Macro assignment
artpos
GeneralEquilibriumMacroeconomics_markup.pdf
Macroeconomic Theory
1 Interlude: General Equilibrium Macroeconomics
All of the models that we have used thus far have been partial equilibrium, as we have looked at the
behavior of agents one side of each market in isolation. We now move to a general equilibrium
setting where we study states of the model where each market has reached a `market-clearing'
state where quantity supplied equals quantity demanded.
⇒ Each of our three macro markets � labor market, �nancial market, and goods market � will have an associated Equilibrium Condition that describes a situation where markets clear
such that quantity supplied equals quantity demanded.
1.1 Labor Market Equilibrium
Representative household supplies labor and representative �rm demands labor, the interaction
of which determines market-clearing equilibrium real wage rate wt∗ and labor nt*.
Household Labor Supply Optimality Condition: ∂u/∂lt ∂u/∂ct
= wt (1)
Firm Labor Demand Optimality Condition: ∂f
∂nt = wt (2)
→ Setting Equation (2) and (1) equal, yields the labor market equilibrium condition.
∂u/∂lt ∂u/∂ct
= ∂f
∂nt (3)
Economic Intuition: When the representative household and �rm face the same real wage rate,
quantity of labor supplied and demanded are equal because the labor supply function is strictly
increasing in the wage rate and that the labor demand function is strictly decreasing in the wage
rate, i.e. they intersect at that wage rate. In the labor market equilibrium, households are paid
the marginal product of their labor.
→ Graphically, labor market equilibrium in (nt,wt) space:
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1.2 Financial Market Equilibrium
Representative household supplies savings and representative �rm demands savings for invest-
ment, the interaction of which determines market-clearing equilibrium real interest rate rt+1∗ and supply of savings st* and investment invt∗.
Household Savings Supply Optimality Condition: ∂u/∂ct
β∂u/∂ct+1 = 1 + rt+1 (4)
Firm Capital Demand Optimality Condition: ∂f
∂kt+1 = rt+1 + δ (5)
→ Using Equation (4) and (5) to eliminate the real interest rate, yields the �nancial market equilibrium condition:
∂u/∂ct β∂u/∂ct+1
= 1 + ∂f
∂kt+1 −δ (6)
Economic Intuition: When the representative household and �rm face the same real interest rate,
quantity of savings supplied is equal to the quantity of savings demanded for investment because
the savings supply function is strictly increasing in the real interest rate and that the investment
demand function is strictly decreasing in the real interest rate, i.e. they intersect at that real
interest rate. In the �nancial market equilibrium, households receive an interest rate on their
principal equal to the marginal product of capital less the marginal value of capital depreciated.
→ Graphically, �nancial market equilibrium in ({st, invt},rt+1) space:
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1.3 Goods Market Equilibrium
The interaction aggregate demand and aggregate supply of goods and services, which determines
market-clearing equilibrium goods price Pt∗ and level of total real output qt ∗.
• Aggregate Demand (AD): The desired quantity demanded for all goods and services by households, �rms, and government.
ADt ≡ ct ∗+invt∗ (7)
Note that if there were taxes in the model to fund government spending, gt, we would add gt to
the expression for aggregate demand.
→ How does ADt vary with Pt?
Household Savings Supply Nominal Optimality Condition: ∂u/∂ct
β∂u/∂ct+1 =
Pt Pt+1
(1 + it+1)
→ If Pt ↑, then ct∗ ↓ if substitution e�ect dominates income e�ect
Firm Investment Demand Nominal Optimality Condition: ∂f
∂kt+1 =
Pt Pt+1
(1 + it+1)
→ If Pt ↑, then kt+1∗ and invt∗ ↓
⇒ Because ct∗ and invt∗ decrease as Pt increase, ADt is downward sloping in (qt,Pt) space:
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• Aggregate Supply (AS): The desired quantity supplied by �rms.
ASt ≡ f(kt∗,nt∗) (8)
→ In our current framework, the AS function has no relationship with Pt, and therefore is vertical in (qt,Pt) space. This is called the `classical' AS function.
→ Setting ADt = ASt as in Equations (7) and (8) yields the goods market equilibrium condition:
f(kt∗,nt∗) = ct ∗+invt ∗ (9)
⇒ When the goods market equilibrium condition (9), the labor market equilibrium condition (3), and the �nancial market equilibrium condition (6) simultaneously hold, the model is said
to be in general equilibrium.
→ Along with budget constraints of all economic agents, these conditions completely char- acterize the general equilibrium solution.
→ In other frameworks the AS function is completely horizontal (Keynesian AS) or upwards sloping (New-Keynesian AS) over Pt. The slope of the AS curve in these models depends on
how quickly Pt adjusts relative to other prices (wt and rt+1). If Pt does not change at our model
frequency (referred to as `sticky prices') then the AS function is completely horizontal. The
implication of the classical AS function is therefore that Pt adjusts instantaneously in response
to changes to costs of production.
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