Download Discussion Section Notes Ricardo

Weeks 1 and 2: Outline
• 3 conditions for General Equilibrium
– Producer Optimization (Px/Py = MRT)
– Consumer Optimization (Px/Py = MRS)
– Market Clearing
• Goods Markets: Xc = Xp, Yc = Yp
• Factor Markets: Lx + Ly = L, Kx + Ky = K
• Get the production point, consumption
point, input allocations and prices.
• Need to determine factor rewards.
– Competitive: Factors paid their VMP.
– Know Prices, MPP. Wage = VMPL = P(MPL)
Important Relationships in GE
•
•
•
•
•
Px/Py= MRT = MPLy/MPLx = MPKy/MPKx
Px/Py= MRS
Wage = VMPL = Px MPLx = Py MPLy
Rent = VMPK = Px MPKx = Py MPKy
Real Wage
– In terms of X: Wage/Px= MPLx = (Py/Px)
MPLy
– In terms of Y: Wage/Py= MPLy = (Px/Py)
MPLx
Producer Optimization
•
STEP 1: Draw a PPF in XY space
– Maximum Y that can be produced given X
•
What does the PPF look like?
(a) Downward Sloping if MPL and MPK>0.
Will be satisfied for all production functions
we use. Implies if increase good X then
need more inputs for X. So must take some
inputs from Y. Implies must reduce Y.
Producer Optimization
(b) Concave if production functions satisfy
Diminishing Marginal Returns. Implies as we
keep putting more inputs into X production,
output of X increases more and more slowly.
dMPL/dL < 0 and dMPK/dK < 0
(c) Linear if production functions satisfy Constant
Marginal Returns. Implies as we keep putting
more inputs into X production, output of X
continues to increase at the same rate.
dMPL/dL = 0 and dMPK/dK = 0
Producer Optimization
(d) How to draw the PPF?
– Give all inputs to X and none to Y. Plot this
point.
– Give all inputs to Y and none to X. Plot this
point.
– Now join the 2 points: As a concave curve if
the PPF is concave. Or, as a straight line if the
PPF is linear.
(e) What is the slope of the PPF? MRT
Consumer Optimization
• STEP 2: Draw community Indifference
Curves in XY space.
– Think of a nice Cobb-Douglas utility function
and draw this map.
• What is the slope of the ICs? MRS
Y
Slope of IC at
point A or the
MRS at point A
A
Indifference Curve
X
STEP 3: Prices
• From the 3 conditions for a GE
MRT = Px/Py = MRS
• Implies the equilibrium is at the tangency
point of the PPF and the IC
• The tangent is the price line.
STEP 4: Market Clearing
• Working with PPFs.
– On the PPF. Implies all inputs used. So get
factor market clearing.
– IC tangent to PPF (On the PPF). Implies
consumed goods equal produced goods. So
get goods market clearing.
Example: FT 2 (Section 2)
• 2 Goods: Cloth and Wheat
• Only 1 input L
• Home Country
– 1 worker produces 4 bushels of wheat or 2 yards of
cloth.
– Has total 25 workers.
• Foreign Country
– 1 worker produces 1 bushel of wheat or 1 yard of
cloth.
– Has total 100 workers.
• How will we get the GE for each economy?
Home Country: MPL
• 1L produces 4W or 2C: w = 4L, c = 2L
– MPLW = 4, MPLC = 2.
– Both MPLW and MPLC positive. So
downward sloping PPF.
– MPLW and MPLC same for all workers
(given).
dMPLW/dL = 0, dMPLC/dL = 0
So CONSTANT Marginal Returns. So get a
Linear PPF.
Home Country: PPF points
• Give all L to W. Implies w = 4L = 4(25) =
100.
No inputs left for C. Implies c = 2L = 2(0) =
0.
• Give all L to C. Implies c = 2L = 2(25) = 50.
No inputs left for W. Implies w = 4L = 4(0)
= 0.
• Plot (w1,c1)=(100,0) and (w2,c2)=(0,50).
Home Country: Drawing the PPF
• Now plot the two PPF points. Put W on the xaxis and C on the y-axis. (Just to match FT, no
other reason – you can reverse if you want).
• Remember the PPF is downward sloping. Your
plotted points should also tells you this.
• Now use the earlier derived fact that the PPF is
linear. Draw a straight line to join the 2 points.
• The PPF!
Home Country: Slope of the PPF
• Slope of the PPF = MRT = MPLc/MPLw
• With linear PPFs, the slope is constant. So the
MRT is constant.
• What is the slope? Remember it is a straight
line.
– Slope = (w2-w1)/(c2-c1) = (50-0)/(0-100) = - ½
– MRT = ½
– Also note that the Slope = (w2-w1)/(c2-c1) =
2(25)/4(25) = MPLc*L/MPLw*L = MPLc/MPLw.
