References and Literature

The large literature documenting productivity and technology differences across firms and the patterns of technology diffusion were discussed in Section 18.1 and the relevant references can be found there.

The model in Section 18.3 is inspired by Howitt (2000), but is different in a number of important respects. First, Howitt uses a model of Schumpeterian growth rather than the baseline expanding input variety model used here. This difference is not important, and the choice here was motivated to simplify the exposition. Second, Howitt uses a model without scale affects. Since our interest here is not with scale effects, the added complication necessary to remove scale effects was deemed unnecessary.

The ideas of appropriate technology discussed in Section 18.4 have a long pedigree. Many development economists in the 1960s realized the importance of the issues of appropriate technology. The classic work here is Stewart (1977), though similar ideas were also discussed in Salter (1966) and David (1974). A classic treatment was provided Atkinson and Stiglitz (1969), who suggested a simple and powerful formalization of how technological change can be localized and thus not transfer from one productive units to another (or from one country to another). Atkinson and Stiglitz’s idea is incorporated into a growth model by Basu and Weil (1998), which was the basis of one of the models in Section 18.4. The last part of this section draws on Acemoglu and Zilibotti (2001), who develop a model of appropriate technologies due to skill differences across countries and combine it with directed technical change to show how there will be a bias towards technologies inappropriate to the needs of poorer nations. Acemoglu and Zilibotti also provide evidence that these effects could be quantitatively large and patterns of sectoral differences are consistent with the importance of this type of technology-skill mismatch. Acemoglu (2002b) shows that technology-skill mismatch applies in a more general model of directed technical change than the one in Acemoglu and Zilibotti (2001) discussed here (see Exercise 18.26).

Finally, the model presented in Section 18.5 draws upon Acemoglu, Antras and Helpman (2007). A number of other models also generate endogenous productivity or technology differences across countries as a result of differences in the organization of production. Some of these will be discussed in Chapter 21.

18.6. Exercises

Exercise 18.1. Derive equation (18.1).

Exercise 18.2. Show that if the restriction that λj ∈ [0,g) in Section 18.2 is relaxed, the requirement that Aj (t) ≤ A (t) can be violated.

Exercise 18.3. Derive equation (18.4).

Exercise 18.4. Complete the proof of Proposition 18.1.

Exercise 18.5. Derive the effect of an increase in λj on the law of motion of aj (t) and kj (t). How does this differ from the effect of an increase in σj ? Explain why these two parameters have different effects on technology and capital stock dynamics.

Exercise 18.6. In the model of Section 18.2, show that if g = O, then all countries converge to the same level of technology. Explain carefully why g > 0 leads to steady-state technology differences, while these differences disappear when g = O.

Exercise 18.7. (1) Set up the world equilibrium problem in subsection 18.2.2 as one

in which the Second Welfare Theorem holds within each country. Under this as­sumption, carefully define an equilibrium path. Explain the significance of this assumption.

(2) Now set up the world equilibrium problem without appealing to the Second Welfare Theorem. Explain why the mathematical problem is identical to that in part 1 of this exercise.

(3) Prove Proposition 18.2.

Exercise 18.8. (1) Why is the condition ρ — nj > (1 — θ) g necessary in Proposition

18.3?

(2) Complete the proof of Proposition 18.3.

Exercise 18.9. In the model of Section 18.2 with consumer optimization, suppose that preferences in country j are given by

where ρj differs across countries.

(1) Show that a unique steady-state world equilibrium still exists and all countries grow at the rate g.

(2) Provide an intuition for why countries grow at the same rate despite different rates of discounting.

(3) Show that this steady-state equilibrium is globally saddle-path stable.

Exercise 18.10. * Consider the model of Section 18.2 with F corresponding to the produc­tion function of an individual firm j (with a slight abuse of notation) and (18.3) corresponding to the law of motion of the technology of the firm, with σj = σ (hj), where hj is the average human capital of the workers of firm j and σ is a strictly increasing and differentiable func­tion. To simplify the discussion, suppose that each firm employs a single worker (which is without loss of any generality given constant returns to scale).

(1) Derive the wage of the worker of human capital hj. [Hint: this consists of the workers value of marginal product in production plus the increase in the productivity of the firm because of the improvement in the firm’s technology due to the higher human capital of the worker].

(2) Show that an increase in g (at any point t) will increase worker wages. Derive the implications of changes in g on the returns to human capital. Contrast an increase in the returns to human capital driven by an increase in g with those discussed in Chapter 15.

Exercise 18.11. Complete the proof of Proposition 18.4.

Exercise 18.12. Consider the model in subsection 18.3.1 and suppose that all countries have the same labor force size Lj = 1 and the same ηj = η, and only differ in terms of their ζj’s. Imagine that the range of ζj’s is the same as used in the quantitative evaluation of the neoclassical growth model in Chapter 8.

(1) Evaluate the impact of these differences in ζj ’s on cross-country technology and income differences for different values of φ.

(2) What value of φ is necessary so that a fourfold difference in ζj’s translates into a thirtyfold difference in income per capita?

(3) How would you interpret the economic significance of such a value of φ? Would this be a satisfactory model of cross-country technology and income differences? If yes, explain why it is more attractive than the neoclassical model and other alternatives

we have seen so far. If not, suggest what important features are missing and how they might be introduced.

Exercise 18.13. * Consider the model in subsection 18.3.1. Suppose that preferences are

Show that an equivalent of Proposition 18.4, with a unique globally saddle-path stable world

equilibrium where all countries grow at the same rate, applies.

