Consider an all$-$equity firm that is currently worth $$600,000$ with $10,000$ shares outstanding. Today, management came across a new potential investment. This investment would cost $$40,000$ today. In one year, the investment would pay off $$50,000.$ The relevant one$-$year discount rate is $7\%.$

$($a$)$ Should the firm undertake the new investment? If the firm completely finances the investment by issuing new equity, what is the value of the firm? How many new shares would need to be issued? Are the $$old$$ shareholders $($shareholders BEFORE issuing new equity$)$ better off? If so$,$ by how much? What is the evolution of the price per share $($i$.$e$.,$ what is the price per share initially, after the announcement of the new investment opportunity, after the new equity issuance, and a year from now$)?$ Assume perfect capital markets.