Exercises

Exercise 15.1. Deriveeq. (15.1).

Exercise 15.2. Complete the proof of Proposition 15.1. In particular, verify that in any BGP, (15.27) must hold and derive the equilibrium growth rate as given by (15.29).

Exercise 15.3. Prove Proposition 15.2. [Hint: use (15.9) to show that when N_h (0) /Nl (0) does not satisfy (15.27), (15.20) and (15.21) cannot both hold as equalities].

Exercise 15.4. Derive equation 15.30.

Exercise 15.5. Explain why in Proposition 15.1 the effect of γ on the BGP growth rate, (15.29), is ambiguous. When is this effect positive? Provide an intuition.

Exercise 15.6. Derive equation 15.31.

Exercise 15.14. Characterize the Pareto optimal allocation in the model with knowledge spillovers and state dependence (Section 15.4). Show that the relative technology ratio in the stationary Pareto optimal allocation no longer coincides with the BGP equilibrium. Explain why this result differs from that in Section 15.3.

(1) Define the equilibrium and BGP allocations.

(2) Specify the free-entry conditions for each machine variety.

(3) Characterize the BGP equilibrium, show that it is uniquely defined and determine conditions such that the growth rate is positive and the transversality condition is satisfied.

(4) Show that the equivalents of Propositions 15.3 and 15.4 hold in this environment.

(5) Characterize the transitional dynamics and show that they are similar to those in Proposition 15.2.

(6) Characterize the Pareto optimal allocation in this economy and show that the Pareto optimal ratio of technologies in the stationary equilibrium are also given by (15.27).

Exercise 15.19. * Consider a version of the baseline directed technological change model introduced above with the only difference that technological change is driven by quality improvements rather than expanding machine varieties. In particular, let us suppose that the intermediate goods are produced with the production functions:

(6) Characterize the transitional dynamics and show that they are similar to those in Proposition 15.2.

(7) Characterize the Pareto optimal allocation in this economy and show that the Pareto optimal ratio of technologies in the stationary equilibrium are also given by (15.27).

(8) What are the pros and cons of this model relative to the baseline model studied in Section 15.3?

Exercise 15.20. As a potential application of the models of directed technological change, consider the famous Habakkuk hypothesis, which claims that technology adoption in the US economy during the 19th century was faster than in Britain because of relative labor scarcity in the former (which increased wages and encouraged technology adoption).

(1) First, consider a neoclassical-type model with two factors, labor and technology, F (A,L), where F exhibits constant returns to scale and A is a technology term, chosen by each firm with costs Γ (A) in terms of the final good. Assume that Γ is continuous, differentiable, strictly increasing and convex. Show that an increase in wages (caused by a decline in labor supply or an exogenous increase in wages because of the minimum wage) cannot increase A.

(2) Next, consider the directed technological change model studied in this chapter with H interpreted as land and assume that N∏ is fixed (so that there is only R&D for increasing Nl). Show that if σ > 1, the opposite of the Habakkuk hypothesis ob­tains. If in contrast, σ < 1, the model delivers results consistent with the Habakkuk hypothesis. Interpret this result and explain why the implications are different from the neoclassical model considered in part 1 above.

Exercise 15.21. Consider the baseline model of directed technological change in Section 15.3 and assume that it is in steady state.

(1) Show that in steady state the relative price of the two intermediate goods, p, is proportional to (H∕L)^β.

(2) Now assume that the economy opens up to world trade, and faces a relative price of intermediate goods p' < p. Derive the implications of this for the endogenous changes in technology. Explain why the results are different from those in the text. [Hint: relate your results to the price effect].

Exercise 15.22. (1) Prove Proposition 15.13. In particular, show that in any BGP

equilibrium (15.37) must hold, and that this equation is inconsistent with capital accumulation.

(2) Prove that there exists no equilibrium allocation in which consumption grows at the constant rate. [Hint: show that a relationship similar to (15.37) must hold, and this will lead to an increase in Nk (t), which then implies that the interest rate cannot be constant].

Exercise 15.23. Deriveeq. (15.47).

Exercise 15.24. Deriveeq. (15.49).

Exercise 15.25. * Complete the proof of Proposition 15.14 and show that there cannot exist any other CGP equilibrium.

Exercise 15.26. * Show that if σ < 1, the CGP equilibrium in Proposition 15.14 is globally stable. Show that if σ > 1, it is unstable. Relate your results to Proposition 15.7.

Exercise 15.27. Now let us use the results of Proposition 15.14 to revisit the discussion of the experiences of continental European economies provided in Blanchard (1997).

Exercise 15.28. * The analysis in the text has treated the supply of the two factors as exogenous and looked at the impact of relative supplies on factor prices. Clearly, factor prices can also affect relative supplies. This exercise looks at the joint determination of relative supplies and technologies.

Let us focus on a model with the two factors corresponding to skilled and unskilled labor. Suppose a continuum υ of unskilled agents are born every period, and each faces a flow rate of death equal to υ, so that population is constant at 1 (as in Section 9.8 above). Each agent chooses upon birth whether to acquire education to become a skilled worker. For agent x it takes K_x periods to become skilled, and during this time, he earns no labor income. The distribution of K_x is given by the distribution function Γ(K) which is the only source of heterogeneity in this economy. The rest of the setup is the same as in the text. Suppose that Γ(K) has no mass points. Define a BGP as a situation in which H/L and the skill premium remain constant.

(1) Show first that in BGP, there is a single-crossing property: if an individual with

cost of education K_x chooses schooling, another withmust also acquire

skills. Conclude from this that there exists a cutoff level of talent, K, such that all

do not get education.

(2) Show that, along BGP relative supplies take the form:

Explain why such a simple expression would not hold away from the BGP.

(3) How would you determine[Hint: agent with talent K has to be indifferent between acquiring skills and not].

Show that as υ → 0, the relative supply of skills as a function of the skill premium becomes

where r* and g* refer to the BGP interest-rate and growth rate.

(4) Determine the BGP skill premium by combining this equation with (15.27) and

(15.30). Can there be multiple equilibria? Explain the intuition.

with δ < 1 and S_n (t) and S_m (t) are the number of scientists allocated to the two types of goods with S_n (t) + S_m (t) ≤ S. Denote the wage of scientists at time t by ω (t). A firm that invents a new good becomes the monopolist producer, but can be displaced by a new monopolist who finds a way of producing the good using unskilled labor.

(1) Denote the unskilled wage by w (t) and the skilled wage by v (t). Show that, as long as v (t) is sufficiently larger than w (t), the instantaneous profits of a monopolist producing skill-intensive and labor-intensive goods at time t are

Interpret these equations. Why is the condition that v (t) is sufficiently larger than w (t) necessary?

(2) Assume moreover that a firm that undertakes R&D to replace the skill-intensive good has an equal probability of replacing any of the existing n — m skill-intensive goods. Define a BGP as an allocation where n and m grow at the same rate g. Show that this implies that output and wages of skilled and unskilled workers must grow at the rate g/ (ε — 1). [Hint: use the equation for the numeraire setting the price of the final good equal to 1 at each date].

(3) Show that on the BGP the wages of scientists also grow at the same rate as the wages of skilled and unskilled workers.

(4) Show that the BGP must satisfy the following condition

where μ ? m/n. [Hint: note that a monopolist producing a labor-intensive good will never be replaced, but a monopolist producing a skill-intensive good faces a constant