Capital Misallocation: A Macroeconomic Perspective on Productivity and Growth, Lecture notes of Economic Analysis

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Course Overview and

Lecture 1 Capital Misallocation

Shengxing Zhang

LSE

September 26, 2017

Course Overview - an example

I (^) Alice

I (^) receives today £100,

I (^) save £100 in a bank

I (^) receives next year £

I (^) Bob

I (^) needs £100 to buy seeds and borrows from the bank

I (^) sell plants for £110 next year

I (^) repays £101 to the bank

I (^) Social surplus generated by the financial contract

I (^) £110-£100=£

Course Overview - What is finance?

I (^) Financing investment

I (^) Financial market ( equity, bond, etc)

I (^) Financial intermediaries ( loan )

I (^) Sharing risk through financial institutions

I (^) Mutual funds in the financial market

I (^) Commercial banks

I (^) Overcoming asymmetric information

I (^) Moral hazard: hidden knowledge of actions

I (^) Lemon: hidden knowledge of attributes

Course Overview - Finance and Macroeconomics

I (^) Fostering growth

I (^) reduce capital misallocation

I (^) Contributing to fluctuations over the business cycle

I (^) financial crisis

I (^) economic boom

I (^) Key: incompleteness of the financial market and resulting

inefficiency

Productivity of an economy

I (^) Suppose that the production function is an economy is

f ( k |{z}

capital

, l |{z}

labor

|{z}

other input

I (^) Imagine that the society produces output using an aggregate

production function

Y

|{z}

Total Output

= Af ( K |{z}

Total Capital Supply

, L

|{z}

Total Labor Supply

where A is the total-factor productivity (TFP)

I (^) But the production of the society is operated by many

individual firms

Y =

X

i

Ai f (ki , li ,.. .)

So the allocation of resources may affect the productivity of

the economy.

Resource allocation and productivity: examples

Productivity

A ⌘

P

i

Ai f (ki , li ,.. .)

f (K , L,.. .)

Suppose the production function is f (k) = k.

I (^) Under utilization of capital.

There is only one firm in the economy but the demand of

capital from the firm is K . So, the output of the economy

is K .Then the productivity of the economy is 1

K

I (^) Allocation of resources across firms.

Suppose there are N firms. There is no financial market to

reallocate capital. Firm i produces Ai ki units of output. The

productivity of the economy is

N X

i= 1

Ai

ki P

j

kj

Equilibrium

I (^) An equilibrium is consumption of the household c, firms’ labor

demand, l 1 , l 2 , firms’ profit ⇡ 1 , ⇡ 2 and wage w , such that

I (^) Given w , l i solves firm^ i’s problem.

I (^) The labor market clears 1 = l 1 +^ l 2.

I (^) c = ⇡ 1 +^ ⇡ 2 +^ w^.

Equilibrium characterization

I (^) From the firm’s optimization problem, l i =^ ki

h Ai ( 1 ↵)

w

i 1 /↵

I (^) From the market clearing condition, l 1 +^ l 2 =^ 1. We have

w =

X

i

ki [Ai ( 1 ↵)]

1 /↵

↵

I (^) The output of the economy is

X

i

Ai k

↵ i

l

1 ↵

i

X

i

Ai k

↵ i

ki A

1 /↵

i P

j

kj A

1 /↵

j

A

1 ↵

I (^) TFP of the economy (let K ⌘

P

i

ki )

P

i

Ai k

↵

i

l

1 ↵

i

K

↵

X

i

Ai

ki

K

ki A

1 /↵

i P

j

kj A

1 /↵

j

A

1 ↵

Social planner’s solution

maxY 1 ,Y 2 ,K 1 ,K 2

Y (Y 1 , Y 2 ) =

Y

1 1

+ Y

1 2

1

subject to Y 1 = A 1 K 1 , Y 2 = A 2 K 2 ,

K 1 + K 2  K ,

K 1 , K 2 0.

Lagrangian

L = (A 1 K 1 )

1 (^) + (A 2 K 2 )^

1 (^) + (K K 1 ^ K 2 )

First order condition:

@L

@K 1

K 1

(A 1 K 1 )

1 (^)

@L

@K 2

K 2

(A 2 K 2 )

1 (^)

K 1

K 2

A 1

A 2

1

Optimal capital allocation

K 1

K 2

A 1

A 2

K 1 = K

A

1 1

A

1 1

+ A

1 2

K 2 = K

A

1 2

A

1

1

+ A

1

2

Decentralization of the optimal allocation

I (^) Two types of firms, producing intermediary goods of type one

and type two

I (^) A market for renting capital

I (^) The representative household owns the capital stock

Utility maximization of the household

maxY 1 ,Y 2

Y (Y 1 , Y 2 ) =

Y

1 1

+ Y

1 2

1

subject to P 1 Y 1 + P 2 Y 2 = RK + ⇡ 1 + ⇡ 2

Pi = Y

1 i

Y

1 1

+ Y

1 2

1

Normalize the price of the general good so that

Pi (Yi ) = CY

1

i

where C =

Y

1 1

+ Y

1 2

1

.

Definition of equilibrium

An equilibrium is interest rate R and capital allocation K 1 and K 2

such that

I (^) Given interest rate R, K 1 and^ K 2 solve the profit maximization

problem firm 1 and firm 2

I (^) The demand function solves the representative agent’s

consumption given any price, P(Yi )

I (^) Capital market clears

K 1 + K 2 = K

Equilibrium condition

From the profit maximization problem of firms

K 1 = A

1

1

R

K 2 = A

1 2

R

K 1

K 2

A 1

A 2

1

Using the market clearing condition K 1 + K 2 = K

K 1 = K

A

1

1

A

1

1

+ A

1

2

K 2 = K

A

1

2

A

1

1

+ A

1

2

The decentralized equilibrium implements the first best allocation.