701 TA session 3

2023 Midterm2

FOC and SOC , What do we have to have in maximization problems?

1. Unconditional max

FOC:

SOC

We need $H$$H$ is negative semidefinite at the crital point, which requires

Typically, if we assume $f(x_1,x_2)$$f(x_1,x_2)$ is concave, then for the unconstrained maxmization, SOC is sufficient.

For a minimization problem,

FOC:

SOC:

We requre $H$$H$ to be a positive semidefinite matrix, which implies $f_{11}\le 0,f_{11}{f}_{22}-f_{12}{f}_{21}\ge 0$$f_{11} \leq 0, f_{11}f_{22}-f_{12}f_{21}\geq0$.

2. Conditional max/min

For constrained optimization problem

FOC

SOC: (Bordered Hessian matrix)

For max problem, we need $det(\mathrm{△})\ge 0$$\det(\triangle ) \geq 0$

For min problem, we need $det(\mathrm{△})\le 0$$\det(\triangle ) \leq 0$.

Substitution and output effect in producer theory - Refers to quation. 3 in 2023 midterm 2.

In consumer theory, we have Slutsky equation,

In producer theory, we have conditional and unconditional input demand, which should follow the relationship like this,

So, take the derivitive with respect to any price , we will have,

, where the first term is the substitution effect, and the second term denotes the output effect.

Total surplus in a monopoly market

Total surplus:

If we ask a monopolistic producer to maxmize the social surplus by choosing $q$$q$ (equivalent to choose $p$$p$),

Our first order condition,

Assume $\lambda =0$$\lambda = 0$, $p(q)-c=0$$p(q) - c = 0$,

then $\pi =pq-cq-F<0$$\pi = pq - cq - F<0$, which contradicts to our constaints, so $\lambda \ne 0$$\lambda \not = 0$. The inequality constraint must be binding.

Taxation

Suppose the cost function is linear,

, and the demand curve has a constant elasticity $ϵ$$\epsilon$, which means the demand curve showing like this $q=p^{-ϵ}$$q = p^{-\epsilon}$.

1. Consider the proportional tax

   $p_s=(1-\tau )p_d$$p_s = (1-\tau )p_d$

2. Consider the unit tax

   $p_d=p_s+t$$p_d = p_s + t$

The question is to find the tax-equivalent condition between $t$$t$ and $\tau$$\tau$.

Let's start with the profit max problem, the key feature here is to distinguish between two price and demand (consumer and supplier)

• Case I. The proortional tax case

  $$\begin{array}{}\text{(19)}& \begin{array}{rl}\underset{p_d}{max}\pi & =p_{s}q(p_d)-cq(p_d)\\ & =(1-\tau )p_{d}q(p_d)-cq(p_d)\end{array}\end{array}$$

  FOC:

  $$\begin{array}{}\text{(20)}& \frac{\partial \pi }{\partial p_d}=(1-\tau )p_d{q}^{\prime }+(1-\tau )q-c{q}^{\prime }=0\end{array}$$

  We can derive,

  $$\begin{array}{}\text{(21)}& p_d=\frac{c}{(1-\tau )(1+\frac{1}{ϵ})}\end{array}$$

• Case II. The unit tax.

  $$\begin{array}{}\text{(22)}& \begin{array}{rl}\underset{p_d}{max}\pi & =p_{s}q(p_d)-cq(p_d)\\ & =(p_d-t)q(p_d)-cq(p_d)\\ & =p_{d}q(p_d)-(c+t)q(p_d)\end{array}\end{array}$$

  Then we can get

  $$\begin{array}{}\text{(23)}& p_d=\frac{c+t}{1+\frac{1}{ϵ}}\end{array}$$

•  

If we let the two price to be equal to each other, then we get $t=\frac{\tau }{1-\tau }c$$t = \frac{\tau}{1-\tau} c$.

JR 4.27

A per-unit tax to $q=p^{-ϵ}$$q = p^{-\epsilon }$

show that the monopoly will increase price by more than the amount of per-unit tax.

For the monopoly, we have

, which is equivalent to $mc=p(1-\frac{1}{ϵ})$$mc = p(1-\frac{1}{\epsilon})$.

Without any tax, the monopolistic producer will choose the price based on,

It implies that monopolistic producer must choose elastic part of the demand curve.

With tax, the price should be chosen by,

, which shows the monopolistic producer will increase price to the level that a little bit more than the case without any tax.

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