# Financial Intermediary¶

Domestic household wealth, $$W^d_{t}=B_{t}$$ and foreign ownership of domestic assets $$W^f_{t}$$ are invested in a financial intermediary. This intermediary purchases a portfolio of government bonds and private capital in accordance with the domestic and foreign investor demand for these assets and then returns a single portfolio rate of return to all investors.

Foreign demand for government bonds is specified in section Government bond market clearing of the Market Clearing chapter:

(63)$D^{f}_{t+1} = D^{f}_{t} + \zeta_{D, t}(D_{t+1} - D_{t}) \quad\forall t$

This leaves domestic investors to buy up the residual amount of government debt:

(63)$D^{d}_{t} = D_{t} - D^{f}_{t} \quad\forall t$

We assume that debt dominates the capital markets, such that domestic investor demand for capital, $$K^{d}_{t}$$ is given as:

(64)$K^{d}_{t} = B_{t} - D^{d}_{t} \quad\forall t$

Foreign demand for capital is given in Private capital market clearing, where $$K^{f}_{t}$$ is an exogenous fraction of excess capital demand at the world interest rate:

(65)$K^{f}_t = \zeta_{K, t}ED^{K,r^*}_t \quad\forall t$

The total amount invested in the financial intermediary is thus:

$\begin{split}W_{t} & = W^d_{t} + W^f_{t} \\ & = D^d_t + K^d_t + D^f_t + K^f_t \\ & = D_t + K_t\end{split}$

Interest rates on private capital and government bonds differ. The return on the portfolio of assets held in the financial intermediary is the weighted average of these two rates of return. As derived in (38) of Section Positive Profits from Government Infrastructure Investment, the presence of public infrastructure in the production function means that the returns to private factors of production ($$r_t$$ and $$w_t$$) exhibit decreasing returns to scale.1 It is assumed that competition ensures a zero profit condition among firms and the returns to public infrastructure through the returns of firms are captured by the financial intermediary and returned to share holders. The return on capital is therefore the sum of the (after-tax) returns to private and public capital.

(66)$r_{K,t} = r_{t} + (1 - \tau^{corp}_t )MPK_{g,t}\left(\frac{K_{g,t}}{K_t}\right)$

The return on the portfolio of assets held by the financial intermediary is thus a weighted average of the return to government debt $$r_{gov,t}$$ from (52) and the adjusted return on private capital $$r_{K,t}$$ from (66).

(67)$r_{p,t} = \frac{r_{gov,t}D_{t} + r_{K,t}K_{t}}{D_{t} + K_{t}} \quad\forall t$

## Footnotes¶

1

See also for a similar treatment of government infrastructure investment, positive profits, and returns to the owners of capital.