Micro Lab 5

11/08/2024

Producer theory

0. Some important concepts and key questions

1. Important Concepts

   • Production set

   • Production function

   • Return to scale

   • Isoquant

   • Marginal rate of technological substitution

   • Elasticity of substitution

   • Constant elasticity of substitution function

   • Cost function

   • Profit function

   • Input demand function - conditional/unconditional

   • Cost - marginal/average/fixed/variable/sunk

2. Key questions

   • How do firms make optimal decisions?

   • How different types of production functions affect firms' behavior?

   • What's the relationship between cost function and production function?

   • What's the relationship between cost minimization and profit maximization?

   • What's the difference between short-run and long-run decisions?

1. Return to scale

Global: Return to scale

A production function $f(\mathbf{x})$$f(\mathbf{x})$ has the property of (globally):

1. Constant returns to scale if $f(t\mathbf{x})=tf(\mathbf{x})$$f(t \mathbf{x})=t f(\mathbf{x})$ for all $t>0$$t>0$ and all $\mathbf{x}$$\mathbf{x}$;

2. Increasing returns to scale if $f(t\mathbf{x})>tf(\mathbf{x})$$f(t \mathbf{x})>t f(\mathbf{x})$ for all $t>1$$t>1$ and all $\mathbf{x}$$\mathbf{x}$;

3. Decreasing returns to scale if $f(t\mathbf{x})<tf(\mathbf{x})$$f(t \mathbf{x})<t f(\mathbf{x})$ for all $t>1$$t>1$ and all $\mathbf{x}$$\mathbf{x}$.

Local: Elasticity of scale

The elasticity of scale at the point $\mathbf{x}$$\mathbf{x}$ is defined as

Returns to scale are locally constant, increasing, or decreasing as $\mu (\mathbf{x})$$\mu(\mathbf{x})$ is equal to, greater than, or less than one. The elasticity of scale and the output elasticities of the inputs are related as follows:

Question

What is the elasticity of scale of the CES technology, $\mathrm{f}(x_1,x_2)=$$\mathrm{f}\left(x_1, x_2\right)=$ ${(x_1^{\rho }+x_2^{\rho })}^{\frac{1}{\rho }}$$\left(x_1^\rho+x_2^\rho\right)^{\frac{1}{\rho}}$ ?

2. Cost and Conditional Input Demands when Production is Homothetic

JR Theorem 3.4

1. When the production function satisfies Assumption 3.1 (The production function is continuous, strictly increasing, and quasiconcave) and is homothetic, (a) the cost function is multiplicatively separable in input prices and output and can be written $c(\mathbf{w},y)=h(y)c(\mathbf{w},1)$$c(\mathbf{w}, y)=h(y) c(\mathbf{w}, 1)$, where $h(y)$$h(y)$ is strictly increasing and $c(\mathbf{w},1)$$c(\mathbf{w}, 1)$ is the unit cost function, or the cost of 1 unit of output; (b) the conditional input demands are multiplicatively separable in input prices and output and can be written $\mathbf{x}(\mathbf{w},y)=h(y)\mathbf{x}(\mathbf{w},1)$$\mathbf{x}(\mathbf{w}, y)=h(y) \mathbf{x}(\mathbf{w}, 1)$, where $h^{\prime }(y)>0$$h^{\prime}(y)>0$ and $\mathbf{x}(\mathbf{w},1)$$\mathbf{x}(\mathbf{w}, 1)$ is the conditional input demand for 1 unit of output.

2. When the production function is homogeneous of degree $\alpha >0$$\alpha>0$, (a) $c(\mathbf{w},y)=y^{1/\alpha }c(\mathbf{w},1)$$c(\mathbf{w}, y)=y^{1 / \alpha} c(\mathbf{w}, 1)$; (b) $\mathbf{x}(\mathbf{w},y)=y^{1/\alpha }\mathbf{x}(\mathbf{w},1)$$\mathbf{x}(\mathbf{w}, y)=y^{1 / \alpha} \mathbf{x}(\mathbf{w}, 1)$.

Question 2020 Oct Prelim Part1 Q3

A factor of production $i$$i$ is called inferior if the conditional demand for that factor decreases as output increases; that is, $\partial x_{ı}(\mathbf{w},y)/\partial y<0$$\partial x_{\imath}(\mathbf{w}, y) / \partial y<0$. (a) Draw a diagram indicating that inferior factors are possible. (b) Show that if the technology is constant returns to scale, then no factors can be inferior. (c) Show that if marginal cost decreases as the price of some factor increases, then that factor must be inferior.

