Question

1) Consider the production function for gummy bears for the following questions. Justify your answers.

Q= LK+L= L(K+1), (MPL=K+1, MPK=L)

a) Calculate this firm’s isoquant.

b) Calculate capital demand for this firm.

c) Calculate labour demand for this firm.

d) Calculate total cost for this firm. Simplify.

e) Calculate average cost (simplify) for this firm.

Answer 1

An isoquant is a curve that shows all the combinations of inputs (K, L) that yield the producer the same level of output. The shape of an isoquant reflects the ease with which a producer can substitute among inputs while maintaining the same level of output. Therefore, an isoquant represents a constant quantity of output.

The absolute value of the slope of the isoquant through the input pair (K, L) is called the marginal rate of technical substitution (MRTS) between input 1 and input 2 at (K, L). ... For a typical production function, with isoquants is convex to the origin, the MRTS diminishes as more of input 1 is used.

Marginal Rate of Technical Subsititution is the rate at which a firm must subsitutute one input for another in order to keep a production at a given level. It is –slope of isoquant

MRTS is ratio of Change in Capital and Change in Labour

It’s the rate at which capital can be exchanged for 1 more or less of labour.

MRTS     = marginal rate of technical substitution

               = the rate at which a firm must substitute one input for another in order to keep production at a given level

= - slope of isoquant

= the rate at which capital can be exchanged for 1 more (or less) unit of labor

MP_K        = the marginal product of K  

MP_L        = the marginal product of L  

Q  = LK + K

where, MPL  = K + 1 & MPK  = L

MRTS [l,k] = MPL/MPK  = K+1/L is the isoquant of the firm

Assuming Q as zero i.e. constant as discussed before,

LK  = - L

therefore, K  = -L/L  = -1  

Capital Demand of the Firm is -1

Labour Demand of the Firm is as follows:

MPL is K + 1 where K = -1

therefore, MPL is -1+1  = 0 i.e. L  = 0

thus, lablour demand is zero

Total Cost of the Firm: L+K is 0-1 is -1

Simplified as in MPL/MPK  = 0 as per isoquant curve

i.e. [K+L]/L  = 0

L  = K+1

L+K  = -1

e. Average Cost of the Firm:

AC  = TC/Q

TC  = L + K  = 0-1  = -1

Q  = 0 based on isoquant curve

AC  = -1/0  = 0

To simplify this, AC  = TC/Q

Since Q  = L [K+1]

TC  = L + K

since, AC is zero

therefore, L+K / L[K+1]  = 0

thus, L + K = 0

Hence Proved !!