A Real Intertemporal Model with Investment
Yu-Chi Chu
National Taiwan University
May 6, 2023
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Environment
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Model Environment
1. Demographics:
• The economy lasts for two periods.
• There is a representative consumer, a representative firm, and
a government.
2. Preferences: U (C, C 0 , l, l0 )
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3. Technology: this is a production economy. The representative
firm uses inputs K and N to produce Y .
• The current aggregate supply of goods is Y = zF (K, N ) and
the future supply of goods is Y 0 = z 0 F (K 0 , N 0 ).
• Assume
2
∂ F
∂K∂N
∂F
∂K
> 0,
∂F
∂N
> 0,
∂2F
∂K∂K
< 0,
∂2F
∂N ∂N
< 0, and
> 0.
4. Markets: goods markets, factor markets, and government
bond markets.
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The Household’s Problem
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Consumer Optimization
The consumer
• takes w, w0 are r as given. The lump-sum taxes T and T 0 are
also taken as given; The dividends incomes π and π 0 are also
taken as given.
• has h units of time in each period and divide this time
between work and leisure in each period;
• chooses current consumptionC, future consumption C 0 ,
current leisure l, future leisure l0 , and saving S p to make
himself as well off as possible, given his budget constraints in
the two periods.
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Consumer Optimization
• The budget constraints in period 1 and in period 2 are:
C + S p = w(h − l) + π − T,
(1)
C 0 = w0 (h − l0 ) + π 0 − T 0 + (1 + r)S p .
(2)
• π: dividends received from the representative firm.
• S p : private saving, can be either positive or negative.
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Consumer Optimization
• Substitute for S p in equation (1) using equation (2) to obtain
the lifetime budget constraint:
C+
C0
w0 (h − l0 ) + π 0 − T 0
= w(h − l) + π − T +
. (3)
1+r
1+r
• Different from the two-period model in Chapter 9, the
consumer now can choose l and l0 to change his life-time
wealth.
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Consumer Optimization
• The maximization problem is
max u(C, C 0 , l, l0 )
C,C 0 ,l,l0
subject to the lifetime budget constraint (3).
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Consumer Optimization
• Work-leisure decision:
MRSl,C
= w;
(4)
MRSl0 ,C 0
= w0 ;
(5)
MRSC,C 0 = 1 + r.
(6)
• Consumption-saving decision:
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Consumer Optimization
• An example:
U (C, C 0 , l, l0 ) = log(C) + α log(l) +
1
1+ρ
log(C 0 ) + α log(l0 )
Find a set of equations that characterize the solution for
0
(C, C 0 , l, l0 , N s , N s ).
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Consumer Optimization
Q How does the consumer substitute current leisure for future
leisure?
MRSl,l0 =
(1 + r)w
.
w0
(7)
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Current Labor Supply
Three factors affect the current labor supply.
• Current real wage:
• Substitution effect: w ↑ → l is more expensive relative to C
→ l ↓→ N s ↑.
• Income effect: Given the same labor supply, w ↑→ wage
income ↑. Because l is normal good, an increase in the lifetime
wealth implies that l ↑→ N s ↓.
• Here, we assume that SE>IE, so labor supply increases when
the current wage increases.
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Current Labor Supply
Three factors affect the current labor supply.
• Real interest rate:
• Substitution effect: r ↑ → from (7), l is more expensive
relative to l0 → l ↓ → N s ↑. (This is called the intertemporal
substitution of leisure)
• Income effect:
• saver: r ↑ has positive effects on the lifetime wealth, so l ↑
• borrower: r ↑ implies that the household has less lifetime
wealth, so l ↓.
• If we assume that SE>IE, then labor supply increases when the
real interest rate increases.
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Current Labor Supply
• Lifetime wealth:
• As leisure is assumed to be a normal good, when there is an
increase in life-time wealth, there is an increase in current
leisure (income effect) and thus, a decrease in current labor
supply.
• An change in lifetime wealth could be caused by a change in
the present value of taxes.
• Draw labor supply curve: Figure 11.1, 11.2 and 11.3.
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Current Demand for Consumption Goods
• In Chapter 9, we demonstrated the impact of changes in
lifetime wealth and interest rates on current consumption.
• We continues to assume SE > IE, so it follows that an
increase in r leads to an increase in saving and a decrease in
current consumption.
• Draw consumption demand function: Figure 11.4-11.6
• The slope of the curve C d (r) is the MPC, which is the amount
by which current consumption increases when there is a unit
increase in aggregate income.
