Heterogeneity and growth, donated by Rodolfo Manuelli Consider an economy populated by a large number of households indexed by i. The utility function of household i is Where 0 i0 > 0. Let total capital in the economy at time t be denoted kt and assume that total labor is normalized to 1. Assume that there are a large number of firms that produce output using capital and labor. Each firm has a production function given by F(k, n) which is increasing, differentiable, concave and homogeneous of degree one. Firms maximize the present discounted value of profits. Assume that initial ownership of firms is uniformly distributed across households. a. Define a competitive equilibrium. b. i) Economist A argues that the steady state of this economy is unique and independent of the ui functions, while B says that without knowledge of the ui functions it is impossible to calculate the steady-state interest rate. ii) Economist A says that if k0 is the steady-state aggregate stock of capital, then the pattern of “consumption inequality” will mirror exactly the pattern of “initial capital inequality” (i.e., ki0), even though capital markets are perfect. Economist B argues that for all k0, in the long run, per capita consumption will be the same for all households. Discuss i) and ii) and justify your answer. Be as formal as you can. c. Assume that the economy is at the steady state. Describe the effects of the following three policies. i) At time zero, capital is redistributed across households (i.e., some people must surrender capital and others get their capital). ii) Half of the households are required to pay a lump sum tax. The proceeds of the tax are used to finance a transfer program to the other half of the population. iii) Two thirds of the households are required to pay a lump sum tax. The proceeds of the tax are used to finance the purchase of a public good, say g , which does not enter in either preferences or technology.
Heterogeneity and growth, donated by Rodolfo Manuelli Consider an economy populated by a large number of households indexed by i. The utility function of household i is
Where 0 i0 > 0. Let total capital in the economy at time t be denoted kt and assume that total labor is normalized to 1. Assume that there are a large number of firms that produce output using capital and labor. Each firm has a production function given by F(k, n) which is increasing, differentiable, concave and homogeneous of degree one. Firms maximize the present discounted value of profits. Assume that initial ownership of firms is uniformly distributed across households.
a. Define a competitive equilibrium.
b. i) Economist A argues that the steady state of this economy is unique and independent of the ui functions, while B says that without knowledge of the ui functions it is impossible to calculate the steady-state interest rate. ii) Economist A says that if k0 is the steady-state aggregate stock of capital, then the pattern of “consumption inequality” will mirror exactly the pattern of “initial capital inequality” (i.e., ki0), even though capital markets are perfect. Economist B argues that for all k0, in the long run, per capita consumption will be the same for all households. Discuss i) and ii) and justify your answer. Be as formal as you can.
c. Assume that the economy is at the steady state. Describe the effects of the following three policies.
i) At time zero, capital is redistributed across households (i.e., some people must surrender capital and others get their capital).
ii) Half of the households are required to pay a lump sum tax. The proceeds of the tax are used to finance a transfer program to the other half of the population.
iii) Two thirds of the households are required to pay a lump sum tax. The proceeds of the tax are used to finance the purchase of a public good, say g , which does not enter in either preferences or technology.
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