• ½ yard of Cloth is the OPPORTUNITY COST of
Wheat. Increase W by 1. Need ¼ L for 1W. So
reduce ¼ L in C. ¼ L can produce ½ C. Reduces
C by ½ yard.
Home Country: GE
• Draw the ICs.
• Draw the tangency point.
• You will note that since the PPF is a straight line,
the PPF is also the tangent and hence the price
line.
• We already know the slope of the PPF so we
know the prices too. The price ratio is Pw/Pc =
MRT = ½ = MRS.
• Tangency point is the production and the
consumption point. It is on the PPF so get both
labor and goods market clearing.
Home Country: Wages
• We have the Prices.
• We have the MPLs.
• So what are the Wages? Remember there
is perfect competition.
• Wage = VMPL = Pw MPLw = Pc MPLc
• Real Wage in terms of Wheat = Wage/Pw
• Real Wage in terms of Cloth = Wage/Pc
Home Country: Real Wages
• Wage = VMPL = Pw MPLw = Pc MPLc
• Wheat: Wage/Pw = MPLw = (Pc/Pw)MPLc
• Cloth: Wage/Pc = MPLc = (Pw/Pc)MPLw
• Wheat: Wage/Pw = 4 = (2)2
• Cloth: Wage/Pc = 2 = (1/2)4
Foreign Country: PPF
• Repeat the previous steps. Use the same
axes. Now MPLw = 1, MPLc = 1.
• You will find that MRS = Pw/Pc = 1 = MRT
= MPLc/MPLw = Opportunity Cost of
Wheat.
• Real Wages
– Wheat: Wage/Pw = 1 = 1(1)
– Cloth: Wage/Pc = 1 = 1(1)
Comparing Home and Foreign
• Opportunity Cost of Wheat: lower at home
– Home: ½ < 1: Foreign
• Prices: Wheat cheaper at home
– Pw/Pc Home = ½ < 1 = Pw/Pc Foreign
• Real Wages: Both higher at home
– Wage/Pw Home = 4 > 1 = Wage/Pw Foreign
– Wage/Pc Home = 2 > 1 = Wage/Pc Foreign
– Reflects that Labor is more SCARCE at home
Comparative Advantage
• OCW: lower at home
– Home OCW = ½ < 1 = Foreign OCW
• Home has a lower OCW. Home has a CA
in Wheat.
• OCC = 1/OCW: lower in foreign
– Home OCC = 2 > 1 = Foreign OCC
• Foreign country has a CA in Cloth.
Price Changes and the PPF
• Suppose world prices are given to you.
• PPF procedure is the same. Technical.
• IC procedure is the same. Preferences.
• What about the tangency?
• If cannot trade then prices determined at
home. Tangency procedure is the same.
World prices do not affect you since the
home country is a closed economy.
Price Changes and the PPF
• What about the tangency if the economy opens
up to trade?
• Tangency procedure is not the same.
– Draw the post-trade world price line (assume this is
given to you for now).
– Remember the PPF and ICs are the same.
– Find the tangency between the New Price Line and
the original PPF. Gives the new PRODUCTION point.
– Find the tangency between the new price line and the
highest IC of your original IC MAP. Gives the new
CONSUMPTION point.
International Trade
• New Production Point = (Xp,Yp)
• New Consumption Point = (Xc,Yc)
• Look at the x-axis.
– If Xp > Xc, then you are exporting good X.
– If Xp < Xc, then you are importing good X.
– If Xp = Xc, then you are not trading good X.
• Look at the y-axis. Repeat.
International Trade: FT Ex.
• Suppose World Price after Trade = Pw/Pc = 2/3.
• Recall Home Pw/Pc = ½ < Trade Pw/Pc = 2/3 < Foreign
Pw/Pc =1.
• New Home Production Point:
– PPF same as earlier.
– New Price Line is steeper.
– New “tangency point” of PPF and the new Price line must be on
one of the corners.
– Look for the corner where the new price line is above the PPF.
– This is the New Production point = (100,0)
– Home does not produce any cloth. Produces as much wheat as
possible.
– Makes sense because OCW is lower at home.
International trade: Consumption
• New Home Consumption Point:
– IC MAP same as earlier.
– New Price Line is steeper.
– Find the New Tangency Point.
– This is the New Consumption Point.
– With nice preferences, the New Consumption
Point will be to the Left of the Autarky
Tangency Point.
– Example: New (40,40) Autarky (50,25)
International trade: Home
• New Production Point = (100,0)
• New Consumption Point = (40,40)
• Look at the x-axis.
– 100 > 40. So Xp > Xc, then you are exporting good X
which is Wheat.
