Q_baseP_Current / Q_baseP_base * 100%

CPI₁ = 100%
CPI₂ = Q1P2 / Q1P1

Basic Consumer Budget Constraint

C = w(h-l) + π

Production Function

y = zF(K^d,N^d)

Profit Function

π = y - wN^d - rK^d

Y = zN
Prove z=w

Y = zN
π = zN - wN
π = (z-w)N

If z>w, you would want infinite N.
If z<w, you would want 0 N.

In equilibrium, z=w

Market Clearing Conditions

Labor Market: N^s = N^d, h-l = N^s
Consumption Goods Market: C = Y or C + G = Y
Capital Market: K^s = K^d

Endogenous Variables

Variables which change:
c, l, N^s, N^d, π, τ, w

Exogenous Variables

Variables which don’t change
z, h, θ, G

How to solve for Competitive Equilibrium

1) Solve for consumption (c*), leisure (l*) and labor supply (N^s)
2) Maximize the firms profit (expect w=z, π=0)
3) Solve market clearing conditions (N^s = N^d, C+G=Y, G = gov’t earnings)
4) Solve for CE Consumption (c^ce), CE leisure (l^ce), and CE Labor Supply (N^ce).

Middle of the night equation(s)

N^s = h - l
h = N^s + l
l = h - N^s

How to construct PPF

1) Write down either Y = C or Y = C + G
2) Substitute Y with the production function
3) In the production function, substitute N with (h-l) and K with K₀

CE Is Pareto Optimal When:

There are no distortionary taxes (only lump-sum taxes and no government are pareto optimal)

Inflation

(CPI₂ - CPI₁) / CPI₁ × 100%
(ID₂ - ID₁) / ID₁ × 100%