$1.(30$ Points$)$ State whether six $(6)$ of the following statements are true or false and explain why. Your answers should be short and clear. You may use graphs where necessary. Only true or false answers will not receive points.

$($a$)$ If labor and capital are perfect complements in production, the firm's short$-$run supply curve will be a vertical line.

$($b$)$ When wages fall, long$-$run labor demand may increase or decrease depending on whether capital and labor are substitutes or complements.

$($c$)$ When wages fall, the firm will increase production in the long run, but we cannot say for sure whether long$-$run capital use will increase or decrease.

$($d$)$ If all firms are the same, an increase in market demand will not affect prices in the long run.

$($e$)$ As a result of the fall in the price of capital, the number of firms in the sector may increase or decrease in the long run.

$($f$)$ If all individuals in a consumer group have the same preferences, this group can be considered to act as a single representative consumer.

$($g$)$ According to the first welfare theorem, market equilibrium is always efficient.

$($h$)$ In a sector consisting of firms all having the same cost structure, consumer surplus will be zero in the long$-$run equilibrium.

$($i$)$ Any distribution resulting from a trade that is profitable for both consumers in an Edgeworth box is efficient.

$($i$)$ According to the second welfare theorem, any point inside the Edgeworth box can be brought into market equilibrium.