ECON60662 Economic Growth

Q1. An infinitely lived agent maximizes his lifetime utility with consumption c in each time period. Her elasticity of marginal utility of consumption is θ, and the rate of time preference isp. The government imposes a proportional income tax τ, and a proportional consumption tax σ, and provides a lump-sum subsidy ϕ . The firms produce output with a production function y  = AKα , where y is output per capita and k is capital per capita.

(a) Write the optimisation problem for the agent in her lifetime.

(b) Solve the above problem to obtain the growth rate of consumption. Explain the conditions necessary to generate long run growth.

(c) What is the effect on the growth rate of an increase in either of the following variables: τ, σ, and ϕ . Explain the results.

(d) If the subsidy ϕ is proportional on output, how will it change the results above?

Q2. In an overlapping generations economy, a two-period-lived agent born at time tis endowed with 1 unit of time, and does the followings in her lifetime:

I. at her young age, t:

i. She spends lt(t) amount of time on work for output production with the state of technology At : yt(t) = Atlt(t)                                                                                           (2.1)

ii. She spends the rest of her time for gaining leisure.

iii. She gains utility from consuming ct(t) , enjoying leisure.

II. at her old age, t+1:

i. She spends lt(t)+1  amount of time on work for output production with the state of technology

At+1:

yt(t)+1  = At+1lt(t)+1                                                                                                               (2.2)

ii. She spends the rest of her time for gaining leisure.

iii. She gains utility from consuming ct(t)+1, enjoying leisure.

Note that β ∈ (0, 1) is the weight on future consumption and leisure compared to their current ones, andy > 0 is the weight on leisure over consumption in the same period.

(a) Write the amount of time she allocates for leisure at her young and old age. (b) Write the optimisation problem for the agent in her lifetime.

(c) Solve the above problem to find the optimal labour supplies in the first and second periods of agent’slife.

(d) Suppose that technology evolves according to At+1  =  (1 + Lt )At, where Lt  is total labour supply at time t. Assuming that the population of agents is N, derive an expression for the growth of technology.