Is there any situations where we can achieve both equity and efficiency? I'm thinking of Covid 19 vaccine program which is run by Goverment. Although the cost for the program is paid by the money from taxes, it still helps with economic growth right? (in my mind, it's like if people are saved they will have a normal life and buy stuffs, which will boost the production) Or is there anything wrong in my mind?
Is it always a trade off between efficiency and equity?
No there is not always trade-off between efficiency and equity, but equity does not simply mean government spends some money on something in presence of progressive tax system (that is extremely naïve view and there are myriad of examples where programs that do that exacerbate inequality).
Equity is one of the words that laymen throw around so much that it is practically meaningless in non-technical debate, but in economics equity is defined as having more equal or fair outcomes (see Feldman 1987). In economics, when it comes to equity, the most focus is placed on income, wealth or consumption inequality (see the discussion in Atkinson (2015) Inequality).
There are some examples where equity and efficiency go hand in hand. For example, if we consider equity in global sense (not just national) we have solid evidence that free migration is both good for efficiency. It could by reasonable estimates double world's GDP (see Clemens, 2011) which would be good for efficiency, and from global perspective it would help to reduce world's income inequality, (albeit it may worsen national-level ones) as it would significantly raise incomes and wealth of poor migrants and thus likely reduce global income inequality (see some layman friendly discussion of that in Peterson 2013).
Alternative simple textbook example of situation where there is no equity-efficiency tradeoff, would be Pigouvian tax on something consumed predominantly by rich, like flying. Flying creates an awful lot of pollution, and Pigouvian tax would thus be efficient as it would correct for the externality (that airlines without tax can pollute for free), and to an extent that richer people fly more often, it would probably reduce consumption inequality somewhat.
Of course, the ones above are just example, there are more cases. Atkinson (2015) source mentioned above has a nice overview of some cases where there is little or none efficiency equity tradeoff in chapter 9 (although note that in earlier chapters he warns that not all examples are generally accepted by all economists since some are controversial).
However, covid-19 vaccine program is not necessarily good example of that. Covid-19 is particularly dangerous to people who are older (Piroth et al 2020), for example wealth inequality increases with life expectancy as older people are expected to have more wealth due to growth dynamics (Vandenbroucke, 2016), and when old people pass to an extent they have more than one child they spread out their wealth around making society more equitable.
What even more, as Deaton (2021) argues:
There is a widespread belief that the COVID-19 pandemic has increased global income inequality, reducing per capita incomes by more in poor countries than in rich. This supposition is reasonable but false. Rich countries have experienced more deaths per head than have poor countries; their better health systems, higher incomes, more capable governments and better preparedness notwithstanding. The US did worse than some rich countries, but better than several others. Countries with more deaths saw larger declines in income. There was thus not only no trade-off between lives and income; fewer deaths meant more income. As a result, per capita incomes fell by more in higher-income countries.
If equity would be goal, then from global perspective the best course of action would be to ship as much vaccines to poor countries and let covid-19 ravage richer countries. What even more, in general research shows that pandemics like that of bubonic plague in the past or other large catastrophes (like large scale wars) were great equalizers (e.g. see Alfani 2017; Milanovic 2016 ; Piketty and Saez 2014), so these actually help equity (at least along income and wealth lines). There are some that argue that covid-19 might be somewhat different (e.g. see Sayed & Peng 2021) but it is more likely than not that it will reduce inequality globally, at least along wealth and income. However, it is precisely in the rich countries where pandemic and lockdowns that were imposed as a response to it create the largest efficiency loses so from pure efficiency perspective vaccines are more important.