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Microeconomic Foundations
of International Trade
Mankiw: Ch. 4, 5, 7, 9, 13-16, and 21
Bradely: Ch. 6, 7, and 8
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Table of Contents
1. Consumer’s Behavior
1.1. Budget Constraint
1.2. Indifference Curve
1.3. Utility Maximization: Consumer’s Optimal Choice
1.4. Deriving the Demand Curve
1.5. Consumer Surplus
2. Producer’s Behavior
2.1. Basics
2.2. Cost Minimization
2.3. Profit Maximization: Firm’s Optimal Choice
2.4. Deriving the Supply Curve
2.5. Producer Surplus
3. Production Revisited: A Country's Point of View
3.1. Isovalue Line
3.2. Production Possibilities
3.3. Maximizing Production Values
4. Summary
5. Worked Examples
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1. Consumer’s Behavior
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1. Consumer’s Behavior
Maximize Objective Function
Subject to Constraint
U = U (x , y )
xPX + yPY = M
1.1. Budget Constraint
§ Budget constraint: the limit on the consumption bundles
that a consumer can afford.
§ The slope of the budget constraint equals
- the opportunity cost of good X in terms of good Y
- the relative price of X
y =-
PX
M
x+
PY
PY
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1. Consumer’s Behavior
1.2. Indifference Curve
Indifference curve shows consumption bundles that give the
consumer the same level of satisfaction.
► 4 Properties of IC
1. Higher indifference curves are preferred
to lower ones.
2. Indifference curves are downward
sloping. (Diminishing MRS)
☞ Marginal Rate of Substitution (MRS)
: The absolute value of the slope of the
indifference curve
: The MRS is the marginal benefit of X
good in terms of Y good (i.e., amount
of Y good the consumer must be given
to compensate for the loss of one X
good)
3. Indifference curves are bowed inward
(i.e., convex): Consumer’s greater
willingness to give up a good that he
already has in large quantity
☞ Diminishing MRS means that
consumers like varieties.
4. Indifference curves do not cross.
☞ The slope of the budget constraint is
the opportunity cost of X good in terms
of Y good.
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1. Consumer’s Behavior
1.2. Indifference Curve
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1. Consumer’s Behavior
1.2. Indifference Curve
■ Marginal utility
MU X =
DU U (x + Dx , y )
=
Dx
Dx
■ The slope of the indifference curve
MU X Dx + MUY Dy = DU º 0
■ Solving for the slope of the indifference curve, we have
Dy
MU X
=º MRS X ,Y
Dx
MUY
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1. Consumer’s Behavior
1.2. Indifference Curve
■ Marginal Rate of Substitution (MRS): the rate at which a
consumer is willing to trade one good for another
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1. Consumer’s Behavior
1.3. Utility Maximization: Consumer’s Optimal Choice
u The optimal bundle is at the point where the budget constraint
touches the highest indifference curve.
u The indifference curve and budget constraint have the same slope.
u Relative price = MRS at the optimum:
é
PX ù é
MU X ù
êSlope of Budget Line º - P ú = êSlope of Ind . Curve º MRSX ,Y º - MU ú
Y û
Y û
ë
ë
Thus,
æ
P ö æ
MU X ö
çç Re lative Market Value º X ÷÷ = çç Individual Value º
÷
P
MUY ÷ø
Y ø
è
è
Or
MU X MUY
=
PX
PY
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1. Consumer’s Behavior
1.3. Utility Maximization: Consumer’s Optimal Choice
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1. Consumer’s Behavior
1.4. Deriving the Demand Curve
■ Left figure: price of Pepsi falls from $2 to $1
■ Right figure: Pepsi demand curve
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1. Consumer’s Behavior
1.5. Consumer Surplus
■ A buyer’s Willingness to Pay for a good is the maximum
amount the buyer will pay for that good.
■ WTP measures how much the buyer values the good.
■ At a price equal to his WTP, the buyer would be indifferent
whether to buy or not to buy the good.
■ If the price is exactly the same as the value he places on that
good, he would be equally happy to buy it or not to buy it
and keep his money.
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1. Consumer’s Behavior
1.5. Consumer Surplus
■ Consumer Surplus (CS): the amount that a buyer is willing
to pay for a good minus the amount the buyer actually pays
for it.
CS = WTP - P
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2. Producer’s Behavior
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2. Producer’s Behavior
2.1. Basics
2.1.1. Production Function (Total Product) and Marginal Product
u A production function shows the relationship between the quantity
of inputs used to produce a good and the quantity of output of that
good.
Qi = f (Li ,K i )
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2. Producer’s Behavior
2.1. Basics
2.1.1. Production Function (Total Product) and Marginal Product
u The marginal product of any input denotes the increase in
the amount of output from an additional unit of that
input, holding all other inputs constant.
