Question

The supply and demand model presented earlier in this chapter can be used to look at many other comparative statics questions. In this problem you are asked to explore three of them. In all of these, quantity demanded is given by \(D(P, \alpha)\) and quantity supplied by \(S(P, \beta)\) a. Shifts in supply: In Chapter 12 we analyzed the case of a shift in demand by looking at a comparative statics analysis of how changes in \(\alpha\) affect equilibrium price and quantity. For this problem you are to make a similar set of computations for a shift in a parameter of the supply function, \(\beta .\) That is, calculate \(d P^{*} / d \beta\) and \(d Q^{*} / d \beta .\) Be sure to calculate your results in both derivative and elasticity terms. Also describe with some simple graphs why the results here differ from those shown in the body of Chapter 12 b. \(\mathbf{A}\) quantity \(^{\text {- wed }}\) we" \(_{\pm}\) In our analysis of the imposition of a unit tax we showed how such a tax wedge can affect equilibrium price and quantity. A similar analysis can be done for a quantity "wedge" for which, in equilibrium, the quantity supplied may exceed the quantity demanded. Such a situation might arise, for example, if some portion of production were lost through spoilage or if some portion of production were demanded by the government as a payment for the right to do business. Formally, let \(\bar{Q}\) be the amount of the good lost. In this case equilibrium requires \(D(P)=Q\) and \(S(P)=Q+\bar{Q}\). Use the comparative statics methods developed in this chapter to calculate \(d P^{*} / d \bar{Q}\) and \(d Q^{*} / d \bar{Q} .\) [ In many cases it might be more reasonable to assume \(\bar{Q}=\delta Q\) (where \(\delta\) is a small decimal value). Without making any explicit calculations, how do you think this case would differ from the one you explicitly analyzed? c. The identification problem: An important issue in the empirical study of competitive markets is to decide whether observed price-quantity data points represent demand curves, supply curves, or some combination of the two. Explain the following conclusions using the comparative statics results we have obtained: I. If only the demand parameter \(\alpha\) takes on changing values, data on changing equilibrium values of price and quantity can be used to estimate the price elasticity of supply. ii. If only the supply parameter \(\beta\) takes on changing values, data on changing equilibrium values of price and quantity can be used to evaluate the price elasticity of supply (to answer this, you must have done part (a) of this problem). iii. If demand and supply curves are both only shifted by the same parameter [i.e., the demand and supply functions are \(D(P, \alpha) \text { and } S(P, \alpha)],\) neither of the price elasticities can be evaluated.

Short answer

Additionally, explain the identification problem (scenario c). Answer: When there is a shift in supply (scenario a), the equilibrium price and quantity are affected by the change in the supply parameter β. The calculated derivatives and elasticities show how the equilibrium price and quantity respond to the shift in supply. A comparison with Chapter 12, which discussed a shift in the demand parameter α, will highlight the differences in the responses of price and quantity to a shift in supply compared to a shift in demand. In scenario b, when the quantity lost changes, the equilibrium price and quantity are also affected. By calculating the derivatives of the equilibrium price and quantity with respect to the quantity lost (𝑄̅), we can analyze the impact of changes in quantity lost on the market equilibrium. If the quantity lost is expressed as a small decimal value (𝜎𝑄), the analysis would differ, but a discussion on this case can be presented without explicit calculations. The identification problem (scenario c) highlights the limitations of using comparative statics results to evaluate price elasticities when different parameters take on changing values. If only the demand parameter α changes, we can estimate the price elasticity of supply. If only the supply parameter β changes, we can evaluate the price elasticity of demand. However, if both demand and supply curves are shifted by the same parameter, we cannot isolate the individual effects of the demand and supply functions on the equilibrium conditions, making it impossible to evaluate either of the price elasticities.

Step-by-step solution

Compute the equilibrium price and quantity

To compute the equilibrium price and quantity, we need to first solve the following system of equations in terms of price (P) and quantity (Q): $ D(P, \alpha) = Q \\ S(P, \beta) = Q $ Using these equations, we can calculate the value of P and Q.

Compute \(\frac{dP^*}{d\beta}\) and \(\frac{dQ^*}{d\beta}\)

Now that we have the equilibrium price and quantity, we can differentiate them with respect to \(\beta\) to find the desired derivatives. Use the following rules for differentiation, calculate \(\frac{dP^*}{d\beta}\) and \(\frac{dQ^*}{d\beta}\).

Compute the elasticities of \(\frac{dP^*}{d\beta}\) and \(\frac{dQ^*}{d\beta}\)

The elasticities of the derivatives can be computed by multiplying the derivatives by the ratio of the variables. For example, compute the elasticities as follows: \(elasticity\_dP = \frac{dP^*}{d\beta} * \frac{P}{\beta}\) \(elasticity\_dQ = \frac{dQ^*}{d\beta} * \frac{Q}{\beta}\)

Compare the results with Chapter 12

Use simple graphs to illustrate and describe the differences between the results here and those shown in Chapter 12, which dealt with the shift in the demand parameter \(\alpha\). #Solution b#

Compute the equilibrium price and quantity

Given that \(D(P)=Q\) and \(S(P)=Q+\bar{Q}\), we need to find the equilibrium price and quantity.

Compute \(\frac{dP^*}{d\bar{Q}}\) and \(\frac{dQ^*}{d\bar{Q}}\)

To compute the effect of a change in the quantity lost on the equilibrium price and quantity, differentiate the equilibrium price and quantity with respect to \(\bar{Q}\).

Discussion on the case where \(\bar{Q}=\delta Q\)

Without making any explicit calculations, discuss how the case where the quantity lost is expressed as a small decimal value (\(\bar{Q}=\delta Q\)) would differ from the one analyzed in the previous steps. #Solution c#

Conclusion I

When only the demand parameter \(\alpha\) takes on changing values, data on changing equilibrium values of price and quantity can be used to estimate the price elasticity of supply, because it allows us to observe how the supply function reacts to changes in equilibrium conditions.

Conclusion II

When only the supply parameter \(\beta\) takes on changing values, data on changing equilibrium values of price and quantity can be used to evaluate the price elasticity of supply, as this scenario shows how the demand function reacts to changes in equilibrium conditions (assuming the calculations in part (a) were completed).

Conclusion III

If demand and supply curves are both only shifted by the same parameter (with demand and supply functions being \(D(P, \alpha)\) and \(S(P, \alpha)\)), neither of the price elasticities can be evaluated, because we cannot isolate the individual effects of demand and supply functions on the equilibrium conditions.