# 3.3 Optimal sequencing of interventions

From the previous analysis, it is clear that a government can rely on di¤erent instruments to ...ll the investment gap. The question that follows is which instrument has to be preferred and under what circumstances. This is what we analyze next, under the assumption that the government can perfectly tailor policies according to the characteristics fxi; �jg of any speci...c project k. In other words, the government can implement reforms that only apply to projects with given values of xi and �j, and the same is true for subsidies and public investments.<sup>8</sup> In what follows, we characterize the allocation of instruments that maximizes total welfare. We denote such allocation as the optimal policy allocation, OPA. Lemma 1 below fully characterizes such an allocation as a function of the the degree of private sector advantage (xi) and the externalities (�j) associated with each speci...c project.

Lemma 1 (1):If reforms are not e¢ cient enough (� <<sup><u>1</u></sup> c<sup>):(i)if�jissu¢cientlylow,noinvest-</sup> ment inducing policy is welfare improving; (ii) for su¢ ciently high values of �j, and low values of xi, it is optimal for the government to implement public projects; (iii) for su¢ ciently high values of �j and xi, subsidies are instead optimal.

(2): If reforms are e¢ cient enough (� �<sup><u>1</u></sup> c<sup>):(i)if�jissu¢cientlylow,noinvestmentinducingpol-</sup> icy is welfare improving; (ii) for high �j and low xi, it is optimal for the government to implement public projects; (iii) for projects with intermediate �j and su¢ ciently high xi, reforms are optimal; while (iv) for su¢ ciently large values of �j and xi, subsidies are optimal.

## Proof: See Appendix.

� Figure 3a summarizes our results for di¤erent values of .<sup>9</sup> The ...rst, easy, takeaway from the ...gure is that, no matter how e¢ cient reforms are, when dealing with high externalities/low private sector advantage-type of projects, the best option is public ...nancing. When private sector advantage is high, instead, the best option is a subsidy. The reason for the optimality of a subsidy is that, when xi is large, a very small subsidy is su¢ cient to crowd in private investors. Hence, b the costs of the subsidy are of second order (si tends to zero when xi tends to c � 1), while those associated with public investments and reforms are always of ...rst order (except in the limit cases in which xi = 0, or �= 1). Thus, while there are always situations in which public investments or subsidies are optimal, for reforms to have a place in an OPA, they should be e¢ cient enough, that is, we need that � ><sup><u>1</u></sup> c<sup>.E¢ciencyhereismeasuredagainstthecostscassociatedwiththe</sup> subsidy. In addition, the appeal of reforms weakens in the presence of large externalities because of the deadweight loss associated.

A pretty trivial, but important point, often forgotten in the cascade debate, is that

> 8 For public investments this is always the case since the government decides which projects to ...nance.

> 9 Also in this ...gure, we set c = 3=2. Hence, in the ...rst two panels of Figure 3a, we have that � < <u>1c</u><sup>,and��</sup><sup><u>1</u></sup> c in the last one.

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