u Marginal Product of Labor to produce good i (MPL i ) = Slope of
the production function
MPL i =
DQi
DLi
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2. Producer’s Behavior
2.1. Basics
2.1.1. Production Function (Total Product) and Marginal Product
u Diminishing marginal product: the marginal product of an input
declines as the quantity of the input increases (other things equal)
TRi = Pi Qi
ARi =
TRi
Qi
MRi =
DTRi
DQi
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2. Producer’s Behavior
2.1. Basics
2.1.2. Costs
u Total Cost ( TCi )
TCi = FCi + VCi
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2. Producer’s Behavior
2.1. Basics
2.1.2. Costs
u Marginal cost ( MCi ) is decreasing initially and then increasing in TCi
from producing one more unit.
MCi =
DTCi
DQi
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2. Producer’s Behavior
2.1. Basics
2.1.2. Costs (wheat producer)
u Assume that every producer is rational and wants to maximize his profit.
u To increase profit, should he produce more or less?
• To find the answer, each producer needs to “think at the margin.”
• If the cost of additional wheat ( MCi ) is less than the revenue, he would
get from selling it ( MRi ), then his profits rise if he produces more.
u Average total cost ( ATCi ) equals total cost divided by the quantity of
output:
ATCi =
TCi FCi + VCi
=
= AFCi + AVCi
Qi
Qi
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2. Producer’s Behavior
2.2. Cost Minimization
u Producer’s choice of factors that minimizes production costs
u Finding the point on the isoquant that has the lowest
possible isocost line associated with it.
u w denotes the wage rate and r is the rental cost of capital
Minimize Ci (L,K ) = wLi + rK i
(
Subject to Qi = f Li ,K i
)
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2. Producer’s Behavior
2.2. Cost Minimization
2.2.1. Cost Function
uThe minimum costs of producing Qi units of output i,
when factor prices are w (for labor) and r (for capital).
Ci = wLi + rK i
2.2.2. Isoquant Curve
u Isoquant curve shows the set of all possible combinations
of two inputs ( Li and K i ) that are just sufficient to produce
a given amount of output ( Qi ).
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2. Producer’s Behavior
2.2. Cost Minimization
2.2.2. Isoquant Curve
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2. Producer’s Behavior
2.2. Cost Minimization
2.2.2. Isoquant Curve
u The Slope of the Isoquant Curve
: Consider the case of giving up a little bit of capital ( DK i )
and using just enough more of labor ( DLi ) to produce
the same amount of output Qi
MPLi DLi + MPK i DK i = DQi = 0
u Solving for the slope of the isoquant curve, we have
DK i
MPLi
= º MRTSLi ,K
DLi
MPK i
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2. Producer’s Behavior
2.2. Cost Minimization
2.2.2. Isoquant Curve
u Marginal Rate of Technical Substitution (MRTS)
: the rate at which the firm will have to substitute one input
for another in order to keep output i ( Qi ) constant.
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2. Producer’s Behavior
2.2. Cost Minimization
2.2.3. Isocost Lines
u From the cost function, we can derive isocost functions,
Ci = wLi + rK i ==>K i = -
w
C
Li + i
r
r
u As we let the number Ci vary, we get a whole family of
isocost lines.
u Every point on an isocost curve has the same cost, Ci ,
and higher isocost lines are associated with higher cost.
u Note that the slope of the isocost curve is the wage- w
rental ratio (or the ratio of labor and capital prices): r
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2. Producer’s Behavior
2.2. Cost Minimization
2.2.3. Isocost Lines
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2. Producer’s Behavior
2.2. Cost Minimization
2.2.4. Producer's Optimal Choice
u The optimal bundle of two inputs is at the point where the
isoquant curve touches the lowest isocost curve.
u The slope of the isoquant curve (MRTS) must be equal to the
slope of the isocost curve in equilibrium
u Relative factor price at the optimum (factor price ratio) = MRTS
ö
æ
wö æ
MPLi
º MRTSLi ,K ÷÷
çç slope of Iso cos t º - ÷÷ = çç slope of Isoquant = r
MPK
è
ø è
i
ø
w MPLi in equilibrium.
Thus,
=
r MPK i
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2. Producer’s Behavior
2.2. Cost Minimization
2.2.4. Producer's Optimal Choice
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2. Producer’s Behavior
2.3. Profit Maximization
u We can also say that that firm’s goal is to maximize profit.
Maximize
p i = TRi - TCi
where TRi is the amount a firm receives from the sale of its output Qi
while TCi is the market value of the inputs a firm uses in production
u When Qi increases by one unit, revenue rises by MRi , cost rises by MCi .
If MRi > MCi , then increase Qi to raise profit.
If MRi < MCi , then decrease Qi to raise profit.
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2. Producer’s Behavior
2.3. Profit Maximization
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2. Producer’s Behavior
