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Microeconomics for International Trade Theory. ECON0301 January 2011. Basic Trade Model. General equilibrium multiple agents (firms and consumers) all optimizing The aggregation problem One shot long run income=expenditure => balance of trade Market structure

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basic trade model
Basic Trade Model
  • General equilibrium
    • multiple agents (firms and consumers) all optimizing
    • The aggregation problem
  • One shot
    • long run
    • income=expenditure => balance of trade
  • Market structure
    • Constant returns to scale, perfect competition
    • Variable returns to scale, market power (new trade theory in 80s; new new trade theory in this past decade)
basic prerequisites
Basic Prerequisites
  • Consumption
  • Production
  • Perfectly Competitive Market
  • General Equilibrium
consumption problem
Consumption Problem
  • Consumption choice problem
  • Results: Demand functions
  • Equalization of marginal utility per dollar
  • Marginal rate of substitution = relative price
marginal rate of substitution relative price

B

A

C

Marginal rate of substitution = relative price

At point A, the budget line and some IC are tangential to each other

Omitting the negative signs,

The slope of the IC = MUx/MUy

The slope of the budget line = Px/Py

marginal rate of substitution relative price1

Y

U1

U0

A

C

X

Marginal rate of substitution = relative price

At point C, MUx/MUy=Px/Py. But obviously not an optimal bundle

Some properties on the shapes of ICs are required and will be assumed to hold.

cobbs douglas utility function
Cobbs-Douglas Utility Function
  • CD utility function:
  • Marginal utilities:
  • MRS = relative price
  • Expenditure shares constant
cobbs douglas utility function1
Cobbs-Douglas Utility Function
  • Very nice demand functions
  • Without loss of generality, assume
  • If not, we can always represent the old CD utility function by a new CD function
measurability of utility

V=2001

V=200

V=100

U=30

U=20

U=10

Measurability of Utility

An order-preserving re-labeling of ICs does not alter the preference ordering.

positive monotonic order preserving transformation
Positive monotonic (order-preserving) transformation
  • They are called positive monotonic transformation
  • They all refer to the same preferences, leading to the same choice
two properties of cd utility functions
Two properties of CD utility functions
  • Unit income elasticity
    • 1% increase in income => 1% increase in consumption
  • The aggregation problem
    • Given total income, income distribution does not affect market demand
the aggregation problem
The aggregation problem
  • Suppose there are N agents, each with the same CD utility function
  • Suppose their incomes are I1,I2, … , IN, summing up to I.
  • The total market demand for x equals
  • Given the total income, the income distribution among the agents does not affect the market demand. As long as the latter is concerned, we do not need to know income distribution.
  • A property need not generally hold for other utility functions
an example where income distribution matters
An example where income distribution matters
  • Two goods: necessity (x) and luxury (y)
  • Two agents, where I1+I2 = I = 10
  • each will consume luxury only after x0 <5 units of necessity is consumed.
  • Suppose prices px=1, py=1.
  • Equal income, the market quantity demanded for x is 2x0; the market quantity demanded for y is 10 - 2x0
  • Unequal income; suppose I1 < I2 and I1 < x0. Then market quantity demanded for x is I1 + x0 < 2x0; market quantity demanded for y is 10-(x0 + I1).
  • Unequal income diverts resource to luxuries while basic necessities are not fully provided => income distribution matters
the aggregation problem1
The aggregation problem
  • The Cobb-Douglas utility function is among the family of utility functions for which income distribution per se does not affect market demand
  • We can further define the concept of “the relative demand for x” by an individual
  • This relative demand is independent of the individual’s income
the aggregation problem2
The aggregation problem
  • The relative demand for x in the whole economy (with individual incomes I1, I2, …, IN) is just the same as the relative demand for x by any individual
  • Take out: With CD utility function, we can talk about
    • demand for x by the economy without knowing how income is distributed among individuals
    • Relative demand for x by the economy without knowing the total income
production function1
Production Function
  • Output elasticity and returns to scale
production function2
Production function
  • When returns to scale do not change with scale, for any t>1, the technology exhibits
production function3
Production function
  • For CRTS technology, there are two nice properties
    • MPK and MPL depend on K/L only, but not on the absolute scale
      • E.g., MPK the same when you hire K=3 and L=5, compared with when you hire K=6 and L=10.
    • MPK*K+MPL*L= f(K,L)
      • When factors are hired up to r = p*MPK and w = p*MPL, the profit is just zero!
production function4
Production function
  • We show the second property:
profit maximization problem
Profit maximization problem
  • Each firm chooses
  • If an optimum exists, will hire K and L such that
cost minimization problem
Cost minimization problem
  • Sometimes it is easier to re-phase the problem as a cost minimization problem, following by the output choice problem
  • Given input prices, choose the input combination that minimizes cost
  • Cost function
cost function
Cost function
  • When RTS does not change with scale, for any t>1, the technology exhibits
isoquant
Isoquant – the locus of K and L such that the output level is constant

