Question

1. To produce ice cream cones, Marcus requires capital K and labour L. Neither by itself produces good ice cream cones. Suppose that the production technology can be captured by the production function q=20L^0.5K^0.5, where q is number of traps, MP_L = 10L^-0.5K^0.5, and MP_K =10 L^0.5K^-0.5.
   1. What can you say about the returns to scale for this production function?
   2. What is the equation of the isoquants?
   3. What is the equation for a slope of an isoquant? What is this called? What does it indicate?
   4. Set up the cost minimization problem and solve for the conditional capital and labour demands as functions of w (the labour costs), r (the capital costs), and q (number of ice cream cones).
   5. What is the equation of the expansion path? Discuss your findings.
   6. Discuss the demand functions you derived in d). Are production inputs the normal inputs? What happens to optimal amount labour as w increases? What happens to optimal amount of capital as r decreases?
   7. Now, assume that the cost of the labour is $10 while cost of capital is $40. What is your “optimal production plan”?
   8. Derive the cost function and simplify the function as much as you can.
   9. What is your “optimal production plan” if you wish to produce 100 ice cream cones?
   10. What is the cost of producing 100 ice cream cones?

Now suppose your investor fixes capital to 10 units in short run.

11. What is your “optimal production plan” with fixed capital if you wish to produce 100 ice cream cones?
    12. What is the short-run cost function?

Answer 1

Answer -1

Given production function-

q = 20 L^0.5 K^0.5

To find the returns of scale of the given production function,we multiply the each inputs of the production function by a positive constant , say t.

On transformation, we get

q(tK, tL) = 20 (tL)^0.5 (tK)^0.5

q(tK, tL) = 20 t ^0.5 L^0.5 t^0.5 (K)^0.5

q(tK, tL) = 20 t^(0.5+0.5) L^0.5 K^0.5

q(tK, tL) = 20 t L^0.5 K^0.5

q(tK, tL) = t q(K,L)

Thus, on multiplying each input by t , the production function gets exactly multiplied by t. Hence, production function exhibits CONSTANT RETURNS TO SCALE (CRS).

B)

Equation for isoquant-

Isoquant represents various combinations of two inputs that can be used to produce a particular level of output.

Lets write the given production function in terms of K

K^0.5 = q/(20L^0.5)

Squaring both sides -

K= q^2 / (400.L)

The above equation represents the equation for an isoquant for a particular level of output say q

For q= 10

The equation for isoquant can be written as

K= 100/400L

or K = 0.25L

C)

Given—

MPL = dq/dL = 10 K^0.5 L^(-0.5)

MPK = dq/dK = 10 K^(-0.5)L^0.5

we can find the slope of isoquant as a ratio of marginal product of inputs.

Slope of isoquant = MRTS = MPL/MPK

MRTS = {10 K^0.5 L^(-0.5) } / {10 K^(-0.5)L^0.5 }

MRTS = K/L .... Slope of isoquant

Marginal Rate Of Technical Substitution ( MRTS) is known as the slope of isoquant. It shows how a firm substitutes between two inputs to produce a particular level of output. As we move along the isoquant curve, MRTS diminishes. In particular, MRTS decreases as the firm substitutes labor for capital along the isoquant. In other words, as the firm substitutes labor for capital, the same amount of capital needs to be replaced by ever larger amounts of labor in order to keep output constant.

D)

In order to establish the relationship between the prices of inputs and the quantities of inputs used by the firm, we set up the cost - minimising problem which will give us the demand function of labour and capital as functions of output (q) , wage (W) and rent (R) .

Kindly refer image -1