2.3. Profit Maximization
u Profit Max. Rule:
MRi = MCi at the profit-maximizing Qi .
Pi = MCi for firms in a competitive market ( MRi = Pi )
u Assume that two goods are produced and profits are
maximized in two goods market:
PX = MC X
thus,
and
PY = MCY
PX MC X
=
PY
MCY
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2. Producer’s Behavior
2.4. Firm’s Supply Decision
u If price rises to P2 , then the profit-maximizing quantity rises to Q2 .
u The MC curve determines the firm’s quantity of output at any price.
u Hence, the MC curve is the firm’s supply curve.
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2. Producer’s Behavior
2.5. Producer Surplus
u A seller will only produce and sell the good, if the price (P )
exceeds his/her cost (MC ).
u Hence, marginal cost is a measure of willingness to sell.
u Producer surplus (PS) is the amount a seller is paid for a good
minus the seller’s cost.
PS = P - cost
u Total producer surplus equals the area above the supply curve
*
below the price from 0 to Q .
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2. Producer’s Behavior
2.5. Producer Surplus
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3. Production Revisited :
A Country's Point of View
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3. Production Revisited:
A Country's Point of View
Maximize Objective Function: V = xPX + yPY
Subject to Constraint: production possibility set
3.1. Isovalue Curve
u Suppose that the country produces two goods, good X and good Y.
u Note that the slope of the isovalue curve equals the relative price of
good X.
y =-
PX
V
x +
PY
PY
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3. Production Revisited:
A Country's Point of View
3.2. Production Possibilities
A production possibilities set shows the combinations of two goods
( X and Y ) the economy can possibly produce given the available
resources ( L , K ) and the available technology ( A ).
x = f (LX ,K X ), y = f (LY ,KY )
LX + LY = L,
K X + KY = K
ü Production Possibility Frontier is the boundary of the production possibilities
set.
ü A bow shaped (bowed outward=concave to the origin to) PPF denotes the
increasing opportunity cost.
ü Note that the slope of the PPF is the ratio of the marginal cost of good X to
the marginal cost of good Y (i.e. MCX/MCY). By reducing the amount of good
X by 1 unit, we can save MCx. In order to transform this into units of good Y,
we must divide by MCY.
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3. Production Revisited:
A Country's Point of View
3.2. Production Possibilities
u The slope of PPF: MRT(Marginal Rate of Transformation)
u MRT (or just opportunity cost) measures the rate at which one
good (good X or Soybeans) can be transformed into the other
(good Y or Wheat).
ü MRT measures the tradeoff
of producing more X good in
terms of Y good.
ü Therefore, the slope of the
PPF is the marginal cost ratio
of producing the goods.
MRTX ,Y =
DY
MC X
=
DX
MCY
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3. Production Revisited:
A Country's Point of View
3.3. Maximizing Production Values
u The optimal production of two goods (X and Y ) is at the point where the
production possibility frontier touches the highest isovalue line.
u The slope of isovalue curve must be equal to the slope of the PPF (MRT)
in equilibrium.
Relative price ratio = MRT at the optimum
æ
ö
P ö æ MC X
ççworld relative price º - X ÷÷ = çç º MRTX ,Y ÷÷
PY ø è MCY
è
ø
Thus, in equilibrium
PX MC X
=
PY
MCY
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3. Production Revisited:
A Country's Point of View
3.3. Maximizing Production Values
This is the same as the profit maximization condition.
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4. Summary
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4. Summary
► Consumer's optimal choice (utility maximization) condition is:
PX MU X
=
º MRS X ,Y
PY
MUY
► Producer's profit (or country's production-values) maximization
condition is:
PX MC X
=
º MRTX ,Y
PY
MCY
æ
ç
è
► Thus, ç MRS X ,Y º
MU X ö PX æ
MC X ö
÷÷ =
÷
= çç MRTX ,Y º
MUY ø PY è
MCY ÷ø
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4. Summary
u When relative price (
PX
) is fixed, optimal consumption bundle
PY
of two goods ( C X* and CY* ) and optimal production bundle
of two goods ( QX and QY ) are determined.
*
*
u Determination of QX* and QY* directly leads to determination of
two factors, L*x and K x* as well as L*Y and KY* , given relative
factor price ( w/r ).
u In autarky, C X* = QX* and CY* = QY* .
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5. Worked Examples
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5. Worked Examples
5-1. The Utility Maximization Problem and the Lagrangian Method
l The utility maximization problem is as follows
max (, )
,
s. t.  +  = 
l This problem is constrained because consumers face a budget
constraint.
l We can convert our constrained problem into an unconstrained
one by constructing an “artificial” Lagrangian function, (..).
l Use Lagrange multipliers to derive the identity