Question JR 3.31

The output elasticity of demand for input $x_i$$x_i$ is defined as

$$\begin{array}{}\text{(3)}& {ϵ}_{iy}(w,y)\equiv (\partial x_i(\mathbf{w},y)/\partial y)(y/x_i(\mathbf{w},y)).\end{array}$$

(a) Show that ${ϵ}_{iy}(\mathbf{w},y)=\varphi (y){ϵ}_{iy}(\mathbf{w},1)$$\epsilon_{i y}(\mathbf{w}, y)=\phi(y) \epsilon_{i y}(\mathbf{w}, 1)$ when the production function is homothetic. (b) Show that ${ϵ}_{iy}=1$$\epsilon_{i y}=1$, for $i=1,\dots ,n$$i=1, \ldots, n$, when the production function has constant returns to scale.

Another practice

Question - JR 3.28

A firm's technology possesses all the usual properties. It produces output using three inputs, with conditional input demands $x_i(w_1,w_2,w_3,y),i=1,2,3$$x_i\left(w_1, w_2, w_3, y\right), i=1,2,3$. Some of the following observations are consistent with cost minimization and some are not. If an observation is inconsistent, explain why. If it is consistent, give an example of a cost or production function that would produce such behavior. (a) $\partial x_2/\partial w_1>0$$\partial x_2 / \partial w_1>0$ and $\partial x_3/\partial w_1>0$$\partial x_3 / \partial w_1>0$. (b) $\partial x_2/\partial w_1>0$$\partial x_2 / \partial w_1>0$ and $\partial x_3/\partial w_1<0$$\partial x_3 / \partial w_1<0$. (c) $\partial x_1/\partial y<0$$\partial x_1 / \partial y<0$ and $\partial x_2/\partial y<0$$\partial x_2 / \partial y<0$ and $\partial x_3/\partial y<0$$\partial x_3 / \partial y<0$. (d) $\partial x_1/\partial y=0$$\partial x_1 / \partial y=0$. (e) $\partial \left(x_1/x_2\right)/\partial w_3=0$$\partial\left(x_1 / x_2\right) / \partial w_3=0$

Partial equilibrium

Perfect competition

In the short run:

In the long run:

Question 2021 Aug Prelim Part1 Q3 - [perfect competition]

[15 points] Suppose there are a large number of identical firms in a perfectly competitive industry. Each firm has the long-run average cost curve :

$$\begin{array}{}\text{(4)}& AC(q)=q^2-12q+50\end{array}$$

where $q$$q$ is the firm's output. (a) What condition must be satisfied in a long-run equilibrium if we assume no barriers to entry or exit?

In a perfect competitive market, in the long run we have,

$$\begin{array}{}\text{(5)}& p=AC=MC\end{array}$$

In other words, $AC=MC$$A C=M C$ implies that $p=minAC$$p=\min A C$.

 

(b) What condition must be satisfied in a perfectly competitive industry?

$$\begin{array}{}\text{(6)}& p=MC.\end{array}$$

 

(c) Derive the long-run supply function for this industry.

In the short run, For each firm, the supply function is

$$\begin{array}{}\text{(7)}& p=MC=3q^2-24q+50\end{array}$$

In the long run, the supply should be $6N$$6 N$, where $N$$N$ is the number of firms operating in the market.

 

(d) How much will each individual firm produce in this equilibrium?

$q=6$$q=6$. for each firm.

 

(e) What do you need to know to determine how many firms will exist in this perfectly competitive long-run equilibrium?

We need to know the market demand function.

Monopoly

Profit maximization yields:

Since we have $r(q)=p(q)q$$r(q)=p(q) q$, differentiate on the both side,

This implies that

Since marginal cost is always non-negative, and the price is also non-negative, we must have $\left|\epsilon \left(q^{\ast }\right)\right|\ge 1$$\left|\varepsilon\left(q^*\right)\right| \geq 1$. Therefore, a monopolist will never produce in the inelastic portion of the demand curve.

Rearranging $\text{(10)}$$\eqref{eq10}$:

Question JR 4.23 (Ramsey Rule )

Building from the preceding exercise, suppose a monopolist faces negatively sloped demand, $p=p(q)$$p=p(q)$, and has costs $C=cq+F$$C=c q+F$. Now suppose that the government requires this firm to set a price $\left(p^{\ast }\right)$$\left(p^*\right)$ that will maximise the sum of consumer and producer surplus, subject to the constraint that firm profit be non-negative, so that the regulation is sustainable in the long run. Show that under this form of regulation, the firm will charge a price greater than marginal cost, and that the percentage deviation of price from marginal cost $\left((p^{\ast }-c)/p^{\ast }\right)$$\left(\left(p^*-c\right) / p^*\right)$ will be proportional to $1/{ϵ}^{\ast }$$1 / \epsilon^*$, where ${ϵ}^{\ast }$$\epsilon^*$ is the elasticity of firm demand at the chosen price and output. Interpret your result.