Term 1 Lecture 4

Question 1

Single output case of a firm’s production

Answer 1

Q < f(z), it’s feasible to produce q using if the production function satisfies this equation
- technically efficient when q = f(z), so the greatest possible output is being produced given the inputs

Question 2

General formulation of a firm’s output

Answer 2

• extend concept to a more complex production scenario using a production plan, or a net output vector - y
• inputs are negative and outputs are positive

Question 3

Transformation function:

Answer 3

Feasibility of a production plan represented by f(y) < 0
- for single output case, function becomes f(q,-z)
Q - f(z)
- f(y) = 0 is the transformation frontier, the boundary between feasible and infeasible production plans

Question 4

Properties of production:
- possibility of shutdown
- monotonicity

Answer 4

1. 0 output is possible, so f(0) < 0
2. In an efficient production process, increasing the output of one good requires reducing the output of another, same for inputs.
   - Df/Dzi >/ 0 for specific inputs
   - DF/dyi >/ 0

Question 5

MRT

Answer 5

Rate at which one net output can be transformed into another while keeping the firm on its transformation frontier
- tradeoff between different outputs and inputs while technical efficiency is maintained
- differentiate f(y) = 0 with respect to yi to get MRT

-((DF/Dyi)/(DF/Dyj))

Question 6

Marginal product, MPi

Answer 6

Rate at which any input can be transformed into output, equal to slope of the production function with respect to input
- Df/Dzi
- additional output produced from an incremental increase in input zi

Question 7

MRTS

Answer 7

Rate at which one input must be increased as another is decreased to maintain the same level of output
- slope of the isoquant

Question 8

Convexity of input requirement sets

Answer 8

Convexity implies the set of input combinations is well-behaved, so firms can flexibly combine different inputs to achieve the same output
- in terms of isoquants: Convexity means that isoquants are convex to the origin, leading to a decreasing MRTS as you move along the isoquant
- diminishing utility of subbing one input for another more and more

Question 9

Elasticity of substitution

Answer 9

Measures how easily one input can be subbed for another while maintaining same level output
- captures curvature of isoquants and indicates responsiveness of input choices to changes in economic incentives
- formula expresses how the ratio of inputs changes in response to changes in MRTS between them
Oij = -((Dln(zi/zj)/(Dln(MRTSij)))

Question 10

Homotheticity

Answer 10

If input vectors which produce the same level of output remain proportional when scaled
- if f(Za) = f(Zb), then f(kZa) = f(kZb)
- means isoquants at higher levels are scaled versions of isoquants at lower levels
- MRTS are constant along rays along the origin

Question 11

Homogeneity:

Answer 11

All homogenous functions are homothetic, but not vice versa
- homogenous of degree a if f(Kz) = K^a.f(z)

Question 12

Returns to scale

Answer 12

DRS:
- scaling up inputs leads in a less than proportionate increase in output, Kf(z) > f(kz), for k>1
- implied by concavity
IRS:
- scaling up inputs leads to a more than proportionate increase in output, Kf(z) < f(Kz) for K > 1
- implied by concavity of production function
CRS:
- scaling up inputs leads to a proportionate increase in output, homogeneity of degree 1

Question 13

Perfect substitutes in production key properties:
- q = f(z1,z2) = G(az1 + bz2)
- MP, MRTS, isoquants, Oij, RS, homothetic

Answer 13

• MPz1 = a… and MPz2 = b…, both CONSTANT
• MRTS = a/b
• isoquants are parallel straight lines
• elasticity of substitution is infinite: perfect substitutability
• CRS
• homothetic

Question 14

Perfect complements production function
- q = f(z1,z2) = min(az1, bz2)

Answer 14

MP is only positive if that input is the bottleneck
- MPz1 = a, if az1 < bz2, otherwise 0
- MPz2 = b if az1 > bz2, otherwise 0
L shaped isoquants, where corner lies along the ray through the origin with slope a/b
- CRS, if both inputs are scaled up by the same proportion, output increases by same
- homoethetic

Question 15

Cobb-Douglas production function
- q = f(z1,z2) = Az1^a.z2^b

Answer 15

• MPz1 = aq/z1
• MPz2 = bq/z2
• MRTS = MPz1/MPz2 = az2/bz1, diminishing MRTS -> convex input requirement sets
• oij = 1, constant
• homothetic as MRTS only depends on the ratio between z1 and z2
• RS: IRS if a+b > 1, etc

