SUMMER TERM 2023
CENTRALLY MANAGED ONLINE EXAMINATION
ECON0028: ECONOMICS OF GROWTH
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QUESTIONS
Answer any 5 questions from Section A and all questions from Section B.
Questions in Part A carry 10 per cent of the total mark each, questions in Part B carry 25 per cent of the total mark each (5 per cent for each sub-question).
PART A
Answer any 5 questions from this section.
A1 Consider a Cobb-Douglas production function Y = Y/1 K1一Y LY where K is capital, L is labor and 0 < γ < 1 is a parameter. What are the returns to scale? Show your reasoning
mathematically. Assume that the economy is competitive. Calculate the wage. What is the labor share of income? Describe briefly (in no more than two short sentences) the evolution of the labor share in the United States since World War II. Does this evidence validate or invalidate the predictions of models with Cobb-Douglas production structure?
A2 Consider two economies characterized by the basic Solow model with no population growth and no exogenous productivity growth. The production function is Y = KCL1一C in both economies, with standard notation. The two economies are identical in all respects, except they experience different depreciation rates, perhaps because they use different kinds of machinery. Economy A has capital depreciation rate of 5% per year. Economy B has capital depreciation rate of 10% per year. Derive the formula for the steady state income per capita in an economy. With α = 1/3, how much richer or poorer in per capita terms is economy A in steady state, relative to economy B?
A3 Consider the Solow model with exogenous technological progress that we covered in class. Trace out graphically the impact of a permanent increase in the growth rate of technological progress. Make sure to carefully label the horizontal axis on your diagram. Are the citizens of this economy better off as a result of this change? Explain your answer.
A4 Is the following statement true or false? Explain your answer. You may find it helpful to draw a simple diagram to illustrate.
The Solow model predicts that all countries will ultimately converge to the same steady state level of income per worker, and that the poorest countries tend to grow faster than the richest: there is cross-country convergence.
A5 Consider a basic Malthusian model discussed in class, where technology level B is constant (as a reminder, the model was described by the following equation: = B β − c,
where L is population, X is land in fixed supply, β is a parameter, and c is subsistence level of income). Using a diagram, describe the effects of a discovery of a new technique to fertilize soil. Make sure to discuss what happens to the standards of living in the short-run and in the long-run.
A6 Consider the simplified original Romer (1990) model, in which the ideas production function is A = LA × A, where LA = SA × L is the number of workers employed in the R&D sector. Assume that there is no population growth (L is constant) and treat the share SA as a (policy-determined) parameter. Calculate the long-run (BGP) growth rate of TFP. Then describe the effects of a government policy of subsidizing basic research through increased university funding, which increases the share of the population that decides to take up an academic career (so that more people are involved in idea generation). Comment briefly (max two sentences) on the plausibility of the conclusions of your analysis in light of the evidence on the idea generation process in the real world.
A7 Consider the following model of growth and environment. There is a representative household with preferences U = u(Y) + v(R) where Y is output and R is the state of the environment. Functions u and v are strictly increasing and strictly concave. Output is produced using energy and labor, and burning energy harms the environment: Y = BEYL1-Y , R = R(− 1) − E. R(− 1) is the starting state of the environment. B is the level of technology which grows exponentially over time. Derive the optimal energy intensity of production (i.e. write a utility maximization problem with E as a choice variable. Take the first order condition and solve for the energy intensity). What does this model predict about how attitudes towards the environment change as the economy grows? [Hint: to answer this question you must first consider what the concavity assumption implies for the derivatives of u and v.]
A8 Fill in the blanks:
Suppose that the world economy consumes a constant fraction of total energy resources that are available inside the planet every year. This implies that the amount of energy used over
time ………… . If this was the end of the story, the relative price of energy would tend to
……….. . Instead, we observe relative energy prices ………… over longer periods of time,
except during well-identified energy crises. This is particularly puzzling because the elasticity
of substitution between energy and other inputs is likely quite …………… . One solution to
the puzzle is that there is …………. ……………… ……………. . [use as many words as you
like for the final part].
A9 Consider a Romer model with the ideas production function given by A = LA(λ)Aφ where
λ ∈ (0,1) and ϕ < 1. The λ parameter denotes a duplication externality – perhaps having many people doing research means that scientists unnecessarily duplicate effort / arrive at the same results. Derive the BGP growth rate in this economy assuming that population growth is n. Discuss (in words only – no math needed) what this new externality means for the equilibrium employment in the R&D sector in relation to an allocation that a social planner would choose.
