Question

10 20 30 40 Labor 21.4 (0) Earl sells lemonade in a competitive market on a busy street corner in Philadelphia. His production function is f( ) = where output is measured in gallons, T is the number of pounds of lemons he uses, and is the number of labor hours spent squeezing them. Me (a) Does Earl have constant returns to scale, decreasing returns to scale. or increasing returns to scale?_ decreasing TOY Where w, is the cost of a pound of lemons and wis the wage rate for lemon-squeezers, the cheapest way for Earl to produce lemonade is to se WoW 7 hours of labor per pound of lemons. (Hint: Set the slope of his isoquant equal to the slope of his isocost line.) (c) If he is going to produce y units in the cheapest way possible, then the number of pounds of lemons he will use is (W1, W2,y) =_ and the number of hours of labor that he will use is c2(w), w2, y) =__ (Hint: Use the production function and the equation you found in the last part of the answer to solve for the input quantities.) (d) The cost to Earl of producing y units at factor prices wi and wą i c(W1, W2, y) = W121(W1, W2, y) + W2X2(W1, W2, y) = 21.5 (0) The prices of inputs (11, 12, 13, 14) are (4,1,3,2).

Answer 1

Solution:

Production function is given as: f(x1, x2) = x1^1/3x2^1/3

a) Returns to scale can be identified by raising all inputs by a common factor:

f (t*x1, t*x2) = (t*x1)^1/3(t*x2)^1/3

f (t*x1, t*x2) = t^1/3+1/3*(x1^1/3x2^1/3)

f (t*x1, t*x2) = t^2/3(x1^1/3x2^1/3) = t^2/3*f(x1, x2)

Since, by changing all inputs by factor t, output changes by a lower factor of t^2/3, the function exhibits decreasing returns to scale.

b) Given the wage rates, slope of isocost line = w1/w2

Slope of isoquant can be identified as: marginal product of x1/marginal product of x2

MPx1 = = (1/3)*x1^1/3-1x2^1/3 = (1/3)x1^-2/3x2^1/3

af(x1, x2)/2.21

MPx2 = = (1/3)*x1^1/3x2^1/3-1 = (1/3)x1^1/3x2^-2/3

af(x1, x2)/8x2

So, slope of isoquant = [(1/3)x1^-2/3x2^1/3]/[(1/3)x1^1/3x2^-2/3] = x2/x1

Thus, at optimal point (that is cheapest way of producing lemonade, it must be that: w1/w2 = x2/x1

So, hours of labor per pound of lemons, x2/x1 = w1/w2

c) From above, we already have seen that at cheapest way of production, x2/x1 = w1/w2

So, x2 = (w1/w2)*x1

Then, substituting this in the production function we get: y = x1^1/3*((w1/w2)*x1)^1/3

y = (w1/w2)^1/3*x1^2/3

x1 = (y/(w1/w2)^1/3)^3/2 = (w2/w1)^1/2y^3/2

So, x2 = (w1/w2)*((w2/w1)^1/2y^3/2) = (w1/w2)^1/2y^3/2

d) Total cost to Earl, c(w1, w2, y) = w1*x1 + w2*x2

c(w1, w2, y) = w1*(w2/w1)^1/2*y^3/2 + w2*(w1/w2)^1/2*y^3/2

c(w1, w2, y) = (w1*w2)^1/2y^3/2 + (w1*w2)^1/2y^3/2

c(w1, w2, y) = 2*(w1*w2)^1/2y^3/2