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# micvar09

By Teresa Garcia,2014-03-02 21:45
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micvar09

9 EQUILIBRIUM

Supply

Competitive market: ；完全竞争市场？each economic agent takes the market price as outside of his control.

Each consumer or producer is a small part of the market as a whole, and has a negligible effect on the market price.

Supply curve: for each p, we determine how much of the good will be supplied, S(p).

Price

p

O x quantity

Market Equilibrium

A number of consumers of a good: their individual demand curves constitute a market demand curve.

Market demand curve: D(p)

A number of independent suppliers of this good: their individual

supply curves constitute the market supply curve.

Market supply curve: S(p)

Price D(p) S(p)

p*

O x* quantity

Equilibrium price；均衡价格？: the price at which the supply of the good equals the demand.

The equilibrium price is the price p* which solves the equation

D (p*) = S (p*)

EXAMPLE: Equilibrium with linear Curves

D(p) = a bp

S(p) = c + dp.

The equilibrium price can be found by solving

D (p) = a bp = c + dp = S (p).

ac p*d;b

The equilibrium quantity demanded (and supplied) is

D(p*)abp*

ac abb;d

Two Special Cases

Case 1: The supply curve is vertical.

Price

p

O x quantity

The equilibrium quantity is determined by supply. The equilibrium price is determined by demand.

Case 2: The supply curve is completely horizontal.

Price

p

O x quantity

The equilibrium price is determined by supply. The equilibrium quantity is determined by demand.

Inverse Demand and Supply Curves

P (q) ---- the inverse supply curve s

P(q) ---- the inverse demand curve D

Equilibrium is determined by

P(q*) = P(q*) SD

EXAMPLE: Equilibrium in inverse demand and supply

P(q) = a bq D

P(q) = c + dq. S

The equilibrium price can be found by solving

P(q) = a bq = c + dq = P(q). DS

ac q*d;b

The equilibrium price is

p*abq*

ac ab b;d

Comparative Statics

How will equilibrium change as the demand and supply curves

change?

Price

p*

O q* quantity

What if both curves shift to the right?

Pareto Efficiency (帕累托最优)

An economic situation is Pareto efficient if there is no way to

make any person better off without hurting anybody else.

Price

p*

q* quantity O q

At q: if there is a trading for an extra unit of good, the buyer is willing pay more than the seller requires.

Trading on for that amount cannot be Pareto efficient.

Price

p*

O q* q，， quantity

At q: trading is at the price that the buyer is willing pay less than the seller requires.

Trading on for that amount cannot be Pareto efficient.

The competitive market produces a Pareto efficient amount of output.

At the equilibrium q*:

Consumers MRS between the good and all other goods

p*MRS = - = - p* 1

All consumers are with the same MRS.

In the case of two goods

When two markets are in equilibrium, every consumer’s optimal choice (x*, x*) satisfies 12

MUp11MRS = = MUp22

10 TECHNOLOGY

Inputs and Outputs

Factors of production: inputs to production.

----land, labor, capital, and raw materials.

Capital goods, physical capital: inputs that are themselves produced goods.

Describing Technological Constraints

Production set (生产集): the set of all combinations of inputs and outputs that comprise a technologically feasible way to produce.

The case of one input

If y units of output can be produced by x units of input, point (x,

y) is in the production set.

Output

f(x)

y

O x input

Production function ；生产函数？---- the boundary of production set.

The maximal units of output produced with x units of input

The case of two inputs:

y = f (x, x). 12

An isoquant；等产量线？: the set of all possible combinations of inputs 1 and 2 that are just sufficient to produce a given amount of

output.

input 2

O input 1

Examples of Technology

Fixed Proportions

f(x, x) = minx, x 1212

input 2

O input 1

General form

f(x, x) = minax, bx 1212

Perfect Substitutes

f (x, x) = x+ x. 121 2

input 2

O input 1

General form

f (x, x) = ax+ bx. 121 2

Cobb-Douglas production function ab f(x,x)Axx1212

Properties of Technology

Monotonic: if you increase the amount of at least one of the

inputs, it should be possible to produce at least as much output as you

were producing originally.

----Property of free disposal

Convex: if you have two ways to produce y units of output, (x, x) 12

and (z, z), then their weighted average will produce at least y units 12

of output.

input 2

O input 1

Long and Short Runs

In the short run；短期？, there are some fixed factors of

production at predetermined levels, ---- immediately feasible.

In the long run；长期？, all the factors of production can be

varied ----eventually feasible.

xSuppose that factor 2 is fixed at in the short run. 2

f(x,x)The production function for the short run is. 12

y

y= f(x, ) x12

O x 1

The Marginal Product

At point (x, x) : 12

The marginal product；边际产量？of factor 1:

f(x;x,x)f(x,x)y11212MP (x, x) = 112xx11

The marginal product of factor 2:

f(x,x;x)f(x,x)y12212MP (x, x) = 212xx22

By derivatives:

y(fMP = = lim1x01(xx11

y(fMP = = lim2x02x(x22

The Technical Rate of Substitution

Consider a change in the use of factors 1 and 2 that keeps output

fixed

(x, x) ( x+?x, x+?x) 121122

Input 2

x 1

x 2

O Input 1

Technical rate of substitution (技术替代率): slop of isoquant

x2 TRS(x,x)12x1

Change it in two steps:

Step 1: (x, x) ( x+?x, x) 12112

Change in output: ?y = MP?x 111

Step 2: (x, +?x x) ( x+?x, x+?x) 1121122

Change in output:?y = MP?x 222

It must be

?y = ?y + ?y 12

= MP?x+MP?x= 0. 1122

which we can solve to get

xMp(x,x)2112TRS(x,x) 12xMP(x,x)1212

Diminishing Marginal Product

Monotonicity: nonnegative marginal product. Law of diminishing marginal product: the marginal product of a

factor will diminish as we get more and more of that factor.

----other inputs are kept unchanged.

Output

MP

O input

Diminishing Technical Rate of Substitution

Diminishing technical rate of substitution: when the amount of factor 1 increases along the isoquant, the technical rate of substitution declines.

Returns to Scale

Let’s increase the amount of all inputs to the production function. How much will output increase?

Constant returns to scale；规模报酬不变？: Output increases by

the same proportion as inputs do.

In general, if we scale all of the inputs up by some amount, the output will increase by the same times.

t f (x, x) = f (tx, tx). 1212

Increasing returns to scale；规模报酬递增？: if we scale up both

inputs by some factor t, we get more than t times as much output:

f (tx, tx) > t f (x, x) 1212

for all t >1.

Decreasing returns to scale；规模报酬递减？？

f (tx, tx) < t f (x, x) 1212

for all t >1.

A technology can exhibit different kinds of returns to scale at different levels of production:

----for low levels of production, the technology exhibits increasing returns to scale

----for larger levels of output, increasing scale by t may just

increase output by the same factor t.

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