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The Representative Firm
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The Representative Firm
• Firm produces output in both the current and future periods,
and the firm can invest in the current period by accumulating
capital so as to expand the capacity to produce future output.
The essence of the firm’s investment decision is how much
current profits are foregone in exchange for more future
profits.
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• Assume that it requires one unit of consumption goods in the
current period to produce one unit of capital. Thus, the
future capital stock is given by:
K 0 = (1 − d)K + I,
(8)
where I is investment in the current period.
• At the end of the future period, the amount of capital left is
(1 − d)K 0 , and the firm liquidates it.
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The Firm’s Profits
• The current profits in units of the current consumption goods
are:
π = Y − wN − I.
(9)
• The future profits for the firm are:
π 0 = Y 0 − w0 N 0 + (1 − d)K 0 .
(10)
• The firm takes all prices w, w0 , r as given to chooses N , N 0 ,
I to maximize the present value of total profits, π +
π0
1+r ,
subject to (8).
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The Firm’s Optimization
max
zF (K, N ) − wN − I +
0
N,N ,I
z 0 F (K 0 , N 0 ) − w0 N 0 + (1 − d)K 0
1+r
subject to
K 0 = (1 − d)K + I
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Current Labor Demand
• Optimal decision on current labor demand:
z
|
∂F (K, N )
= w.
∂N
{z
}
(11)
M PN
• Three factors that affect the demand curve for current labor:
N d (|{z}
w , |{z}
K , |{z}
z ).
−
+
+
• Both K and z positively affect the marginal productivity of
labor. At the same wage, the firm wants to hire more labors.
• See Figure 11.7, 11.8
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Future Labor Demand
• Optimal decision on future labor demand:
z0
|
∂F (K 0 , N 0 )
= w0 .
0
∂N
{z
}
(12)
0
M PN
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Investment
• Optimal decision on investment:
1=
where M P 0 K =
z 0 ∂F (K 0 ,N 0 )
∂K 0
+1−d
,
1+r
(13)
z 0 ∂F (K 0 ,N 0 )
.
∂K 0
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Two ways to understand investment decision
Marginal Costs equals Marginal Benefits
• Cost: π ↓
• I ↑ 1, π ↓ 1.
• MC of I is one.
• Benefit:
π0
1+r
↑
• I ↑ 1, K 0 ↑ 1, so π 0 ↑ M Pk0 + (1 − d)
• MB of I is
M Pk0 +1−d
1+r
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Two ways to understand investment decision
No-arbitrage is allowed
• We can rewrite Equation (13) to
M P 0 K + 1 − d = 1 + r.
(14)
• Investment one dollar yields two returns:
• Buy government bonds: 1 + r.
• Buy future capital: M P 0 K + 1 − d.
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Optimal Investment Schedule
• I = I( r , z 0 , K )
|{z} |{z} |{z}
−
+
−
• If r increases, MB decreases, so I decreases.
• z 0 ↑→ M P 0K ↑, so MB increases, I increases.
• If K increases, M P 0K ↓, so MB decreases, I decreases.
• See Figure 11.9 and 11.10
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Investment: including capital price
Q. Assuming q units of current consumption goods can purchase
one unit of capital and q’ units of future consumption goods
can purchase one unit of future capital. (q units C in
exchange for one unit of K, so q is the capital price)
π = Y − wN − Iq
π 0 = Y 0 − w0 N 0 + (1 − d)K 0 q 0
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Investment: including capital price
1. The firm’s problem is
max
zF (K, N )−wN −qI+
0
N,N ,I
z 0 F (K 0 , N 0 ) − w0 N 0 + (1 − d)K 0 q 0
1+r
subject to
K 0 = (1 − d)K + I
2. The optimal decision on investment changes to
q=
0 + (1 − d)q 0
M PK
.
1+r
• MC is q
• MB is
0
M PK
+(1−d)q 0
.
1+r
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Investment and the interest rate spread
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Investment and the interest rate spread
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Why investment falls in the financial crisis
Q Suppose that the firm has to borrow to invest, and the
borrowing rate is
rl = r + x,
where x is the default premium (or the interest rate spread)
Write down the first-order condition to determine I.
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Why investment falls in the financial crisis
The optimal investment schedule for the firm is
0
1 + r + x = 1 + M PK
− d.
• The LHS is the cost of funds.
• The RHS is the return of capital.
It is clear that the model predicts a negative relationship between
x and the investment.