=
 

which
satisfies consumer’s utility maximization.
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5. Worked Examples
5-1. The Utility Maximization Problem and the Lagrangian Method
l Lagrangian function is constructed as follows: it is equal to our
objective function (the utility) plus a variable called a Lagrangian
multiplier times the “slack” of the budget constraint.
 , ,  =  ,  +   −  − 
ü (, , ) denotes the Lagrangian function.
ü  −  −  is the slack of budget constraint (the money
left over). At the optimum this slack will be 0.
ü λ is the Lagrangian multiplier. It tells us by how much the
maximized utility will increase if the income increases by $1.
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5. Worked Examples
5-1. The Utility Maximization Problem and the Lagrangian Method
l Maximizing the (unconstrained) Lagrangian function with
respect to , ,  is equivalent to solving the constrained
problem.
l The Lagrangian is  , ,  =  ,  +   −  −  .
l To find the maximum (or any optimum point!), equate the firstorder partial derivatives to zero
l The first-order conditions for , ,  are
(, , ) (, )
=
−  = 0


(, , )  (, )
=
−  = 0


(, , )
=  −  −  = 0.

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5. Worked Examples
5-1. The Utility Maximization Problem and the Lagrangian Method
l The third condition is just the budget constraint:  −  − 
l Rearrange the other first-order conditions:
 = 
 = 
l  > 0,  > 0 and preferences are monotonic, we can be sure
that λ > 0.
l Dividing the two conditions yields


= 

 
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5. Worked Examples
5-2. Progress Exercises on the Utility Maximization Problem
u Use Lagrange multipliers to find the maximum utility for the
utility function U = 5xy, when subject to a budget of $30,
where the price of each unit of X is $5 and each unit of Y is $1.
l SOLUTION
ü The equation of the budget constraint is
 +  =  à 5 +  = 30
ü The equation of the Lagrangian is therefore
 = 5 + (30 − 5 − )
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5. Worked Examples
5-2. Progress Exercises on the Utility Maximization Problem
SOLUTION (Cont.)
Step 1: Find the first-order partial derivatives:
 = 5 − 5,  = 5 − ,  = 30 − 5 − 
Step 2: Equate the first-order derivatives to zero and solve
1 5 − 5 = 0  5 = 5
2 5 −  = 0  5 = 
3 30 − 5 −  = 0
To solve, first eliminate λ from equations (1) and (2).
1
→
5 = 5
2 ×5
→ 25 = 5λ
4 
→ 5 − 25 = 0
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5. Worked Examples
5-2. Progress Exercises on the Utility Maximization Problem
SOLUTION (Cont.)
Now use equations (3) and (4) to solve for x and y.
5
→ (4)⁄5
→
 − 5 = 0
6
→
→
 + 5 = 30
3 rearranged
add
2 + 0 = 30
→
 = 15
Step 3: Substituting y = 15 into equation (5) gives x = 3; therefore,
maximum utility occurs when x = 3, y = 15.
Step 4: The level of maximum utility is U = 5(3)(15) = 225. From
equation (1), λ = 15.
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5. Worked Examples
5-3. The Profit Maximization Problem for a Perfectly Competitive Firm
l The profit maximization problem is as follows
  =   −  ()
l The firm wants to maximize profits by producing  units of
output.
l The profit maximization problem for a perfectly competitive firm
can be solved by taking the first-order and second-order
conditions.
- FOC:
()
()
 
=
−



- SOC:
()
( )
 
