Analyze the effect of an increase in the growth rate of technology 𝑔 coupled with an increase in 𝛿𝐻 on the steady state levels of ℎ̃ ∗ and 𝑘̃ ∗in this case. Compare your results with part (a).

Question 1 – 10 marks
Consider the Solow model with human capital as presented in chapter 6 of the text. Recall that we can analyze the model using a phase diagram constructed with the following two steady state equations:
𝑘̃ ∗ = ( 𝑠𝐾ℎ̃𝑡
𝜑
𝑛 + 𝑔 + 𝑛𝑔 + 𝛿) 1
1−𝛼
ℎ̃ ∗ = ( 𝑠𝐻𝑘̃𝑡
𝛼
𝑛 + 𝑔 + 𝑛𝑔 + 𝛿)
1
1−𝜑

a) Use a phase diagram to analyze the effect of an increase in the investment rate in human capital 𝑠𝐻 on the steady state levels of 𝑘̃ ∗ and ℎ̃ ∗.

b) Now suppose that 𝑠𝐾 + 𝑠𝐻 = 𝑠̅ where 𝑠̅ denotes the total savings rate of households to invest in either physical or human capital. If we assume that 𝑠̅ is fixed, use a phase diagram to analyze the effect of an increase in the investment rate in human capital 𝑠𝐻 on the steady state levels of 𝑘̃ ∗ and ℎ̃ ∗. How does your answer compare with part a)?

c) Recall that the steady state level of output per capita in this model is given by
𝑦𝑡
∗ = ( 𝑠𝐾
𝑛 + 𝑔 + 𝑛𝑔 + 𝛿)
𝛼
1−𝛼−𝜑( 𝑠𝐻
𝑛 + 𝑔 + 𝑛𝑔 + 𝛿)
𝜑
1−𝛼−𝜑

If 𝑠𝐾 + 𝑠𝐻 = 𝑠̅ , derive the optimal value of 𝑠𝐻 as a function of 𝑠̅ . Under what conditions should 𝑠𝐻 exceed 𝑠𝐾? (Hint: convert the above expression into logarithms and then derive a first order condition with respect to 𝑠𝐻.

Question 2 – 10 marks
Consider the Solow model with human capital as presented in chapter 6 of the text. In this case, we allow for the possibility that the depreciation rates on physical and human capital are different, so that:
𝑘̃ ∗ = ( 𝑠𝐾ℎ̃𝑡
𝜑
𝑛 + 𝑔 + 𝑛𝑔 + 𝛿𝐾
) 1
1−𝛼
ℎ̃ ∗ = ( 𝑠𝐻𝑘̃𝑡
𝛼
𝑛 + 𝑔 + 𝑛𝑔 + 𝛿𝐻
)
1
1−𝜑

a) Suppose initially that the two depreciation rates are identical, so that 𝛿𝐾 = 𝛿𝐻 = 𝛿. Use a phase diagram to illustrate the effect of an increase in the technology growth rate 𝑔 on the steady state levels of physical and human capital per effective unit of labour.

b) A potentially unattractive side effect of an increase in technology growth 𝑔 is an increase in 𝛿𝐻. For example, one could imagine that as the pace of technological change increases, existing skills and knowledge depreciate faster. Analyze the effect of an increase in the growth rate of technology 𝑔 coupled with an increase in 𝛿𝐻 on the steady state levels of ℎ̃ ∗ and 𝑘̃ ∗in this case. Compare your results with part (a).

c) In class, we assumed that α = ψ = 1/3, so that 1/3 of national income goes to physical capital, 1/3 to human capital and 1/3 to unskilled labour. Suppose instead that α = 1/3, but that ψ = 2/3 so that all labour income goes to human capital and none goes to unskilled labour. Illustrate the resulting phase diagram carefully, and discuss how the results of this model differ relative to the model discussed in class.

Question 3 – 5 marks
Consider a version of the Solow model with a diminishing productive resource such as oil, described in section 7.2 of the text. In this case the production function is given by
𝑌𝑡 = 𝐾𝑡
𝛼(𝐴𝑡𝐿𝑡)𝛽𝐸𝑡
𝜖
where 𝐸𝑡 is the amount of oil used in production in period 𝑡. Note that the stock of oil remaining in period 𝑡 + 1 is then given by 𝑅𝑡+1 = 𝑅𝑡 − 𝐸𝑡, where 𝐸𝑡 = 𝑠𝐸𝑅𝑡. We also assume that 𝛼 + 𝛽 + 𝜖 = 1.

a) Show that the growth in GDP per capita in this case is given by
𝑔𝑦 = 𝛽
𝛽 + 𝜖 𝑔 − 𝜖
𝛽 + 𝜖 𝑛 − 𝜖
𝛽 + 𝜖 𝑠𝐸

b) One of the problems with assuming that a resource diminishes over time is that new sources and/or substitutes can be discovered, which reverses the perceived drain in the initial resource. This can be modeled in a simple way by assuming that 𝑠𝐸 can take on a negative value. At what (negative) value for 𝑠𝐸 will growth in per capita GDP be the same as in the base model with technological progress (ie, the model of chapter 5)?