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Competitive Equilibrium
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Competitive Equilibrium
Definition:
1. Given prices, household maximizes lifetime utility
2. Given prices, firm maximizes present value of profits
3. Government’s budgets constraints are satisfied.
4. For all markets, demand equals supply at the equilibrium
prices.
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Competitive Equilibrium
• Exogenous variables: K, z, z 0 , G, G0
(note: an exogenous variable is determined outside of the
economic model being analyzed)
0
• Endogenous variables: C, C 0 , l, l0 , N s , N s , S P , I, K 0 , π, π 0 ,
0
0
0
N d , N d , Y d , Y d , Y s , Y s , T , T 0 , B, w, w0 , r
(note: endogenous variables are those that are determined
within the economic model being analyzed.)
• There are 23 endogenous variables
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Competitive Equilibrium
1. The household optimizes:
1.1 (C, C 0 , l, l0 ) are determined by
• The first-order conditions: Equation (4), (5), (6)
• The lifetime budget constraint: Equation (3)
1.2 saving S p is solved from the budget constraint in the current
period: Equation (1)
0
1.3 (N s , N s ) are solved from the two time constraints:
0
h − l = N s and h − l0 = N s .
We have 7 equations
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Competitive Equilibrium
2. The firm maximizes total profits:
0
1. (N d , N d , I) are solved from first-order conditions: (11), (12),
and (13).
2. K 0 is determined from (8)
0
3. Solve Y s , Y s , π, and π 0 using Y s = zF (K, N d ),
0
0
Y s = z 0 F (K 0 , N d ), Equation (9), and Equation (10).
Now we have 15 equations
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Competitive Equilibrium
3. Government budget constraint is satisfied:
G=T +B
G0 = T 0 − (1 + r)B
G+
T0
G0
=T +
.
1+r
1+r
• Now we have 18 equations.
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Competitive Equilibrium
4. All markets clear
1. Current labor market: N s = N d
0
2. Future Labor market: N s = N d
0
3. Current Goods market: Y s = C + I + G.
0
4. Future Goods market: Y s = C 0 + G0 .
5. Current Government bond market: S p = B.
• Finally, we have 23 equations.
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Expressing Competitive Equilibrium Using Diagrams
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The Output Supply Curve (Y s ) in the Current Period
1. Definition.— the output supply curve consists of all
combinations of current output and real interest rates, (Y, r),
for which the current labor market is in in equilibrium.
• See Figure 11.15
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The Output Supply Curve (Y s ) in the Current Period
2. Why the output supply curve is upward-sloping?
• r ↑ ⇒ the labor supply curve shifts right ⇒ there is excess
supply of labor ⇒ w∗ ↓ and N ∗ ↑ ⇒ Y ∗ ↑. (Figure 11.15)
3. What shifts the output supply curve?
• Anything that changes the production function, eg. z and K.
• Anything that changes N ∗
• Factors that affect N d , e.g. z and K
• Factors that affect N s , eg. T +
T0
1+r
• See Figure 11.16, 11.17
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The Output Supply Curve (Y s ) in the Current Period
4. The slope of the output supply curve is determined by the
magnitude of the intertemporal substitution effect of leisure.
• Left for HW.
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The Output Demand Curve (Y d ) in the Current Period
1. Definition. — the output demand curve tells us the
equilibrium demand for current goods given the real interest
rate r (also known as the IS curve).
• The equilibrium here is that total expenditure equals
income.
• This is because every dollar spent on goods and services by
one person becomes the income of someone else in the
economy.
• What is expenditure? C + I + G.
• What is income? Y .
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The Output Demand Curve (Y d ) in the Current Period
•
Expenditure = C d (r, Y ) + I d (r) + G.
(15)
• Figure 11.18: the point at which the curve (15) intersects the
45 degree line is
Y ∗ = C d (r, Y ∗ ) + I d (r) + G.
• Y ∗ is the equilibrium demand for goods when income equals
expenditure.
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The Output Demand Curve (Y d ) in the Current Period
Example: suppose that the consumption function is given by
C d = c1 (Y − G) + c2 (Y 0 − G0 ) − c3 r.
Suppose that the investment demand curve is given by
I = −b1 r + bZ 0 − b3 K.
The government spending is G. Algebraically derive an expression
for the Y d curve.
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The Output Demand Curve (Y d ) in the Current Period
Y = c1 (Y − G) + c2 (Y 0 − G0 ) − c3 r − b1 r + bZ 0 − b3 K + G
(1 − c1 )Y = (1 − c1 )G + c2 (Y 0 − G0 ) − (c3 + b1 )r + bZ 0 − b3 K
!