• Look at the y-axis.
– 0 < 40. So Yp < Yc, then you are importing good Y
which is Cloth.
• Makes sense. Buying Cloth from the foreign
country since its OCC is lower there. Selling
Wheat to the foreign country since your OCW is
lower.
Gains from Trade
• Is the IC higher with Trade or in Autarky?
• If higher, then consumers are better off.
• So economy has got a welfare gain by
opening the economy to trade.
• What is the GFT?
Home GFT
• Real Wages in Autarky at Home:
– Wage = VMPLw = Pw MPLw = Pw 4 = VMPLc = Pc
MPLc = Pc 2
– Wage/Pw = 4
– Wage/Pc = 2
• New Trade Price is Pw/Pc = 2/3
• New VMPLw / VMPLc = (Pw MPLw)/(Pc MPLc)
= (Pw/Pc)(MPLw/MPLc) = (2/3)(4/2) = 4/3 = 1.3
• VMPLw is higher that VMPLc. So all workers
want to move to the Wheat industry.
Real Wages at Home after Trade
• Home only producing Wheat now. So we
can only define VMPLw and not VMPLc.
• Wage = VMPLw = Pw MPLw = Pw 4
• Real Wage
– Wheat: Wage/Pw = VMPLw/Pw
= Pw MPLw/ Pw = MPLw = 4.
– Cloth: Wage/Pc = VMPLw/Pc = Pw MPLw/Pc
= (Pw/Pc) MPLw = (2/3) 4 = 8/3.
Home GFT
• Trade Wage/Pw = 4 = Autarky Wage/Pw
• Trade Wage/Pc = 2.67 > 2 = Aut Wage/Pc
• Workers are at least as well-off after trade.
– In fact, here they are better-off.
– Note that the GFT comes from the higher
ability to purchase Cloth (the imported good
which was cheaper in the Foreign country).
International Trade: Foreign
• Repeat the same for the foreign country.
• New Production Point = (100,0)
• New Consumption Point is to the right of the
Autarky Tangency Point. Say, new (60,60) and
old (50,50).
• BOTH countries Gain from trade.
• Imports Wheat and Exports Cloth.
• Wage = VMPLc
• Real Wage
– Wheat: Wage/Pw = VMPLc/Pw = (Pc/Pw) MPLc =
(3/2) 1 = 3/2 = 1.5
– Cloth: Wage/Pc = VMPLc/Pc = MPLc = 1 = 1
Foreign GFT
• Trade Wage/Pw = 1.5 > 1 = Aut Wage/Pw
• Trade Wage/Pc = 1 = Aut Wage/Pc
• Workers are at least as well-off after trade.
– In fact, here they are better-off.
– Higher ability to purchase Wheat.
Home Export Supply of W
• At Aut Home Price: (Pr,X) = (1/2, 0)
• At Trade Price:
(Pr,X) = (2/3, 60)
• Think of an increase in Trade Price. The price line is steeper now
and consumption moves more to the left. So Home’s Export Supply
Curve is Upward sloping.
– Why? Cloth cheaper. Substitute into cloth. (Assuming that extra income
from wheat does not dominate the substitution). Also Foreign country
wants to sell more Cloth since they are getting a better price.
• Home’s Export Supply Curve is Convex.
– Why? If Price (Pw/Pc) increases by the same amount again, then want
to consume less Wheat. Due to the law of Diminishing Marginal
Utility, you now don’t want to give up as much of Wheat as earlier
because you’re already consuming less of it.
– Similarly, by the law of Diminishing Marginal Returns, if you keep
putting in more resources into Wheat production, the output of Wheat
will rise more slowly.
• What about linear PPFs (with CONSTANT Marginal Returns)?
Home Export Supply of W
•
•
•
•
At Aut Home Price: (Pr,X) = (1/2, 0)
At Trade Price:
(Pr,X) = (2/3, 60)
At Price > 2/3: Say, (Pr,X) = (1, 80)
At Price between ½ and 2/3: Say, (Pr,X) =
(0.55, 55). Check this yourself.
Linear PPF: Export Supply
• What else at Autarky price ½ ?
• When Price is ½ , workers get the same
wages in both Wheat and Cloth (Just like
in Autarky). This is because the slope of a
linear PPF is ½ at all points.
• So can produce anywhere on the PPF and
continue to consume at the desirable
Consumption Point (which is the original
consumption).
Linear PPF: Export Supply of W
•
•
•
•
What else at Autarky price ½ ?
Consumption Point (50,50).
Maximum Wheat Production = 100
Maximum Wheat X = Prod – Cons
= 100 - 50 = 50
• Get a New Point:
(Pr,X) = (1/2, 50)
• Already know Aut Point: (Pr,X) = (1/2, 0)
• Can also choose anything in the middle so join
these 2 points to get a Straight Line at Autarky
Price = ½.
