Exercise: Price elasticity

If you know that the market demand for good X corresponds to the following equation:

$P=-2Q+450$

and the supply curve can be model by the equation:

$P=3Q+200$

Calculate the price elasticity for each curve at the equilibrium point. Furthermore explain which curve is relatively more elastic.

The first step consists of solving for the equilibrium price and quantity:

$\\ (1.) -2Q+450=3Q+200 \\ (2.) Q_e=50 \\ (3.) -2(50)+450=350=P_e$

Remember the formula for the price elasticity is

$E_d,E_s=\frac{\%\Delta Q}{\%\Delta P}$

Therefore pick the two closest points and calculate the price elasticity, in this case let's pick a point that is one quantity less than the equilibrium point on both curves.

$\\(49,352)\; \textit{and }(50,350) \rightarrow E_d=|\frac{\frac{50-49}{49}}{\frac{350-352}{352}}|\approx 3.5 \\ (49,347)\; \textit{and }(50,350)\rightarrow E_s=\frac{\frac{50-49}{49}}{\frac{350-347}{347}}\approx 2.3$

Since we are dealing with linear equations we can use another formula which is specific for these types of curves and is more precise than the method we used so far. The formula is the fraction of the price over the quantity multiplied by the inverse of the slope.

$\\ E_d=|\frac{350}{50} \cdot \frac{1}{2}|=3.5 \\ E_s= \frac{350}{50} \cdot \frac{1}{3}=2.3\bar{33}$

Since Ed>Es, the demand curve at the equilibrium point is relatively more elastic than the supply curve. Note that both curves are elastic since both have an elasticity greater than 1. Furthermore, one could have made an educated guess by just looking at the absolute value of the slope for each curve.

The State wants to impose a price floor on the market for good X where the price elasticity of demand is equal to 6. Derive the equation for the price floor.

To answer this question, let's remember that one way to derive the elasticity for a linear function is by using a formula that combines the price, quantity and slope of a linear curve. Therefore, if we know the elasticity, we can create a linear function that intersects the demand curve at the price floor.

$\\(1.)E_d=6=(\frac{P}{Q}) \cdot \frac{1}{2} \\ (2.)P=12Q \\(3.)12Q=-2Q+450 \\(4.)Q_{pf}\approx 32.13, P_{pf}= 12(32.13)\approx 385.6$

Thus, since the equation of a price floor is only bounded by the price, its equation is equal to P=385.6, at this point the price elasticity of the demand curve is about 6.

Let's visualize the problem:

Comments

Macroeconomics: multiplier and crowding out effects

Multiplier effect: whenever any of the components of AD increases, the increase in GDP will be greater than the initial increase in expenditures. The impact on GDP of a particular increase in spending depends on the proportion of the new income that is taken out of the system to the proportion that continues to circulate in the economy. The multiplier effect tells us the impact a particular change in one the components of AD will have on the total income (GDP). Let k denote the spending multiplier, which is a function of MPC and MPS. The larger the marginal propensity to consume, the larger the spending multiplier. Notice that the larger the MPC, the greater the impact a particular change in the spending variables will have on the nation's GDP. The crowding out effect: If government spending increases without an increase in taxes, the government must borrow funds from the private sector to finance its deficit, thereby increasing the interest rate. This increase in interest ...

Econometrics: Bivariate population model

Hello I'm finally back from my extended break. I thought that we should start studying Econometrics. Let's begin by analyzing a simple bivariate regression. Assume that this equation describes the relationship between two variables X and Y. We say that Y is the dependent variable, whereas X is the independent variable. In other words, we assume that Y (the output) depends on X (the input). Epsilon is the error term, it represents other factors that affect Y. The error term must be uncorrelated with the variable X so that we do not need to include them in our regression, and thus the coefficient of X (beta) should not change, even though Epsilon also determines Y. beta-0 is the constant term. It tells us what would be Y if X=0. beta-1 is the effect on Y if X changes by one unit. To see this, assume X=education is a continuous function, let's take the derivative of Y=wage with respect to X: Thus, if education goes up by one unit, we should expect, on average, wage to go up ...

Exercise: short and long run effects of a fiscal policy

You are in charge of the government budget for the year 2021. You are told that schools and public roads need to be updated, and you estimate an appropriate budget in order to carry out the task. Describe what will happen in the short run to the economy, and what type of inflation do we see? Describe the supply side effect from this policy, explain what happens in the long run. Assume that there is no crowding out effect. First, note that updating public infractures, assuming taxes stay the same, will require an increase in government spending, which will also have an impact on the investment and consumption level in the economy (remember the spending multiplier?). Therefore, the aggregate demand curve will shift to the right. This will cause inflation (i.e positive change in the price level) and a higher output in the short-run. Note that if we were to assume a large crowding out effect, the short run aggregate supply curve would shift to the left (increase in production c...

Economics and Statistics

Search This Blog