Exercise 18.14. Show that (18.14) is necessary and sufficient for a positive world growth rate in the model of subsection 18.3.2. Write down the conditions that characterize the world

equilibrium when this condition is not satisfied.

Exercise 18.15. Prove Proposition 18.5.

Exercise 18.16. * Analyze the local dynamics of the steady-state world equilibrium in Proposition 18.5.

Exercise 18.17. * Consider Proposition 18.5 with the discount rates, the ρj's differing across countries. Prove that a unique steady-state world equilibrium, with all countries growing at the same rate, still exists.

Exercise 18.18. In the model of subsection 18.3.2, replace equation (18.12) with

where G is homogeneous of degree 1.

(1) Generalize the results in Proposition 18.5 to this case and derive an equation that determines the world growth rate implicitly.

(2) Derive an explicit equation for the world growth rate for the specific case in which N (t) = max_j∙ Nj (t).

Exercise 18.19. In the model of subsection 18.3.2, there is a strong scale effect.

(1) Show that if population grows at some constant rate nj > 0 in each country, there will not exist a steady-state equilibrium.

(2) Construct a variation of this model along the lines of the semi-endogenous growth models of Section 13.3 in Chapter 13, where this strong scale effect is removed. [Hint:

(3) Provide a full characterization of the steady-state world equilibrium in this case.

Exercise 18.20. Consider the model in subsection 18.4.2. Suppose that the world consists of two countries with constant and equal populations, and constant savings rates

Suppose that the production function in each country is given by (18.15) with k' correspond­ing to the highest capital-labor ratio in any country experienced until then. There is no technological progress and both countries start with the same capital-labor ratio.

(1) Characterize the steady-state world equilibrium (that is, the steady-state capital­labor ratios in both countries).

(2) Characterize the output per capita dynamics in the two economies. How does an increase in γ affect these dynamics?

(3) Show that the implied income per capita differences (in steady state) between the two countries are increasing in γ. Interpret this result.

(4) Do you think this model provides a good/plausible mechanism for generating large income differences across countries? Substantiate your answer with theoretical or empirical arguments.

Exercise 18.21. Complete the proof of Proposition 18.6. In particular, explicitly derive the expression for the threshold and the skill premium.

Exercise 18.22. Derive the equilibrium expressions (18.20)-(18.23).

Exercise 18.23. Prove Proposition 18.7. [Hint: in steady state the profits from owning a skill-complementary and unskilled labor-complementary machine must be equal].

Exercise 18.24. Prove Proposition 18.8.

Exercise 18.25. Consider the model of appropriate technology in subsection 18.4.3.

(1) Suppose that now research firms can sell their machines to all producers in the world, including those in the South and can charge the same markup. Derive the steady-state equilibrium under these conditions.

(2) Comparing your answer in part 1 to the analysis in the text, derive the implications of intellectual property rights enforcement in the South on equilibrium technologies? What are the implications on income per capita differences between the North and the South?

(3) In view of your answer to 1 and 2 above, could it be the case that Southern economies prefer lack of intellectual property rights enforcement to full intellectual property rights enforcement? [Hint: distinguish between a world in which there is a single Southern country versus one in which there are many].

Exercise 18.26. * Instead of the multi-sector model in subsection 18.4.3, suppose that out­put is given by an aggregate production function of the form

as in Chapter 15, with Yl and Yh being produced exactly is in that chapter. Assume as in the model of subsection 18.4.3 that new technologies are developed in the North for the Northern market only.

(1) Characterize the steady-state (balanced growth path) equilibrium of this economy. [Hint: use exactly the same analysis as in Chapter 15 and subsection 18.4.3].

(2) Show that if σ ? ε — (ε — 1) (1 — β^) is equal to 2, the results are identical to those in subsection 18.4.3.

(3) Derive the equivalents of Proposition 18.8.

(4) Do the implications of inappropriate technologies become more or less important when σ increases?

Exercise 18.27. Prove Proposition 18.9.

Exercise 18.28. Prove Proposition 18.11.

Exercise 18.29. Prove Proposition 18.12.

Exercise 18.30. * Consider the model of Section 18.5. Suppose that there is a total popula­tion of L. Assume that each individual can work as a supplier for one of the M products, or he can work in the process of technology adoption. For this reason, suppose that the cost of technology adoption is given by Γ (N) ? wΓo (N), where w is the wage rate, corresponding to the outside option of each supplier.

(1) Characterize the general equilibrium of the economy by endogenizing A for a given

number of products M. In particular, show that in equilibrium the following market clearing condition must be satisfied:where N* is the equilibrium

technology choice (number of suppliers).

(2) What is the effect of an increase in μ on N*? Explain the result.

(3) Now suppose that the M products differ according to their elasticity of substitution, in particular, each product has a different α, with the distribution of α's across products given by a distribution function G (α) with support within the interval [0,1]. Let N* (α) be the equilibrium technology choice (number of suppliers) for a product with parameter α. Show that the market clearing condition now takes the form:

(4) What is the effect of an increase in μ on the equilibrium in this case?

(5) How would you endogenize Q in this model? What types of insights would this generate?

Exercise 18.31. Consider the model of Section 18.5. What types of organizational forms might emerge when contracting institutions are imperfect (i.e., μ is very low)? In particular, discuss how vertical integration and repeated interactions between suppliers and producers might change the results discussed in that section. How would you model each of these?