Production Theory¶

Constrained optimization: Marginal Benefit = Marginal Cost 1. Maximize objective: Maximize Profit = Revenue - Cost 2. Constraint: Firms get to decide how much to produce

Let - \(q\) be quantity produced by firm - \(Q\) be quantity produced by entire market

Assumptions¶

Assumptions¶

Quantity¶

Factors of Production¶

Long run is the time period for which all inputs are variable

Production Function¶

Function that defines output as a function of input(s)

Total Factor Productivity¶

Productivity increase due to technology/innovation, keeping other factors constant

$$ A_t = \dfrac{q}{k^\alpha \times l^\beta} $$ \(\alpha, \beta =\) two inputs' respective shares of output

Marginal Productivity¶

Diminishing Marginal Productivity¶

With increase in input - total productivity increases - marginal productivity decreases

Graph¶

• y = MP
• x = input (l,k)

The graph will be a straight line with a downward slope

Isoquants¶

• Similar concept to indifference curves
• All the assumptions of indifference curves apply

Inputs	\(q\)
Perfectly Substitutable	\(l+k\)
Perfectly Non-Substitutable Leontief Production Function	\(\min(l, k)\)		Computers & Programmers

MRTS¶

Marginal Rate of Technical Substitution

Slope of Isoquant at a point, hence depends on the position on the isoquant

Diminishing marginal productivity affects this

Returns to Scale¶

What happens to production when you increase all inputs proportionally

Not \(k\) vs \(l\)

Returns to Scale	\(f(a l, a k)\)	Example
Constant	\(a \cdot f(l, k)\)
Increasing	\(> a \cdot f(l, k)\)	Firm with increased inputs can specialize more
Decreasing	\(< a \cdot f(l, k)\)	Firm with increased inputs means more confusion

Production Possibility Frontier¶

deals with the production capacity of an economy

• analyzes available resources
• predicts what can be produced with those resources, with the assumption that time and technology are constant
• analyzes various output mixes (combinations of different commodities)

Production Possibility curve¶

Curve that shows the optimal level of production for different combinations of outputs

what is the maximum output that can be produced of various commodities

Graph¶

• \(y =\) commodity 2
• \(x =\) commodity 1

The graph will be a curve

MRT¶

When we go from left to right, the slope increases from left to right, ie, cost of production of \(x\) increases

• \(\text{MP}_k^y, \text{MP}_l^y\) increases
• \(\text{MP}_k^x, \text{MP}_l^x\) decreases

Example: if mango’s MP is 2 and apple’s MP is 4, then 4 units of apples are to be sacrificed to produce 2 units of mangoes

Points on the curve¶

• inside are inefficient input mix - wasting profit
• on the curve are optimal input mix
• outside is impossible input mix

Shift of PPC¶

It depends on type of technological change that occurs

also depends on change in the quantity and productivity of labor and capital

Output Mix¶

Costs¶

Types¶

Cost Function¶

$$ c = k r_k + l r_l $$ where - \(r_k =\) depreciation - \(r_l =\) wage rate

Substituting for \(k\) or \(l\) from the production function will give the relationship between \(C\) and \(q\)

• AC decreases and then increases
  • Increasing quantity helps amortize the fixed cost

Isocost Curves¶

Efficient Input Mix¶

Efficient input mix is the tangent of the isocost and isoquant curves

Increase in wages

At the optimal input mix - MRTS = \(-\dfrac{r_l}{r_k}\) - \(\dfrac{\text{MP}_l}{r_l} =\dfrac{\text{MP}_k}{r_k}\)

Efficient input mix is said to be

• Economically efficient if the costs associated with an input mix is minimal
• Technically efficient if it is impossible to maintain the same output, without keeping the same inputs if any input decreases, the output also decreases
  • in India, we have a technically inefficient agricultural sector. This is proved by the fact that: even though many people moved to tertiary sector, it did not affect agricultural output

Long-Run Cost Curve¶

Expansion Path¶

How does cost change with production quantity

\(\text{MP}_l = \text{MP}_k\)
\(\text{MP}_l > \text{MP}_k\)
\(\text{MP}_l < \text{MP}_k\)

Increasing Cost of Inputs¶

Long-Run vs Short-Run¶

• Long-run operations are always more efficient than short-run
  • You can optimize for both cost and capital in the long-run

Profit¶

Every company tries to maximize its profit.

Profits can be maximized by cost minimization through optimization of input mix. The input mix depends on 1. productivity of input factor (high is preffered) 2. price of input factor (low is preffered)

Labor is preferred for lower price; capital is preferred due to higher productivity

Assumption: In the short run, no firm enters/exits; capital is a sunk cost

where - \(p\) is unit sale price - \(q(p)\) is units sold - \(c \Big( q(p) \Big)\) is cost - \(c' \Big( q(p) \Big)\) is the marginal cost

$$ \begin{aligned} \dfrac{\partial \pi}{\partial q} &= \dfrac{\partial R}{\partial q} - \dfrac{\partial C}{\partial q} \ 0 &= \text{MR} - \text{MC} \ \implies \text{MR} &= \text{MC} \end{aligned} $$ For perfectly-competitive market - \(\text{MR}(Q)\) = \(P(Q)\) - Hence, at market equilibrium, \(q\) for which \(\text{MC}(Q) = P\)

Average Profit¶

Taxation¶

	Fixed Tax	Per-Unit Tax
Optimal Quantity	No change	Reduced
Profit	Reduced	Reduced
Curves	No change

Shutdown Decision¶

Producer still part of the market, but does not produce anything

Sometimes, producing something is better than nothing, as making low loss is better than extreme loss

Short-run: Only shutdown if - revenue is lower than variable cost, as you have to pay fixed cost regardless - \(Pq < \text{VC}\) - ie, \(P < \text{AVC}\)

Firm vs Market Demand¶

$$ q(P) = Q(P) - S^0(P) $$ - \(q(P)=\) Firm demand - \(Q(P)=\) Market demand - \(S^0(P) =\) Residual demand = how much competition is selling

Even with inelastic \(Q\), \(q\) can be elastic

\(\dfrac{dS^0}{dP} > 0\) as number of suppliers will inc with inc in selling price

If there are \(N\) identical firms - \(Q = Nq, S^0 = (N-1)q\) - \(e_{di} = N E_d - (N-1)E_s\) - \(e_d =\) Firm demand elasticity - \(E_d =\) Market demand elasticity - \(E_d =\) Market supply elasticity

Firm Entry/Exit¶

Only in the long-run

Happens when long-run profit > Barrier of entry

	Entry	Exit

Outcome of Competition¶

• Forces each supplier to produce efficiently
• Forces entry and exit of suppliers until

such that - Price = Marginal Cost = Min of Average Cost - Total profit of market = 0

Different Firms¶

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