=
−
< 0 or



= 0 or  ’ =  −  =0
 ’’ = ( )’ − ( )’ < 0
 ’ <  ’
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5. Worked Examples
5-3. The Profit Maximization Problem for a Perfectly Competitive Firm
l The second derivative of the profit function is a negative
(positive) constant, profit will be maximized (minimized).
l The second-order condition for profit maximization denotes the
derivative of MC is greater than the derivative of MR.
l Note that the derivative of MR is the second derivative of TR and
the derivative of MC is the second derivative of TC.
l
 ’’ = ()’’ − ()’’ = ( )’ − (MC)’ < 0
 ’ < C’
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5. Worked Examples
5-4. Progress Exercises on the Profit Maximization Problem
l The demand function for a good is given by the equation
P=50-2Q, while total cost is given by TC=160+2Q. Find the
maximum profit and the value of Q at which profit is maximum.
l SOLUTION
ü Step 1 (Find TR function): TR=P*Q=(50-2Q)*Q=50Q-2Q2
ü Step 2 (Find profit function): =TR-TC=-2Q2+48Q-160
ü Step 3 (Get derivatives of profit function):
()
=-4Q+48=0 at

turning point.
ü Step 4: At Q=12,  =128.
ü Step 5 (Max. or min.?):  ’’ =-4 < 0. The second derivative
of the profit function is a negative constant, therefore profits
are a maximum.
=128 at Q =12
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5. Worked Examples
5-5. The Cost Minimization Problem and the Lagrangian Method
l The cost minimization problem is as follows
min  + 
,
s. t.   ,  = 
l The firm wants to minimize its costs ( +  ) of producing 
units of output i. The fact that the firm wants produce  units
of output is given by the constraint   ,  =  .
l The cost minimization problem of the firm can be solved by the
Lagrangian method
 , ,  =  +  +   −   , 
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5. Worked Examples
5-5. The Cost Minimization Problem and the Lagrangian Method
l Take the partial derivatives of the Lagrangian with respect to
three arguments, ,  and .
 , , 
  , 
=−
=0


 , , 
  , 
=−
=0


 , , 
=  −   ,  = 0

  ,
  ,
 
 
Ø Note that
and
are the first partial derivatives of


the production function of good i with respect to  and .
l Eliminate λ from the first two equations to get:
   , 
=
   , 
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5. Worked Examples
5-5. The Cost Minimization Problem and the Lagrangian Method
l Note that the partial derivatives of the production function are
simply the marginal products of each input,  and .
l Since   ,  =   and   ,  =   by definition, we
get a cost minimization condition below

 
=  


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5. Worked Examples
5-6. Progress Exercises on the Lagrangian Multipliers and Cost Minimization
u Minimize Costs Subject to a Production Constraint: A production
function is Q=12L0.5K0.5. Labor costs are $25 per unit (wage rate)
and capital costs are $50 per unit (rent). The production
constraint is 240 units of output. Find the values of L and K to
minimize costs
l SOLUTION
The equation of the cost function is
 +  =  è 25 + 50 = 
This is the function which is to be minimized subject to the
production constraint. Thus, the Lagrangian is
 , ,  = 25 + 50 + (240 − 12.  . )
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5. Worked Examples
5-6. Progress Exercises on the Lagrangian Multipliers and Cost Minimization
SOLUTION (Cont.)
Step 1: Find the first-order partial derivatives:
 , , 
= 25 − 12 0.5.  . = 25 − λ6.  .

 , , 
= 50 − 12 0.5.  . = 50 − λ6.  .

 , , 
= 240 − 12.  .

Step 2: Equate the first-order partial derivatives to zero and solve:
1
0 = 25 − λ6.  . → 25 = λ6.  .
2
0 = 50 − λ6.  . → 50 = λ6.  .
3
0 = 240 − 12.  . → 240 = 12.  .
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5. Worked Examples
5-6. Progress Exercises on the Lagrangian Multipliers and Cost Minimization
SOLUTION (Cont.)
Step 3: As usual, eliminate λ from equations (1) and (2). Divide the
corresponding sides of each equation:
(1) 25 λ6.  .
→
=
(2) 50 λ6.  .
1 .  .
=
2 .  .
1 
=
2 

=
2
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5. Worked Examples
5-6. Progress Exercises on the Lagrangian Multipliers and Cost Minimization
SOLUTION (Cont.)
Step 4: Substitute

=  into equation (3) to find L* and K*.

240 = 12. ().
240 =
=
12
2

2 (240)
12
∗ = 28.28 and  ∗ = 14.14
Therefore C = 25L + 50 K = 25(28.28) + 50(14.14) = 1,414.
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