Answer
a)
each firm: $L = (200-W)/4$
market: $L_{market} = 1000-5W$
b) $W = 160$, each firm hires 10 workers and makes 200 dollars profit
c) income to workers increases to 320 dollars, and the profit for owners increases to 400 dollars
d) wages fall to 80 dollars per worker, and the profit decreases to 100 dollars per owner
Work Step by Step
a)
$Q = 100*L-L^2$
$MPL = 100-2L$
value of $MPL = P * MPL$
$VMPL = 2 * (100-2L)$
$VMPL = 200-4L$
Since the firms are in a perfectly competitive market, they set the value of the marginal product of labor equal to the wage.
$W = VMPL$
$W = 200-4L$
$W+4L-W=200-4L+4L-W$
$4L = 200-W$
$4L/4 = (200-W)/4$
$L = (200-W)/4$
There are 20 firms in the market, and each firm has the same labor demand.
$L = (200-W)/4$
$L_{20} = 20*(200-W)/4$
$L_{20} = 5 *(200-W)$
$L_{20} = 1000-5W$
$L_{market} = 1000-5W$
b)
$L_{market} = 1000-5W$
$200 = 1000-5W$
$200+5W = 1000-5W+5W$
$200 +5W = 1000$
$5W =800$
$5W/5 = 800/5$
$W = 160$
$L = (200-W)/4$
$L = (200-160)/4$
$L = 40/4$
$L = 10$
$Q = 100*L-L^2$
$Q= 100*10-10^2$
$Q = 1000 -100$
$Q = 900$
price * quantity - cost of labor
$2 * 900 - 10*160$
$1800-1600$
$200$
c)
$VMPL = 4*(100-2L)$
$W = 400-8L$
$W+8L-W = 400-8L+8L-W$
$8L = 400-W$
$8L/8 = (400-W)/8$
$L_{firm} = (400-W)/8$
$L_{firm}*20 = L_{market}$
$L_{market} = 20*(400-W)/8$
$L_{market} = (8000-20W)/8$
$L_{market} = 1000-2.5W$
$200= 1000-2.5W$
$200+2.5W = 1000-2.5W+2.5W$
$200+ 2.5W = 1000$
$200+ 2.5W-200 = 1000-200$
$2.5W = 800$
$2.5W/2.5 = 800/2.5$
$W = 320$
$L_{firm} = (400-W)/8$
$L_{firm} = (400-320)/8$
$L_{firm} = (80)/8$
$L_{firm} = 10$
$Q = 100*L-L^2$
$Q = 100*10-10^2$
$Q = 1000-100$
$Q = 900$
price * quantity - cost of labor
$4*900 - 10*320$
$3600-3200$
$400$
d)
$MPL*2=100-2L$
$MPL*2/2 = (100-2L)/2$
$MPL = 50-L$
$W = 2*(50-L)$
$W = 100-2L$
$W+2L-W = 100-2L+2L-W$
$2L = 100-W$
$2L/2=(100-W)/2$
$L_{firm} = (100-W)/2$
$L_{firm}*20 =L{market}$
$L_{market} = 20*(100-W)/2$
$L_{market} =1000-10W$
$200= 1000-10W
$200-200+10W = 1000-10W-200+10W
$10W = 800$
$10W/10 = 800/10$
$W = 80$
$L_{firm} = (100-W)/2$
$L_{firm} = (100-80)/2$
$L_{firm} = (20)/2$
$L_{firm} = 10$
$Q = (100*L-L^2)/2$
$Q = (100*10-10^2)/2$
$Q = (1000-100)/2$
$Q = 900/2$
$Q = 450$
price * quantity - cost of labor
$450*2 - 10*80$
$900 - 800$
$100$