1
0
0
0
r=
(1 − c1 )G + c2 (Y − G ) + bZ − b3 K − (1 − c1 )Y
c3 + b1
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The Output Demand Curve (Y d ) in the Current Period
2. Why the output demand curve is downward sloping?
3. What shifts the output demand curve?
• Government spending
• Anything that affects the current consumption C
• A change in the present value of taxes
• An increase in future income Y 0
• Anything that affects the investment decision I,
• eg. z 0 , K
• See Figure 11.19 and left for HW.
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The Output Demand Curve (Y d )
4. The slope of the output demand curve is determined by the
magnitude of the MPC.
• Left for HW.
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The Complete Real Intertemporal Model
Step 1: Equilibrium (Y ∗ , r∗ ), is determined by the intersection of Y d
and Y s .
Step 2: from Y ∗ = zF (K, N ), we can figure out the equilibrium
employment N ∗ .
Step 3. From the labor supply curve, N s (r∗ ), we can figure out the
equilibrium wage, w∗ .
Step 4. Once we know r∗ and w∗ , we can analyze C and I.
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Q How about the labor market?
• The current labor market also clears. This is because on the
Y s curve, the current labor market is in equilibrium.
Q How about the financial market?
• By the Walrus law, when the goods market and the labor
market clear, the credit market clears. Thus, when Y d and Y s
intersect, all three markets clear.
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Equilibrium Analysis
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Drawbacks of Using Partial Equilibrium Analysis
• General equilibrium: All markets in the economy reach
equilibrium, with prices determined by market-clearing
conditions and government budget constraints satisfied.
• Partial equilibrium: Not all markets reach equilibrium
(ignoring price change effects), and government budget
constraints may not be satisfied (ignoring tax change effects).
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Basic Guidelines
• Initially, keep prices constant and consider the impact of an
external variable on both Y d and Y s .
• If Y d and Y s are not equal, prices must change to balance the
market. Then, examine the effect of price on the variables.
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Temporary Increase in G
• Three factors contribute to the effect of a G increase on Y d :
• G↑ → C +I +G↑
• G↑→T +
T0
1+r
↑ → C ↓ → C +I +G↓
• Multiplier effect
• By income-expenditure identity, C ↑→ Y ↑→↑ C →↑ Y....
• These three factors determine the demand multiplier:
md =
∆Y d
.
∆G
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Temporary Increase in G
Rounds
∆Y
∆C
∆I
∆G
1
1-MPC
-MPC
0
1
2
MPC(1-MPC)
MPC(1-MPC)
0
0
3
MPC2 (1 − MPC)
MPC2 (1 − MPC)
0
0
The total change in the demand for current goods is
∆Y = 1 − MPC + MPC(1-MPC) + MPC2 (1 − MPC) + .... = 1
The demand multiplier is one.
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Temporary Increase in G
• T+
T0
1+r
↑ → Ns ↑
• Y s shifts to the right.
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Temporary Increase in G
Impact on the goods market:
• Let’s assume that the shift in Y s is smaller than the shift in
Y d.
• As a result, there will be an excess demand at the original
prices, leading to an increase in r.
See Figure 11.22.
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Temporary Increase in G
Impact on the labor market:
• The supply of labor N s will shift to the right for two reasons:
• the income effect from higher taxes,
• and an increase in r.
• As a result, there will be an excess supply at the original
prices, leading to a decrease in w.
see Figure 11.22.
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What happens to C
C d = C d (r, Y, Y 0 , T +
T0
)
1+r
• Holding r fixed, the net change in lifetime wealth is 0
• the present value of taxes increase by ∆G → the lifetime
wealth decreases by ∆G.
• Y d shifts right by ∆G → the current aggregate income Y
increases by ∆G → the lifetime wealth increases by ∆G.
• C decreases because r rises in equilibrium.
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What happens to I and K 0 ?
• I decreases because r, increases.
• K 0 ↓: an increase in current G negatively affects the
economy’s future productive capacity.
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Takeaway 1
• Increased temporary government spending, although it leads
to higher aggregate output, comes at a cost:
• the household consumes less and takes less leisure.
• the investment is crowed out due to a higher interest rate.
• the capital stock will be lower in the future, and the future
capacity of the economy for producing goods will be lower.
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Total Government Expenditure Multiplier
Total Government Expenditure Multiplier =
Y2∗ − Y1∗
∆G
Q. Why do we care about the total government expenditure
multiplier?
