Firm heterogeneity

Question 1

Facts about firm heterogeneity

Answer 1

There is heterogeneity in:

• Size
• Productivity
• • correlation between size and productivity
• Richer countries tend to have larger firms

Question 2

Lucas model setup

Answer 2

Households

• Single representative household
  
  • Household has a continuum of members of size 1
• HH members differ in the managerial ability z
• Log preferences discounted by rho

Firms

• Organized by an enterpreneur with ability z`
  
  • Notice that previous models assumed a representative firm
• Production requires both labor and managerial ability
• Decreasing returns to scale in capital and labor => MPK depends on firm size
• Firms are price takers and use the same tecnology

Question 3

FOC of the firms implications

Answer 3

• All managers choose the same combination of inputs
• Managers with higher ability run larger firms
• Managers with higher ability obtain more profits
• Factor shares are constant and independent of z
• Capital labor shares are the same for all managers
  
  • It only depens on the share parameters and the ratio of prices
• Since all firms pay the same wage and z increases the productivity of labor, firms with higher z demand more labor

Question 4

Factor demands and output

Answer 4

• More productive managers run firms at larger scale because their higher z compensates the DRS
• Both are linear in z
• Output and profits are also linear in z

Question 5

Household problem

Answer 5

• Household has two decisions:
  
  • Consumption
  • Occupation of its members
• There is perfect insurance: all the members of the household consume the same
• There is a threshold above of which all HH members become managers

Question 6

HH budget constraint

Answer 6

I is non-financial income

The second term corresponds to financial income

Question 7

Solution to HH problem

Answer 7

Remember we need:

• Equation for consumption
• Equation for the occupational decision

We obtain the same type of Euler equation

Question 8

Equilibrium def

Answer 8

Question 9

Measure of average firm size

Answer 9

G is the CDF of the managerial ability

Question 10

Equilibrium productivities

Answer 10

• Firms with higher TFP are larger
• But average labor productivity is the same across all firms (because all firms equalize its MLP to the same wage

Question 11

Stationarity

Answer 11

In equilibrium:

• Threshold
• Distribution of fims sizes

are time independent

Question 12

Aggregation

Answer 12

Equilibrium can be characterized by an aggregate representive firm

Question 13

Steady state

Answer 13

• equilibrium with constant aggregate capital =>
  
  • aggregate consumption and production are constant
  • w and r are also constant

Question 14

Leibniz’s rule

Answer 14

Question 15

Conclusions of the model when allowing technology and population growth

Answer 15

• Per capita variables grow at the rate of technology
• r constant, w grows at the rate of technology
• threshold is constant
• fraction of enterpreneurs and firm size distribution are constant
• number of firms growths at the rate of growth of population