Question 16

CES production function
- q = f(z1,z2) = [Az1^a + Bz2^b]^b

Answer 16

• MP
• MRTS = (A/B)(z1/z2)^a-1
• o = 1/1-a
• homothetic - MRTS only depends on z1/z2, so homothetic
• IRS if ab > 1, etc

Question 17

How to know if production is not homothetic

Answer 17

MRTS is not constant along rays
Is not homogenous of degree 0 in income and prices

Question 18

Firm profit FOC and SOC, link to MRTS

Answer 18

MPi = wi/p
- where w is an input price
- second order condition ensures a concave production function
- taking the FOC for two inputs and eliminating p, we get:
(Df/Dzi)/(Df/Dzj) = wi/wj, so MRTS equals the price ratio

Question 19

Unconditional input demand and supply functions

Answer 19

• solving profit maximisation problem for input demands can be written as D(p,w), decreasing in wi
• putting these demands into the production function f(z) will give output as S(p,w), the firm’s supply function, increasing in p
• both homogenous of degree 0, so only relative prices matter

Question 20

Comparative statics and monotonicity

Answer 20

ChangeP.Changeq - ChangeW.ChangeZ >/ 0
- so if p, output price increases, then output q must also increase or stay the same
- if an input price wi rises, while output price p remains constant, then demand for that input zi must fall.

Question 21

Profit function including supply and demand

Answer 21

pS(p,w) - w’D(p,w)
- homogenous of degree one
- increasing the price of output cannot decrease profit
- increasing the price of an input can not increase profit

Question 22

Hotellings lemma from profit function

Answer 22

Deriving the profit function with respect to output price gives the supply function

Derivative of profit with respect to input prices gives input demand function.

Question 23

Symmetry

Answer 23

Second derivatives of the profit function are symmetric

Impact of a price change of one input on another input’s demand Di must be identical to the impact of a price change in wi on demand for Dj

Impact of output price on input demand is equal and opposite to the impact of input price on output supply

Question 24

Generalised profit maximisation problem

Answer 24

Max r’y subject to F(y) < 0
- r is the price vector of net outputs
- y is the net output vector
- F(y) represents technological constraints on production
- homogenous of degree 1
- differentiating with respect to price gives the supply rule
Dpi/dri = yi, positive if firm is a seller

Question 25

Cost minimisation problem

Answer 25

Min w’z subject to q \< f(z) - given an output level q, firms choose inputs z to minimise costs - cost minimisation is a necessary condition for profit maximisation After taking FOC, MRTSij = wi/wj

Question 26

Conditional input demands

Answer 26

H(q,w) represent input quantities which minimise cost for a given output level q and input prices w - differs from unconditional input demand as here, q is fixed - homogeneity of degree 0 - monotonicity: changeW.changeZ \< 0, so if one input price increases, then demand for that input must decrease

Question 27

Cost function

Answer 27

Minimum cost required to produce a given level of output q using the optimal combination of inputs - C(q,w) = w’H(q,w), where w is the vector of input prices - cost function is increasing in input prices - homogeneity of degree 1

Question 28

Shephard’s Lemma linking cost function and input demand

Answer 28

DC/Dwi = Hi(q,w) - partial derivative of the cost function with respect to an input price wi equals the conditional input demand - symmetry, the way input demand Hi responds to the price of another input wj is symmetric.

Question 29

Relationship between conditional and unconditional input demands:

Answer 29

D(p,w) = H(S(p,w),w) - unconditional input demand is derived from the conditional input demand at the firm’s profit maximising supply - input price changes affect demand through two channels, direct substitution effects from cost min and indirect output adjustments from profit max

Question 30

Responsiveness of input demands to changes in output

Answer 30

If technology is homothetic, then input choices scale proportionally with output - implies cost shares of different inputs remain constant.

Question 31

Returns to scale and cost functions

Answer 31

CRS: - homogenous and homothetic, so cost function is proportional to output - C(q,w) = qk(w) DRS: - impossible to scale production plans up and multiplying output by k>1 will multiply costs by more than k IRS: - possible to scale production plans up multiplying output by k>1 will multiply costs by less than k

Question 32

Relation between marginal and average costs:

Answer 32

MC>AC, AC is rising MC=AC, AC is at a stationary point, usually the MES

Question 33

Profit maximisation model: Pq - C(q,w)

Answer 33

- FOC: P = MC - SOC: second derivative is greater than 0 - since shutdown is assumed possible, firms cannot make negative profits, P >/ C/Q - therefore firm’s supply curve is the portion of the MC curve which lies above the AC curve, ensuring non-negative profits.