• Its magnitude tells us how effective fiscal policy is in
stimulating aggregate demand.
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Total Government Expenditure Multiplier
• The total government expenditure multiplier is less than one.
(G increases by ∆G, GDP increases by less than ∆G. )
• If r were not changed, then GDP increases by ∆G.
• Since r increases, it leads to a decrease in C and I, resulting in
a smaller change in demand, so GDP increases by less than
∆G.
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Total Government Expenditure Multiplier
• A smaller multiplier indicates that fiscal policy is a less
effective way to stimulate aggregate demand.
• Q. What leads to a smaller multiplier ?
• When r increases more.
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Total Government Expenditure Multiplier
• The multiplier is smaller if
• the income effect on labor supply is smaller (this makes the
rightward shift in Y s smaller)
• the intertemporal substitution effect of the real interest rate on
labor supply is smaller (this makes the output supply curve
steeper).
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Total Government Expenditure Multiplier
• In both scenarios, the equilibrium r increases more when G
increases.
• C and I are crowded out more by G.
• If the income effect on labor supply is smaller, the excess
demand in the goods market is larger, so r must increase a lot
to balance the market.
• If the intertemporal elasticity of substitution (IES) is smaller,
this means that r must increase a lot for households to work
more.
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A Decrease in the Current Capital Stock K
• Sometimes major reductions in the aggregate capital stock
occur over a short period of time, e.g. a war, a natural
disaster....
• When K ↓
• M Pn ↓, so labor demand decreases.
• M Pk0 ↑, so investment increases.
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A Decrease in the Current Capital Stock K
Impact on the goods market:
• Y d shifts to the right because of more investment.
• Y s shifts to the left.
• Excess demand in the goods market, so r must increase.
• Y ∗ is ambiguous.
See Figure 11.24.
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A Decrease in the Current Capital Stock K
Impact on the labor market:
• N d shifts to the left because the marginal product of labor
decreases.
• N s shifts to the right because r increases.
• Excess supply in the labor market, so w must decrease.
• N ∗ is ambiguous.
See Figure 11.24.
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A Decrease in the Current Capital Stock K
What happens to C?
• C?
• r increases, so C decreases.
• Y?
What happens to I?
• I is ambiguous.
• Current K decreases, so I goes up
• The increase in r leads to a lower I
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An Increase in Current Total Factor Productivity z
• An increase in z reflects a situation in which more aggregate
output is produced while the same factor inputs are used.
• Eg. weather, government regulations, a new invention, a
decrease in the relative price of energy, a more efficient
allocation of factors of production across firms, or education (if
not including human capital).
• z ↑:M Pn ↑
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An Increase in Current Total Factor Productivity z
Given the original equilibrium prices, w1∗ and r1∗ , an increase in
current total factor productivity z shifts
• the labor demand curve to the right (because of a higher
MPN )
• the output supply curve to the right (because z ↑ and N ↑)
See Figure 11.25
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• Goods market:
• r2∗ < r1∗ and Y2∗ > Y1∗ .
• From Y = zF (K, N ), both z and Y increase, so the
equilibrium N2∗ is ambiguous.
• Labor market:
∗ ∗
• The labor supply curve shifts left because r<
r1 .
• The labor demand curve shifts right because z ↑.
• There is excess demand in the current labor market, w2∗ > w1∗ .
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• C↑
• r decreases, so C increases.
• Y increases, so C increases.
• I↑
• r decreases, so I increases.
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An Increase in Future Total Factor Productivity z 0
• The anticipation of future events can have important
macroeconomic consequences in the present.
• Positive news about future productivity, such as rising stock
prices.
• z 0 ↑, M Pk0 ↑.
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Given the original equilibrium prices, w1∗ and r1∗ , an increase in z 0
shifts
• the output demand curve to the right because the firm
0 and wishes to invest more.
expects a higher M PK
See Figure 11.27
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• Goods market:
• r2∗ > r1∗ and Y2∗ > Y1∗ .
• From Y = zF (K, N ), N2∗ > N1∗ .
• Labor market:
• The labor supply curve shifts right because r2∗ > r1∗ .
• There is excess supply in the current labor market, w2∗ < w1∗ .
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• C is ambiguous
• the increase in r reduces C.
• the increase in Y increases in C.
• the increase in Y 0 increases C.
• I is ambiguous:
• the increase in r reduces I.
• the increase in z 0 increases I.
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