Earl was selling lemonade before - the total cost to produce y gallons of lemona

Question

Earl was selling lemonade before - the total cost to produce y gallons of lemonade, C(w_1, w_2, y) = 2W^0.5_1 w^0.5_2y^1.5 where w_1 was the price of input factor 1 (price of lemon) and W2 was the price of the input factor 2 (wage rate). If w_1 = w_2 = 1 and the price of lemonade is p, what is Earl's marginal cost function? What is his supply function? What is his supply function if w_1 =4 and w_2 = 9 instead? In general, Earl's marginal cost depends on and At prices w_1 and W_2, what is his marginal cost to produce y units of lemonade? What is his supply function then?

Explanation / Answer

If lemons cost $1 per pound, the wage rate is $1 per hour, and the price of lemonade is p, Earl’s marginal cost function is

MC= dTC/q=dTC/y =d(2w11/2w21/2y3/2)/y

MC(y)=3y1/2

and his supply function is S(p)=p2/9. (see below for derivation)

If lemons cost $4 per pound and the wage rate is $9 per hour, his supply function will be

S(p)=p2/324.

b)

In general, Earl’s marginal cost depends on the price of lemons and the wage rate. At prices w1 for lemons and w2

for labor, his marginal cost when he is producing y units of lemonade is MC(w1,w2,y)= 3w11/2w21/2 y1/2.

The amount that Earl will supply depends on the three variables, p ,w1,w2. As a function of these three variables, Earl’s supply is

S(p,w1,w2)=p2/9w1w2

MC= P for profit maximization

Thus, 3w11/2w21/2 y1/2=P

Squaring both sides

P2 = 9w1w2y

Supply function=y=P2/9w1w2

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