# Econ 134 HW #1
Name: Keyi(Serena) Tan, SID: 3037172358

## Question #1
### A:
* Laspeyres Price Index (LPI): $$ LPI = \frac{\sum_{i} p_{2i} q_{1i}}{\sum_{i} p_{1i} q_{1i}} = \frac{2.5\times 20+1 \times 40 + 1 \times 40}{2 \times 20 + 1 \times 40 + 0.5 \times 40} = 1.3 $$
* Paasche Price Index (PPI): $$ PPI = \frac{\sum_{i} p_{2i} q_{2i}}{\sum_{i} p_{1i} q_{2i}} = \frac{2.5\times 16+1 \times 40 + 1 \times 40}{2 \times 16 + 1 \times 40 + 0.5 \times 20} = 1.2195 $$
* Laspeyres Quantity Index (LQI):$$ LQI = \frac{\sum_{i} q_{2i} p_{1i}}{\sum_{i} q_{1i} p_{1i}} = \frac{16\times 2+40 \times 1 + 20 \times 0.5}{20 \times 2 + 40 \times 1 + 40 \times 0.5} = 0.82 $$
* Paasche Quantity Index (PQI):$$ PQI = \frac{\sum_{i} q_{2i} p_{2i}}{\sum_{i} q_{1i} p_{2i}} = \frac{16\times 2.5+40 \times 1 + 20 \times 1}{20 \times 2.5 + 40 \times 1 + 40 \times 1} = 0.7692 $$
* Fisher Price Index (FPI): $$ FPI = \sqrt{LPI \times PPI} = \sqrt{1.3 \times 1.2195} = 1.2591 $$
* Fisher Quantity Index (FQI): $$ FQI = \sqrt{LQI \times PQI} = \sqrt{0.82 \times 0.7692} = 0.7942 $$
* Inflation with Fisher Index: $$ \pi = (FPI - 1) \times 100 \% = (1.2591 - 1) \times 100 \% = 25.91 \% $$
* Real GDP growth with Fisher Index: $$ RealGDPgrowth = (FQI - 1) \times 100 \% = (0.7942 - 1) \times 100 \% = -20.58 \% $$

### B:
##### Price Indexes:
* Laspeyres Price Index (LPI): $1.3$
* Paasche Price Index (PPI): $1.2195$
LPI is larger than PPI which indicates when using base year quantities as weights, the price increase more than using the current period.

##### Quantity Indexes:
* Laspeyres Quantity Index (LQI): $0.82$
* Paasche Quantity Index (PQI): $0.7692$
LQI is larger than PQI which indicates when using base year prices as weights, the quantity decrease less than using the current period.

LPI and LQI give more weights to base year price or quantity, which emphasizes more on the impact of goods in the base year. PPI and PQI give more weights to current year price or quantity, which emphasizes more on the impact of goods in the current year.

In the comparison of LPI with PPI, and LQI with PQI, when the quantities of the goods decrease or the prices increase in the current year, the Laspeyres index will tend to overstate price increases and understate the quantity decreases.

### ==C==:
* Laspeyres Price Index (LPI): $$ LPI = \frac{\sum_{i} p_{2i} q_{1i}}{\sum_{i} p_{1i} q_{1i}} = \frac{16 \times 2.5 + 40 \times 1 + 20 \times 1}{16 \times 2.5 + 40 \times 1 + 20 \times 1} = 1 $$

* Paasche Price Index (PPI): $$ PPI = \frac{\sum_{i} p_{2i} q_{2i}}{\sum_{i} p_{1i} q_{2i}} = \frac{12 \times 2.5 + 60 \times 1 + 10 \times 1}{12 \times 2.5 + 60 \times 1 + 10 \times 1} = 1 $$

* Laspeyres Quantity Index (LQI):$$ LQI = \frac{\sum_{i} q_{2i} p_{1i}}{\sum_{i} q_{1i} p_{1i}} = \frac{12 \times 2.5 + 60 \times 1 + 10 \times 1}{16 \times 2.5 + 40 \times 1 + 20 \times 1} = 1 $$

* Paasche Quantity Index (PQI):$$ PQI = \frac{\sum_{i} q_{2i} p_{2i}}{\sum_{i} q_{1i} p_{2i}} = \frac{12 \times 2.5 + 60 \times 1 + 10 \times 1}{16 \times 2.5 + 60 \times 1 + 20 \times 1} = 0.8333 $$

The price index is downward biased since the price of orange did not increase with the quality improvement. The price for oranges remaines the same from Period 1 to Period 2, while the orange quality increases. Therefore, the price index does not capture the increased consumer value.

### D:
There might be substitution bias and new goods bias.
* Substitution Bias: If consumers start buying Passion Fruits as a substitute for other fruits, the index may overstate inflation if substitution is not capured in the index.
* New Goods Bias: The new products, Passion Fruits, provides additional variety and utility. If the Passion Fruits index is not included, the index fails to capture the value added from the introduction of new products.

### E:
Problems arises:
* Traditional price index might not fully account for these quality improvement. If the price increases but the quality also improved, the inflation might be overstated
* When new product/tech introduce to the market, it is hard to compare the pricing and value of the item. The inflation might be overstated

The economic growth is likely to be underestimated. The new products that introduced to the market have improved a lot of productivity, which are not fully captured in the traditional economic index.

Inflation is likely to be overestimated. If the quality improvements are not fully captured, the increase in price indexs might be pure inflation.

## Question #2
### A:
$$
FPI = \sqrt{LPI \times PPI}
$$
* without "COVID" consumption baskets: The quantities in LPI and PPI are based on pre-pandemic consumption level. This means the weights of difference goods reflect their importance in the normal goods basket
* With "COVID" consumption baskets: The quantities are adjusted to reflect the changes in consumption during pandemic

### B:
I disagree. If people are switching bundles to less expensive goods, $P_L$ will be 1 since $P_0$ ane $P_1$ are same, and $Q_0$ canceled out, since $LPI = \pi_L+1$, therefore, $\pi = 0$. And the same thing will apply to $P_P$, therefore, $\pi_P = 0$.

### C:
Cavallo points out the traditional CPI system cannot reflect quick change such as pandemic. During the pandemic, there was an inflation jump. However, the consumption of certain goods might reduced due to pandemic. If the CPI basket have been updated, certain goods would have lower weight in CPI measurements. Referring to A, the Fisher Price Index are composed of LPI and PPI, which use quantities from the base and current periods. In Part B, even if consumers buy less expensive items, the inflation rate remains 0 when all prices are unchanged. If the CPI put less weights on transportation, the PPI will be lower, which leads to a lower FPI and lower inflation.

### D:
The mismatch will be smaller. If BLS used LPI, it would not capture rapid changes in consumption change, which may underestimate inflation. Fisher index considers both past and curent baskets. Therefore, Fisher index has smaller gap.

## Question #3
### A:
Blinder and Watson analyze the economic performance of US under Democratic presidents and Republican presidents. It shows that economy has performed better under Democratic presidents, with higher GDP growth, lower unemployment rate, and also better stock returns. It is an correlation not a causation since there are a lot of omitted variables.

### B:
Although their analysis shows a correlation between Democratic presidents and better economic performance, it does not means Democratic presidents causes better economic performance. There are omitted varibles such as technology advancement, global conflicts, etc.