USA debt $(20$ points$)$

In class, I mentioned that the returns on investment were higher in foreign af$-$

filiates of US multinationals than in US affiliates of Foreign multinationals. This

exercise proposes that the US comparative advantage in R&D may explain these

patterns. Suppose that statistical agencies calculate the returns on investment of

an establishment $i$ as: ${?}^1$

$\frac{y_i{p}_i-C_i}{K_i+A_i}$

where $y_i$ is the quantity produced, $p_i$ is the price, $C_i$ is the total cost, $K_i$ is the stock

of capital invested in the establishment and $A_i$ is the stock of R&D investments

in the establishment. We refer to this return as the "accounting returns" $($which

is different from the returns on investment in the problem of the firm in part $($c$)).$

$($a$)(4$ points$)$ Use this expression to explain how investments in R&D in the

USA headquarters may affect the accounting returns of affiliates of US firms

in foreign countries. Specify which variables may change and why.

$($b$)(4$ points$)$ Use the same expression to explain how investments in R&D in

the USA affiliates of a foreign multinational affects its accounting returns.

Again, specify which variables may change and why.

$($c$)(7$ points$)$ Denote with $K=(K_1,K_2,$dots,$K_{\text{I}})$ the vector of capital that a firm

uses in each location of its establishments, $i=1,$dots,I, and with $A=(A_1,$dots,$A_{\text{I}})$

as the vector of investments in $R$&$D.$ Assume that the firm chooses its op$-$

timal R&D investment and capital in each location solving the following

problem:

$,ma{x}_{K,A}{\displaystyle \underset{i}{\overset{?}{}}}{y}_i{p}_i-w_i{L}_i-r{K}_i-r_{Ai}{A}_i$

subject $to$: $F_i(L_i,K_i,A)=y_i$

$,p_i=D^{-1}(y_i,A)$

where $F_i$ is the production function of the firm in country $i,$ and $D^{-1}$ is the

inverse demand function that the firm faces. Interpret each line of the prob$-$

lem above. In particular, highlight and explain economically the difference

between $K_i$ and $A_i$ in the problem.

$($d$)(5$ points$)$ Do the different accounting returns in $($a$)$ and $($b$)$ imply that for$-$

eign and US multinationals are not optimally allocating investments in R&D and capital? Use part $($c$)$ to substantiate your answer to this question.