Question

The consumption preference function in an economy is defined as: U(c, l) = A ln(c) + B ln(l)

There is a proportional tax on profits earned by consumers, the consumer's budget constraint function is defined as: c = wN + (1 − τ )π

The production function of the firm is defined as: Y = zKαN 1−α

1. What is the consumer's optional budget of consumption and leisure as a function of f w, τ , π, A, B and h?

2. What is the firms profit function? solve for the firm's optimal choice of labour as a function of z, K, α, and w.

Answer 1

Utility function or consumption preference function is given as,

U(c,l) = Aln(c) + Bln(l)

Also, there is a proportional tax on profits earned by consumers and consumer's budget constraint function is defined as, c = wN + (1 - T) .

let, total no. of hours are there say h, where leisure (l) is determined by l = h - N where N is labour supply.

Then, the budget constraint becomes c = w(h - l) + (1 - T).

1. The problem becomes, Max U = Aln(c) + Bln(l)

Subject to, c = w(h - l) + (1 - T)

Using Lagrange Multiplier,

L = U(c,l) + [ w(h - l) + (1 - T) - c]

L/c = A/c - = 0.......(i)

L/l = B/l - w = 0......(ii)

L/ = w(h - l) + (1 - T) - c = 0....(iii)

Divide (i) and (ii),

Al/Bc = 1/w

l = Bc/Aw

From (iii), wh - w[Bc/Aw] + (1 - T) - c = 0

wh - Bc/A + (1 - T) - c = 0

c(1 + B/A) = wh + (1 - T)

c* = [wh + (1 - T)] / (1 + B/A)

Thus, l* = [B [wh + (1 - T)] / (1 + B/A)] / Aw

l* = [AB [wh + (1 - T)] / (A + B)] / Aw

l* = [B [wh + (1 - T)] / (A + B)] / w

Hence, the consumer's optimal budget set of c* is [wh + (1 - T)] / (1 + B/A) and l* is [B [wh + (1 - T)] / (A + B)] / w.

2. The Production function is Y = zKN

The Total cost of the firm is wN + rK where r is rental price and K is capital.

Firm's profit function = pY - (wN + rK) where p is price of output.

   = p[ zKN ] - wN - rK

The first order condition is,

   / N = p( 1 - )KN - w = 0

N= w / [ p( 1 - )K]

N* = [ w / [ p( 1 - )K] ]  

Hence, firm's optimal choice of labour is N* = [ w / [ p( 1 - )K] ] .