Foreign Excess Demand (Import
Demand) of Wheat
• Usual Excess Demand Function
– Downward sloping for Price not equal to 1.
– Convex for Price not equal to 1. (Law of DMU still
holds which gives the convex shape).
– But the Constant MR changes the shape at Foreign
Autarky Price = 1
• Straight line segment at Price = 1 = Foreign
Autarky Price.
• Special property of Linear PPFs.
• Think of the Autarky Price in the case of a
concave PPF: No straight line portion.
Foreign Import Demand (Excess
Demand) of Wheat
• At Pw/Pc =1 (Same as
Foreign’s Autarky Price):
All C Prod Point
Cloth
Pw/Pc=1
100
A
Cons Point
All W Prod Point
Wheat
50
100
Max IM Dd of W Max Ex Ss of W
at Pw/Pc = 1
at Pw/Pc = 1
– Continue to consume at A
– Can produce anywhere
on the PPF
– If produce all C, then
Import Dd of W
= Cons – Prod
= 50 (A) – 0 (No W) = 50
– If produce all W, then
Export Supply of W
= Prod – Cons
= 100 (all W) – 50 (A) =50
Foreign Import Demand (Excess
Demand) of Wheat
• At Pw/Pc =1 (Autarky Price):
New Prod Point for P’ & P’’
New Cons Point for P’
100
New Cons Point for P’’
P’’<P’
A
P’<Pw/Pc •
50
IM Dd of W at P’’
IM Dd of W at P’
100
– Continue to consume at A
– Can produce anywhere on the
PPF
– If produce all C, then Import
Dd of W = Cons – Prod = 50
(A) – 0 (No W) = 50
– If produce all W, then Export
Supply of W = Prod – Cons =
100 (all W) – 50 (A) =50
At P’<1, Pink Horizontal Line.
(Produce all C at green point ,
Pw/Pc=1
Consume at pink tangency
point).
• At P’’<P’<1, Blue Horizontal
Line (Produce all C at green
point, Consume at blue
tangency point)
Foreign Import Demand (Excess
Demand) of Wheat
Plot the points we just derived. The graph should look like this.
Convince
yourself that
this quadrant
looks like this.
Pw/Pc
1
P’
P’’
Excess Ss
of Wheat
50
50
Excess Dd
of Wheat
Equilibrium Trade Patterns
• Intersection of Home X Ss and Foreign M Dd of
Wheat.
– Home X Ss = Foreign M Dd
• Market Clearing:
• Wheat produced in the world = Wheat
consumed in the world.
• Home Prod + Foreign Prod = Home Cons + Foreign Cons
• Home Prod – Home Cons = Foreign Cons – Foreign Prod
• Home X Ss = Foreign M Dd
Equilibrium Terms of Trade
• ToT = Price of Exports / Price of Imports
• Home exports W so its ToT = Pw/Pc = 2/3
• Can we say something about ToT?
• Recall the Average Cost of producing W at Home =
Amount paid to 1 worker = Wage at Home
• Similarly, AC of producing Cloth in Foreign = Wage
in Foreign
Equilibrium Terms of Trade
• Here AC = AVC.
• If Price < AVC then firms Shuts Down.
• Home produces W if Pw >= ACw = Wage/MPLw
• Foreign produces C if Pc >= ACc = Wage/MPLc
• Suppose Pw = 1.
–
–
–
–
–
Wage at Home = Pw MPLw = 4.
Wage in Foreign = Pw MPLw = 1.
Home: Pc = ACc = 2. Pw/Pc = ½.
Foreign: Pc = ACc = 1. Pw/Pc = 1.
½ < Pw/Pc < 1.
Equilibrium Terms of Trade
• ToT = Price of Exports / Price of Imports
• Home exports W so its ToT = Pw/Pc = 2/3
• Can we say something about ToT?
• Recall the Average Cost of producing W at Home =
Amount paid to 1 worker = Wage at Home
• Similarly, AC of producing Cloth in Foreign = Wage
in Foreign
Equilibrium Terms of Trade
• Here AC = AVC.
• If Price < AVC then firms Shuts Down.
• Home produces W if Pw >= ACw = Wage/MPLw
• Foreign produces C if Pc >= ACc = Wage/MPLc
• Suppose Pw = 1.
–
–
–
–
–
Wage at Home = Pw MPLw = 4.
Wage in Foreign = Pw MPLw = 1.
Home: Pc = ACc = 2. Pw/Pc = ½.
Foreign: Pc = ACc = 1. Pw/Pc = 1.
½ < Pw/Pc < 1.