max U( C, h-Ns )
C, Ns
subject to and
C = wNs + - T C 0 , Ns 0.
In other words, C and Ns must satisfy the FOC,
U2( C, h-Ns ) - wU1( C, h-Ns ) = 0.
2) Nd solves the firm’s problem given z, K and w. Thus the FOC
• Hence, in equilibrium MRT l,C = MRS l,C = w.
Optimality
• Questions: Is the competitive equilibrium efficient ? Are there any other ways to obtain a better outcome ?
– Under certain conditions, a competitive equilibrium is Pareto optimal.
MRT l,C = MPN = - ( Slope of PPF )
• In equilibrium, MPN = w. Thus, we must have
MRT l,C = w.
• This helps us determine the equilibrium values C* and l*.
• Note that we have also derived the same condition for a competitive equilibrium.
Two Welfare Theorems
• The first fundamental theorem of wuilibrium
• In equilibrium, C* and l* are chosen by the representative consumer.
• Note that ADB is the budget constraint. By the FOC of the consumer’s problem MRS l,C = w.
slope dY dY
dl
dN
MPN
Production Possibilities Frontier
• In equilibrium, C = Y - G = zF( K, h - l ) - G a relationship between C and l, given the exogenous variables z, K and G.
C = zF(K, h - l ) - G
• Lagrangian L = U(C, l) + [ zF(K, h - l ) - G - C ].
• First-order ( Necessary ) conditions:
U1C,l 0, U2 C,l zF2 K, h l 0,
• Since there is only one consumer, we can ignore how consumption goods are allocated among consumers.
• Rather, we focus on how production is arranged.
• In equilibrium, we have Nd = Ns = N = h - l.
• Output is given by Y = zF( K, h - l ).
which is a relationship between output and leisure. • Y* is the level when l = 0.
• Besides representative consumer and representative firm, there is an additional agent: the government.
• Government
• The only action of the government is to implement fiscal policy. Fiscal policy refers to the government’s choices over its expenditures, taxes, transfers and borrowing.
Production Possibilities Frontier
• Negative of the slope of the PPF is called the marginal rate of transformation, MRT l,C which is the rate at which one good can be converted into another.
• Graphically, the social planner chooses a consumption bundle that is on the PPF and is on the highest possible indifference curve for the consumer.
Social Planner’s Problem
• A competitive equilibrium is Pareto optimal if there is no way to rearrange production or to reallocate goods so that someone is made better off and no one is made worse off.
• Equilibrium refers to the state when the actions of all consumers and firms are consistent.
Formal Definition: • A competitive equilibrium is a set of endogenous quantities C,
is satisfied.
zF2 K, N d w
3) Market-clearing conditions:
– Labor market clearing Ns = Nd = N
4) Balanced Budget constraint: G = T.
We have: Goods market clearing Y = C + G
zF K, h l G C 0.
• Combining the first two conditions,
MRS l ,C
U2 U1

zF2
MRTl,C
which states that the Pareto optimum is the point where the indifference curve is tangent to the PPF.
• From these, we can solve for the equilibrium C and l.
Graphical Illustrations
Step 1: Derive the production possibilities frontier ( PPF ), which describes the technological possibilities for the entire economy, in terms of the production of C and l.
Social Planner’s Problem
• Consider a social planner who runs the representative firm and chooses the quantities C and l so as to maximize consumer’s utility.
Chapter 5 A Closed-Economy One-Period Macro Model(中级
宏观经济学,香港中文大学)
Chapter 5
A Closed-Economy One-Period Macro Model
The Model
• The economy that we consider is a closed economy, i.e. one that does not trade with other economies.
• This is the PPF which captures the trade-off between leisure and consumption given the production technology.
• Only the points on DB are available ( where C 0 ).
Step 2: Put the PPF together with the consumer’s indifference curves, so that we can analyze a competitive equilibrium in a single diagram.
Production Possibilities Frontier
• Comparison:
– Representative consumer faces a linear or kinked budget constraint.
– Social planner faces a concave PPF.
• The Pareto optimum is at B where the equality holds
• The government must observe the balanced budget constraint, G = T.