A10 Consider an industry in an economy with interest rate equal to zero and with constant population. A firm that is currently the industry leader is making profits of $1million per year, and it is estimated that these profits represent only around half of consumer surplus from selling the product. Innovation from a competing firm arrives with a constant probability of 10% each year. When a competitor innovates successfully, the incumbent producer is replaced. Calculate the private value (i.e. an expected profit stream) of coming up with a new product that is 5% better than the existing product, in the sense that it makes 5% more profit and increases consumer surplus by 5%. Then calculate the social value of such an invention. Comment briefly (one sentence max) on this difference. Which phenomenon is at play here?
PART B
Answer all questions from this section.
B1 Consider an economy with a neoclassical production function and labor augmenting technological progress, so that output is Y(t) = F EK(t), A(t)L(t)F, where K(t) is capital, L(t) is labour and A(t) is the level of labour-augmenting technology adopted at time t (you are welcome to drop the (t) notation in your answer if you want). The level of adopted technology is not the same as the cutting edge of technology because adoption of newly invented technologies is slow. Specifically, the knowledge frontier is T(t) ≥ A(t). The frontier T(t) grows exogenously at a constant exponential rate λ, so that T(t) = e λtTo where To is given and can be normalized to 1. Adoption of technology depends on the gap between the current level of technology and the frontier, as well as on educational attainment and human
capital of the workforce ℎ , so that ideas adopted each period are given by: A t) =
ϕ(ℎ)ET(t) − A(t)F where function ϕ(h) satisfies ϕ > 0, ϕ, > 0.
(a) Write down a differential equation that characterizes the evolution of T(t) based on the information provided (this equation should feature ). Which of the models covered in this course is able to deliver such an evolution of the frontier of technology as an endogenous (equilibrium) outcome?
(b) Briefly interpret the equation A t) = φ(ℎ)ET(t) − A(t)F and the restrictions φ >0, φ , > 0.
(c) Write down an equation that characterizes the behavior of the growth rate of adopted technology A(t). Then assume that the economy is on a balanced growth path with a constant level of ℎ. Calculate the BGP growth rate of adopted technology A(t). Explain carefully how you reach your answer. Hint: if you get stuck, you may find it helpful to do the final sub-question (e)first.
(d) Derive the expression for the technological gap, , on the balanced growth path, as a function of exogenous λ and ϕ(ℎ). How does the gap depend on the level of human capital ℎ? How does it depend on λ? Provide some economic intuition.
(e) For this sub-question, do not assume the economy is on a BGP. Plot the two differential equations you obtained in (a) and (c), with the technology gap on the horizontal axis. Mark the point which denotes the BGP of the economy, and draw the arrows denoting the transition dynamics starting from above and below the BGP level of the gap.
B2 Consider the Romer model with idea production function A = θLAAΦ with θ > 0 and ϕ < 1. A is the stock of ideas, LA is the number of researchers. Population L grows at rate n. Final output is produced with labor and with a range of intermediate inputs by a perfectly competitive final goods sector, with technology Y = Kc xi(1) 一adi where K is the economy’s
capital stock. Each intermediate input xi is produced by a firm who holds a patent for that particular variety, granting it the exclusive perpetual rights to produce this variety. Patents are produced in the R&D sector, according to the idea production function above.
(a) For this sub-question only, suppose that α = 0. How substitutable are the varieties? What can you say about the mark-up that monopolistically competitive intermediate producers will be able to charge in equilibrium? Consequently, what can you say about the incentives to innovate in this economy and ultimately the equilibrium long-run growth rate? You do not need to do any math; succinct economic reasoning is sufficient.
(b) From now on assume that a ∈ (0,1). Write down the profit maximization problem of the representative final goods producer, denoting the price of each variety with pi and the rental rate of capital with T. Make sure to specify what the choice variables are. Then derive the first order conditions.
(c) Intermediate producers employ workers to manufacture their products. The
production function is xi = λli , where xi is the output of producer i, li is labor employed by this producer, and λ is a labor productivity parameter, common across all producers. Each worker earns a wage w.
What are the returns to scale in the production of each variety? What is the marginal cost of each producer? Write down the cost function of a producer of variety i. Finally, write down the labor market clearing condition, denoting the labor supply in this economy with L. (Hint: this condition states that the supply of labor is equal to demand for labor in the economy – when doing this recall that workers are also employed in the R&D sector).
(d) Since all intermediate firms charge the same prices and face identical demand curves, they will also produce the same quantity in equilibrium. Using this fact, derive the equilibrium aggregate production function. Denote labor employed in the entire intermediate sector with Lx . Hint: if total labor employed in that sector is Lx and there is measure A of firms, how many workers work in each firm?
(e) Suppose that aside from the endogenous profit driven innovation (that increases A), there is also exogenous growth in worker ability, so that = gλ , where gλ > 0 is a parameter.
Define total factor productivity in this economy. What is the growth rate of total factor productivity on the balanced growth path? Assuming that a constant fraction of output is saved and invested, as in the Solow model, what is the growth rate of income per capita on the BGP?