Bending toward the origin

K

Optimal input mix to produce Q=10

Q=20

Iso-cost line

Q=10

rK+wL=constant

L

isoquant
equilibrium condition
Main assumptions: CRTS technology, perfect competition, all inputs variable (long run equilibrium).

The min. costs to produce $1 worth of a good are exactly $1.

Given output and input prices, we can determine the optimal mix of inputs

Or, given output price and K/L ratio, we can determine the relative input prices

K

Q=1/p’

Iso-cost line

Q=1/p

rK+wL=constant

L

Equilibrium condition
equilibrium condition1
Equilibrium condition
  • For CRTS technology and in LR equilibrium, we cannot tell the output level of a particular firm, because every output level will lead to the same profit (which is zero) given fixed input and output prices
cobb douglas production function1
Cobb-Douglas Production Function
  • What does α+β=1 mean?
    • α+β>1; IRTS
    • α+β=1; CRTS
    • α+β<1; DRTS
the aggregation problem3
The aggregation problem
  • Consider an industry with all firms having the same CRTS technology: Q=f(K,L); output & input markets perfectly competitive; and firms maximizing profits. Collectively, the industry employs K* and L*.
  • What is the total output of the industry?
    • f(K*,L*). The output produced would be the same as if there were one single firm employing K* and L*. The “fine structure” of the industry (i.e., the number of firms, the sizes, etc.) is irrelevant.
the aggregation problem4
The aggregation problem
  • Let K1, K2, …, KN be the amount employed in the N firms.
  • Let L1, L2, …, LN be the amount employed in the N firms.
  • Cost minimization requires that K1/L1=K2/L2=… = KN/LN=K*/L*=a.
  • Total output
the aggregation problem5
The aggregation problem
  • Question: What is the marginal product of capital (labor) in each firm?
  • Despite possibly different scales, each firm’s marginal product of capital is simply equal to ∂F(K*,L*)/∂K, i.e., marginal product of capital of a fictitious firm which employs just K* and L*.
  • Similarly, all firms have the same marginal product of labor = ∂F(K*,L*)/∂L.
the aggregation problem6
The aggregation problem
  • Question: What is the rental rate of capital paid by each firm? What is the wage rate of labor paid by each firm?
  • Let P be the price of the good produced in the industry. The rental rate of capital is just P*∂F(K*,L*)/∂K, while the wage rate of labor is just P*∂F(K*,L*)/∂L.
  • The real rental rate (≡ w/p) and real wage rate (≡ r/p) should be ∂F(K*,L*)/∂K & ∂F(K*,L*)/∂L.
the aggregation problem7
The aggregation problem
  • To know the output of an industry, as well the wage rate of labor and the rental rate of capital being paid by firms in the industry, there is no need to know the internal structure of the industry.
  • We can simply work out the problem by assuming all inputs (K* are L*) are hired by a single firm (which is nonetheless price taking in both input and output markets).
  • In this case, we can understand f(K,L) as an industry production function.
the aggregration problem
The aggregration problem
  • Takeout: given same CRTS technology, & LR competitive equil, the total K* and L* employed in the sector fully describe the output level as well as the real rental rate and real wage rate.